Topics — What a work is about. Nucleation holds whatever reading the owner files here, whether or not an Anthology of the SOTA topic could also hold it; instruction for ML practice belongs to the anthology (ADR-048). The words are seeded from the clusters the reading-time triage of 2026-09-25 found, and grow by decision.
Anthology candidate — a curation flag, not a subject — somebody judged the Anthology of the SOTA may want this work as well; it stays here unless a transfer is decided, and is never a primary topic (ADR-005, ADR-048; group: flags).
301 of 2004 LIT documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| LIT-056 v3 | Consciousness in Artificial Intelligence: Insights from the Science of Consciousness | Butlin et al. (2023), ARXIV-2308.08708. The report assumes computational functionalism and takes a “theory-heavy” approach. From it the authors derive 14 indicator properties: RPT-1–2, GWT-1–4, HOT-1–4, AST-1, PP-1 and AE-1–2. The claim for them is only that more of them raises the probability of consciousness. They assess a handful of systems against subsets of the list and conclude that no current system “appears to be a strong candidate”. Transformer LLMs have “only a relatively weak case” for any GWT indicator, because a single forward pass has no recurrence or global broadcast. Perceiver arguably has GWT-1, GWT-2 and half of GWT-4 but lacks broadcast. Among the embodied agents, AdA is the likeliest to meet AE-2. Most indicators could be built with standard ML techniques. The abstract’s “no current AI systems are conscious” is stronger than anything the body shows. | Active — read in full 2026-09-25 (NOTE-052); the reference statement of the indicator-property (“theory-heavy”) approach to AI consciousness, worth reading for §1–§2 (method and theory survey) and Tables 1–2. Its §3.2 system verdicts are architecture-dated and cover only GWT and agency/embodiment. The 2026 Trends in Cognitive Sciences paper by an overlapping 20-author group may now be the citable version; I have not read it, so this should not yet be marked Superseded. |
| LIT-058 v3 | Mutual Information and Minimum Mean-square Error in Gaussian Channels | Guo et al. (2004), arXiv:cs/0412108. Let Y = √snr·X + N with N ~ N(0,1) independent of X, and let X have any distribution with E X² < ∞. Then d/dsnr I(X; √snr X + N) = ½·mmse(X | √snr X + N) in nats, equivalently I(snr) = ½∫₀^snr mmse(γ) dγ (Thm 1, Eq. 47). The vector form is d/dsnr I(X; √snr HX + N) = ½·E‖HX − H·E[X|Y]‖² (Thm 2). In continuous time, the same identity combined with Duncan’s I = (snr/2)·cmmse gives cmmse(snr) = (1/snr)∫₀^snr mmse(γ) dγ (Thm 8). | Active — read in full 2026-09-25 (NOTE-086); the founding statement and proof of the I-MMSE identity. It is fully proved (five proofs, with the interchange of limits justified in App. IV) and it remains the reference. Read §II-C for the incremental-channel proof and §V-B for the one place it fails (feedback). |
| LIT-085 v3 | The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo | Hoffman & Gelman (2011), ARXIV-1111.4246. NUTS removes HMC’s step-count L. It doubles a leapfrog trajectory forwards and backwards in random directions until some balanced subtree satisfies (θ⁺−θ⁻)·r⁻ < 0 or (θ⁺−θ⁻)·r⁺ < 0. It keeps detailed balance by slice sampling and by excluding states that could not have regenerated the tree. With ε set by a dual-averaging scheme (γ = 0.05, t₀ = 10, κ = 0.75, target δ ≈ 0.6), it matched the best hand-tuned HMC in ESS per gradient on two logistic-regression posteriors. It beat that HMC by about 3× on a 250-D Gaussian and on a 3001-D stochastic-volatility model (§4.4). | Active — read in full 2026-09-25 (NOTE-081); This is the source of the default sampler in Stan, PyMC and NumPyro, and it is worth reading for its derivation. The algorithm as run today, with multinomial rather than slice selection and adapted mass matrices, is documented elsewhere. A reader who wants current practice should read this paper together with that later account. |
| LIT-178 v2 | Borges and AI | Bottou & Schölkopf (2023), ARXIV-2310.01425. A short essay, not an argument with evidence. It claims that a “perfect language model”, defined as sampling each next word from a random occurrence of the current text in the infinite collection of all plausible texts, is a fiction machine governed by “narrative necessity”, to which “neither truth nor intention” matters. On that view hallucinations “are just confabulations”, chat-bot answers take the role the user’s framing implies, and fine-tuning and RLHF are attempts to prune the garden “against its nature”. It warns that power over what LLMs write becomes “power over what we think”. | Active — read in full 2026-09-26 (NOTE-102); worth reading as a short, well-made and much-cited statement of the “LLM as fiction machine” framing, provided it is read as an essay offering imagery. It supplies no evidence, and its central idealization (the perfect LM) is asserted, not argued, to describe trained and aligned chat models. |
| LIT-226 v2 | The Conditional Entropy Bottleneck | Fischer (2020), ARXIV-2002.05379. The Conditional Entropy Bottleneck minimizes I(X;Z|Y) − γI(Y;Z), which is the Information Bottleneck with γ = β − 1, aimed at the “Minimum Necessary Information” point I(X;Y) = I(X;Z) = I(Y;Z). On Fashion MNIST it improves accuracy, adversarial robustness, OoD detection and calibration over VIB and deterministic baselines, and it does not memorize random labels. On CIFAR-10 it improves accuracy and robustness. That excess retained information causes these failures is stated as a hypothesis and supported only empirically. | Active — read 2026-10-14 (NOTE-754) in full; worth reading as the paper that names the Minimum Necessary Information point I(X;Y) = I(X;Z) = I(Y;Z), shows that its objective is the Information Bottleneck with γ = β − 1, and gives variational bounds (the residual information ReX) that make the distance to that point observable. Its CatGen decoder is InfoNCE with the backward encoder as critic, capped by log K. Read it knowing that the causal claim, that retained information causes adversarial vulnerability, miscalibration, poor OoD detection and memorisation, is a hypothesis supported only by Fashion MNIST and one selected CIFAR-10 model per setting, and that the arXiv v1 prints the CatGen inequality (Eq. 30) with the wrong sign. |
| LIT-227 v2 | Contrastive Learning Is Spectral Clustering On Similarity Graph | Tan et al. (2023), ARXIV-2303.15103. Over a finite set of n objects with a symmetric augmentation matrix π, SimCLR’s InfoNCE loss with a Gaussian kernel on the unit sphere equals tr(Zᵀ L(π) Z) + log R(Z), a Laplacian quadratic form plus the log of InfoNCE’s normalizers. The paper calls this spectral clustering under a definition that admits any regularizer, and does not characterize the minimizer. CLIP is treated the same way on a bipartite pair graph. Mixtures of exponential kernels gain 1 to 4 linear-probe points on CIFAR-10/100 and TinyImageNet, not uniformly. | Active — read 2026-10-14 (NOTE-846) in full; worth reading as an exact identity: SimCLR’s InfoNCE loss, over all n objects with a Gaussian kernel on the unit sphere and a symmetric augmentation matrix π, equals tr(Zᵀ L(π) Z) + log R(Z), a Laplacian quadratic form plus the log of InfoNCE’s normalizers. Read it knowing that its “spectral clustering” admits any regularizer (Def. 2.7), so no minimizer is characterized and no eigenvector or density-ratio statement follows. The CLIP extension is a two-sentence proof, the batch-size reading is a remark, and the kernel gains are 1 to 4 points on three small datasets. |
| LIT-228 v2 | Contrastive and Non-Contrastive Self-Supervised Learning Recover Global and Local Spectral Embedding Methods | Balestriero & LeCun (2022), ARXIV-2205.11508. On a finite sample with a positive-pair graph G, idealized VICReg is Laplacian eigenmaps, a Euclidean SimCLR variant is a generalized MDS, and linear Barlow Twins is CCA. All three align the representation’s leading left singular vectors with G’s. SimCLR and Barlow Twins force rank(Z) = rank(G), and VICReg can stay full rank. Any of them is optimal for a linear probe on the training set iff G’s spectrum spans the task’s. | Active — read 2026-10-14 (NOTE-752) in full; worth reading for its method-by-method dictionary from joint-embedding SSL to spectral embedding on a finite positive-pair graph G. VICReg with a squared variance term is Laplacian eigenmaps (LPP when linear). A Euclidean SimCLR variant is a generalized MDS whose optimum is the top eigenvectors of G scaled by √eigenvalue. Linear Barlow Twins is CCA. All of them align Z’s left singular vectors with G’s, and they differ in rank. Read it knowing three things. The closed forms hold for modified losses, on a finite sample, with free or linear Z. Theorems 6, 9 and 10 are sketched or unproved. And the practical advice (prefer low-invariance VICReg under a misaligned G) rests on an unproved bound and synthetic experiments with N = 256. |
| LIT-231 v2 | The Hidden Uniform Cluster Prior in Self-Supervised Learning | Assran et al. (2022), ARXIV-2210.07277. Joint-embedding methods whose anti-collapse term maximizes the volume of the representation carry a prior toward features that split the data into equal-sized clusters: explicitly in MSN (a KL to uniform over mean cluster assignments) and SwAV (an equipartition constraint), through an informal reduction to K-means for VICReg, and by a cited equivalence for SimCLR. With mini-batches of two to eight ImageNet classes, SimCLR, VICReg and MSN lose 11–18 points of linear ImageNet accuracy while MAE and data2vec do not. Replacing MSN’s uniform target with a power law (PMSN) gains about a point on long-tailed iNaturalist-18, three with the true class distribution. | Active — read 2026-10-14 (NOTE-850) in full; worth reading for the clearest statement that a joint-embedding objective’s anti-collapse term is not neutral: MSN’s mean-entropy regularizer is a KL penalty pulling cluster sizes toward uniform, and SwAV enforces equal sizes outright. Read it knowing that the same prior is reached for VICReg only through an informal reduction of a simplified loss to K-means, and for SimCLR only through a cited equivalence; that the imbalance experiment uses two to eight ImageNet classes per batch, not a naturally long-tailed dataset; and that the iNat18 gains are single runs of 1–3 points, the largest using the true class labels. The account is THEORY-306; the practical advice belongs in the anthology. |
| LIT-232 v2 | Knowledge Sheaves: A Sheaf-Theoretic Framework for Knowledge Graph Embedding | Gebhart, Hansen and Schrater (2021; AISTATS 2023), ARXIV-2110.03789. A knowledge-graph embedding is a cellular sheaf of vector spaces on the schema graph, with restriction maps per relation type, plus a 0-cochain of its pullback to the knowledge graph. The summed Structured Embedding score is the sheaf Laplacian form, and TransE, TransR and RotatE are special cases; bilinear models are not. Since the zero cochain is always a section, training needs negatives and a margin. Composite multi-hop queries are scored without extra training by harmonic extension, a Schur complement, which beats naive TransE summation on ip and pi queries in untuned experiments. | Active — read 2026-10-14 (NOTE-843) in full; worth reading as the clean dictionary between coboundary-type knowledge-graph scores (Structured Embedding, TransE, TransR, RotatE) and cellular sheaves on the schema, with the summed score as a sheaf Laplacian form and multi-hop queries scored by harmonic extension, a Schur complement. Read it knowing that bilinear models fall outside it, that the zero cochain is always a section so all content is in the contrastive margin, that “approximate global section” means only small Laplacian energy, and that the experiments are preliminary, untuned and well below query-embedding baselines. It shares the word “sheaf” with the contextuality line and nothing else of substance. The account is THEORY-299. |
| LIT-233 v3 | Data-Dependent Generalization Bounds via Variable-Size Compressibility | Sefidgaran & Zaidi (2023), ARXIV-2303.05369. Letting the coding rate of a block of training runs vary with the realised sample and output gives generalisation bounds that depend on the sample at hand, not only on its distribution. With the rate set to log dP_{W|S}/dP(W) for a fixed prior, the bound is the disintegrated PAC-Bayes bound, and general change-of-measure versions recover most PAC-Bayes and mutual-information bounds. A new trajectory bound is data-dependent except for a coupling term that carries the trajectory’s dependence on the sample, and its experiment drops that term. | Active — read 2026-10-14 (NOTE-806) in full; worth reading as the paper that makes rate-distortion generalisation bounds data-dependent by letting the coding rate vary with the realised sample and output (Theorem 1), and that shows the disintegrated PAC-Bayes bound to be the case where the rate is log dP_{W|S}/dP(W) against a fixed prior, with Xu–Raginsky, Rivasplata et al. and LIT-236’s bounds as other instances (Theorems 3–5). It adds a lossy PAC-Bayes bound finite for deterministic continuous-parameter algorithms (Proposition 1) and a trajectory bound (Theorem 7). Read it knowing that the coding construction is proved only for finite alphabets and the general proofs are change of measure, that the rate is a Shannon code length against a prior and not Kolmogorov complexity, that Theorem 7 is data-dependent except for a coupling term M that carries the trajectory’s dependence on the sample, and that the CIFAR-10 experiment drops M and shows only that two curves fall together, with a vacuous value for FCN4. |
| LIT-234 v3 | The Effects of Regularization and Data Augmentation are Class Dependent | Balestriero et al. (2022), ARXIV-2204.03632. On ImageNet with ResNet-50, strengthening random crop or changing weight decay raises mean test accuracy while some classes fall sharply (e.g. “barn spider” about 68%→46% as the crop lower bound goes from 100% to 8%). Different pretrained architectures agree on which classes crops make unrecognisable. Pretraining crop strength also moves per-class linear-probe accuracy on iNaturalist in opposite directions. The evidence is selected classes and an uncorrected test. | Active — read 2026-10-14 (NOTE-845) in full; worth reading as a clear demonstration, on standard ImageNet ResNet-50 training, that the regularization strength best on average is not best for every class: stronger random crop and changed weight decay raise mean accuracy while some named classes lose 20 points or more, and the shift survives a frozen-backbone transfer to iNaturalist. Read it knowing that the evidence is per-class curves for hand-picked classes over 10–20 runs, that its significance test is uncorrected for 1,000 classes (4.4% rejections at the 5% level), that the weight-decay claim rests on one figure with no numbers, that Theorem 1 does not reach the class-level claim, and that the abstract’s iNaturalist example (class #8889, 70% to 30%) is a drop from moderate to strong cropping, not from none. |
| LIT-235 v2 | When and How Does Known Class Help Discover Unknown Ones? Provable Understanding Through Spectral Analysis | Sun et al. (2023), ARXIV-2308.05017. A spectral contrastive loss over an augmentation graph that adds labelled same-class pairs (NSCL) is equivalent to factorizing its adjacency. At that optimum, a novel labelling’s least-squares residual is exactly its component on the discarded eigenvectors that their labelled rows cannot absorb, which can be zero. In a toy, known classes sharing the target attribute remove the residual, and ones sharing a rival attribute raise it. | Active — read 2026-10-14 (NOTE-833) in full; worth reading as an exact account, within HaoChen et al.’s spectral contrastive framework, of how labelled data from other classes changes which partition of unlabelled data the learned features express: the novel labelling’s residual is its part outside the top eigenspace that the labelled rows of the discarded eigenvectors cannot absorb, and a toy shows known classes can help or hurt. Read it knowing that Theorem 5.5 is printed as ≤ though its proof gives equality, that everything is at the exact optimum of a population graph, that the general bound (Theorem 5.6) is approximate, pools all known classes into one averaged block and has notational slips, and that the help-or-harm examples have one sample per kind. |
| LIT-236 v3 | Rate-Distortion Theoretic Generalization Bounds for Stochastic Learning Algorithms | Sefidgaran et al. (COLT 2022), ARXIV-2203.02474. The expected generalisation error of any algorithm is at most √(2σ²·R/n) + ε, where R is the least I(S;Ŵ) of a surrogate output within ε in expected generalisation error. With ε = 0 this is Xu–Raginsky’s bound, which for finite alphabets is shown to be a lossless compression rate. With ε > 0 it stays finite where I(S;W) is infinite, and for a Lipschitz loss it becomes a rate-distortion function, or dimension, of the output’s law. A tail version takes a supremum over a KL ball. All bounds depend on the unknown data distribution. | Active — read 2026-10-14 (NOTE-780) in full; worth reading as the source of the lossy generalisation bound |E gen| ≤ √(2σ²·R_E(ε)/n) + ε, where R_E(ε) is the least I(S;Ŵ) over surrogate outputs within ε in expected generalisation error (Theorem 4), of which Xu–Raginsky’s bound is the lossless case, and of its Lipschitz, rate-distortion-dimension and tail forms (Corollaries 6–7, Theorem 10). Read it knowing that the block-coding meaning is proved only for finite alphabets, that in general spaces Theorem 4 is Xu–Raginsky applied to a surrogate, that every bound depends on the unknown data distribution, that no bound is evaluated for a learned model, and that the one worked example mixes squared and unsquared distortion, so its d/n rate is unsupported. |
| LIT-240 v3 | The Role of the Information Bottleneck in Representation Learning | Vera et al. (2018), DOI-10.1109/ISIT.2018.8437679. The generalization gap of the cross-entropy loss for an encoder is bounded by a term in the information the representation keeps about the input, so minimizing empirical risk plus that bound is equivalent to the Information Bottleneck on the empirical distribution — offered as an explanation of why noise injection such as dropout helps. | Unavailable — searched 2026-10-14 for a lawful full text of the ISIT 2018 paper and none was reached, so it is not read and no NOTE is filed. OpenAlex: closed, with one other location, the CentraleSupélec HAL record hal-01756003, which the HAL API gives as submitType “notice” (metadata only, no file); Unpaywall: not open access; IEEE Xplore returned a bot check (202), failing on access, not availability, though the paper is subscription-only there; arXiv: a search of Piantanida and Vera lists no paper with this title, only the related, distinct arXiv:1802.05355; CONICET’s repository, which lists an event-paper record for it, returned 502 and 503, so whether it holds a file is unknown; a web search for the title found no author-posted copy. It stays Unavailable until a copy is supplied. |
| LIT-242 v3 | On the Stepwise Nature of Self-Supervised Learning | Simon et al. (2023), ARXIV-2303.15438. For a simplified Barlow Twins loss on a linear or fixed-kernel model, gradient flow from small initialization learns the top eigenvectors of a symmetrized cross-correlation operator one at a time and reaches the minimum-norm zero-loss solution, kernel PCA on K_Γ. Exact from aligned initialization; informal for generic small initialization. ResNets trained with Barlow Twins, SimCLR and VICReg show the same stepwise rank growth, as an experimental finding. | Active — read 2026-10-14 (NOTE-741) in full; worth reading for the dynamics that the optimum-based spectral accounts lack. For a simplified Barlow Twins loss (λ = 1, no batch norm, symmetrized cross-correlation) on a linear or fixed-kernel model, gradient flow from small initialization learns the top eigenvectors of Γ, or of K_Γ, one at a time and reaches the minimum-norm zero-loss solution: kernel PCA. Read it knowing three things. The step for generic initialization (Result 4.6) is an informal derivation. Its operator is built from the model’s kernel and the sample pairs, not from K⁺. And its evidence for SimCLR, VICReg and ResNets is experimental: the initial NTK predicts them poorly, and the final empirical NTK captures about half the embedding subspace. |
| LIT-245 v4 | Minimum Description Length and Generalization Guarantees for Representation Learning | Sefidgaran et al. (2023), ARXIV-2402.03254. The expected generalization error of an encoder–decoder classifier is bounded by 2√((2·MDL + K + 2)/n) + ε for any decoder. Here MDL is the KL divergence between the encoder’s joint law of train-and-ghost latents and a prior symmetric under label-preserving permutations, not I(X;U). The quantity reflects the encoder’s structure and is finite for deterministic encoders under lossy compression. It reduces to sample-wise mutual information for a product prior, and the predicted-label version contains the f-CMI and VC bounds. | Active — read 2026-10-14 (NOTE-742) in full; worth reading as the theorem that bounds the generalisation error of an encoder–decoder learner by the encoder alone: 2√((2·MDL + K + 2)/n) + ε, where MDL is the KL divergence of the joint train-and-ghost latents to a prior symmetric under label-preserving permutations (Theorem 4). The bound holds for any decoder and stays finite for deterministic continuous encoders under lossy compression. With a product prior it reduces to sample-wise mutual information, which is why it can be tighter than IB-style bounds. Read it knowing that the bound is never evaluated numerically, that the prior may depend on the unseen ghost data and is only approximated in practice, that a K + 2 term sets a floor, that the proofs other than Theorem 4’s were not checked here, and that the experiment is one CIFAR-10 CNN at 62–65% accuracy. |
| LIT-246 v3 | On Variational Bounds of Mutual Information | Poole et al. (2019), ARXIV-1905.06922. Every variational lower bound on mutual information in use — Barber–Agakov, Donsker–Varadhan/MINE, NWJ/f-GAN-KL, InfoNCE — is one family, tight at a critic that is a function of the log density ratio log p(y|x)/p(y), and InfoNCE is the multi-sample member that trades variance for a hard ceiling of log K. | Active — read 2026-10-14 (NOTE-739) in full; worth reading as the single derivation that puts Barber–Agakov, Donsker–Varadhan, NWJ, MINE and InfoNCE in one family, and as the exact proof that InfoNCE is a lower bound on mutual information for every critic when the K batch pairs are i.i.d. from the joint, capped at log K. Read it knowing that Eq. 10 is printed with its indices transposed relative to its own derivation (the bound holds either way; the optimal critic log p(y|x) + c(y) belongs to the derived form), that the InfoNCE optimal critic is stated on Ma & Collins’s authority rather than proved, that the optimal critic of the interpolated bound I_α is a conjecture, and that every experiment is on 20-dimensional Gaussian toys or dSprites. |
| LIT-247 v3 | On Mutual Information Maximization for Representation Learning | Tschannen et al. (2019), ARXIV-1907.13625. The success of InfoNCE-style representation learning cannot be attributed to mutual information itself: MI is invariant under invertible reparametrizations, invertible encoders that maximize true MI can be worse than raw pixels, and tighter MI bounds from higher-capacity critics can give worse representations. | Active — read 2026-10-14 (NOTE-749) in full; worth reading as the controlled case that the downstream quality of InfoNCE- and NWJ-trained encoders does not track mutual information: invertible encoders improve while true MI is constant, a higher NWJ bound from a bigger critic gives worse features, and at matched bound values the encoder architecture decides. Also the proof that InfoNCE is neither a lower nor an upper bound when batch pairs are dependent, and the identity between InfoNCE with an inner-product critic and the multi-class K-pair loss. Read it knowing that the experiments are MNIST image halves (CIFAR10 for two of them) under a linear probe, that the tighter-bound result is shown for NWJ only (InfoNCE saturates near log K for every critic), and that the metric-learning account is argued, not tested. |
| LIT-248 v2 | Self-Supervised Learning with Kernel Dependence Maximization | Li et al. (2021), ARXIV-2106.08320. With image identity as a one-hot label, HSIC(Z,Y) = (Δl/N)(E_pos k − E k), and the mean squared MMD between the per-image view distributions is 2(N/Δl)·HSIC(Z,Y) (Thm A.1, App. B.2). The large-batch InfoNCE loss is exactly −HSIC(Z,Y) plus a log-mean-exp of the centred critic (eq. 28), and for a bounded kernel it bounds −HSIC(Z,Y) + γHSIC(Z,Z) up to a second-order remainder (Thm B.1). SSL-HSIC, −HSIC(Z,Y) + γ√HSIC(Z,Z), is proposed as a loss. | Active — read 2026-10-14 (NOTE-830) in full; worth reading as the place where an SSL objective becomes a statement about kernel mean embeddings: with image identity as a one-hot label, HSIC(Z,Y) is the gap between the mean kernel on positive pairs and on all pairs, (1/2N²)Σ MMD²(i,j) = (N/Δl)·HSIC(Z,Y), and the large-batch InfoNCE loss is exactly −HSIC(Z,Y) plus a log-mean-exp of the centred critic. Read it knowing that Theorem B.1 prints its remainder with the wrong sign, that the InfoNCE it analyses is the large-batch form and not the K-negative estimator, that “InfoNCE works because it estimates kernel dependence” is an interpretation no experiment isolates, and that the experiments are single runs. |
| LIT-249 v4 | Provable Guarantees for Self-Supervised Deep Learning with Spectral Contrastive Loss | HaoChen et al. (2021), ARXIV-2106.04156. Minimizing the population spectral contrastive loss L(f) = −2E[f(x)ᵀf(x⁺)] + E[(f(x)ᵀf(x⁻))²] is, up to a constant, a rank-k factorization of the normalized adjacency matrix of the augmentation graph, so its minimizers are the top-k eigenvectors (eigenfunctions, in the infinite case) of that graph up to a row scaling and an invertible linear map — neither of which changes linear-probe accuracy. | Active — read 2026-10-14 (NOTE-740) in full; worth reading as the founding statement that a contrastive loss is a low-rank factorization of the normalized augmentation graph. Lemma 3.2 makes the spectral contrastive loss equal to ‖D^(−1/2)AD^(−1/2) − FFᵀ‖²_F up to a constant, so its population minimizers are the top-k eigenvectors up to a row scaling and a right factor, which Lemma 3.1 shows a linear probe cannot see. Theorem 3.8 then bounds linear-probe error by Õ(α/ρ²_⌊k/2⌋). Read it knowing that everything is proved on a finite augmentation space at the population optimum of the spectral loss alone. The infinite case (App. F) assumes a uniform bound on the density ratio, not only Hilbert–Schmidt, and asserts rather than proves the extension of the main theorems. The symmetry of the pair distribution is used throughout and never stated. |
| LIT-250 v3 | Duality of Bures and Shape Distances with Implications for Comparing Neural Representations | Harvey et al. (2023), ARXIV-2311.11436. The Procrustes shape distance between two representations (optimal rotational alignment of units) equals the Bures distance between their centred linear kernel matrices, so the rotation-invariant geometry of a representation is exactly the geometry of its kernel, and through Uhlmann’s theorem the fidelity of two kernels is the maximum overlap over all representations consistent with them. | Active — read 2026-10-14 (NOTE-753) in full; worth reading as the clean finite-dimensional statement that, for centred linear kernels, the Procrustes distance equals the Bures distance and normalized Bures similarity is the cosine of the Riemannian shape distance, so the kernel is a complete invariant of a representation up to translation and orthogonal maps, and as the proof that CKA and NBS are related only within a rank-dependent envelope. Read it knowing that it covers linear kernels only, assumes the orthogonal group rather than arguing for it, and treats the infinite limits by a law-of-large-numbers sketch and a citation; it does not reach an RKHS or GNS statement. |
| LIT-251 v2 | Contrastive learning, multi-view redundancy, and linear models | Tosh, Krishnamurthy & Hsu (2021), ARXIV-2008.10150. The logistic contrastive problem of telling a same-datum pair of views from an independent pair is solved over all functions by the pointwise mutual information, whose exponential g*(x,z) = p(x,z)/(p(x)p(z)) is the change of measure from p(z) to p(z|x). When the two views are redundant for a label, the infer-then-predict function E[E[Y|Z]|X=x] = ∫ E[Y|Z=z] g*(x,z) p(z) dz is within εX + 2√(εXεZ) + εZ of the Bayes predictor, so linear functions of the landmark embedding (g*(x,Z_1), …, g*(x,Z_m)) are near-optimal, with error O(1/m). | Active — read 2026-10-14 (NOTE-841) in full; worth reading as the redundancy account of contrastive learning: when two views are redundant for a label, the infer-then-predict function µ(x) = E[E[Y|Z]|X=x] is within εX + 2√(εXεZ) + εZ of the Bayes predictor (Lemma 1), and µ is linear in the density ratio g*(x,·) = p(x,z)/(p(x)p(z)), “the change-of-measure from the marginal distribution of Z to the conditional distribution of Z given X = x”. Random landmark evaluations of g* then give near-Bayes linear predictors at rate 1/m. Read it knowing that it is population-level, that the logistic problem has a single negative, that its error propagation assumes a uniform bound on the ratio, and that its low-dimensional direct embeddings are shown to exist, not to be reached. It never assumes the views are exchangeable. |
| LIT-252 v2 | Spectral Inference Networks: Unifying Deep and Spectral Learning | Pfau et al. (2018), ARXIV-1806.02215. For a symmetric kernel k on a space with sampling density p, the span of the top-N eigenfunctions of Kf = E_x′[k(x,x′)f(x′)] on L²(p) maximizes the generalized Rayleigh quotient Tr(Σ⁻¹Π), with Σ = E[u uᵀ] and Π = E[k(x,x′)u(x)u(x′)ᵀ]; a Cholesky-masked gradient orders the eigenfunctions, argued informally, and moving averages of Σ and its Jacobian correct the minibatch bias. Slow Feature Analysis is the graph-Laplacian case on adjacent frames, where the slowest features are the bottom of the Laplacian’s spectrum. | Active — read 2026-10-14 (NOTE-824) in full; worth reading as the generic neural solver for eigenfunctions of a symmetric kernel operator on L²(p): the top-N span maximizes Tr(Σ⁻¹Π), a Cholesky-masked gradient orders the eigenfunctions, and moving averages of Σ and its Jacobian remove the minibatch bias. Slow Feature Analysis is the graph-Laplacian case. Read it knowing that symmetry of k is a precondition placed on the kernel, not derived from a pair distribution, that the ordering argument is informal by the authors’ own statement, that only the hydrogen-atom experiment has ground truth, and that it never mentions positive pairs, contrastive learning or the density ratio. |
| LIT-253 v2 | When Does Closeness in Distribution Imply Representational Similarity? An Identifiability Perspective | Nielsen et al. (2025), ARXIV-2506.03784. For softmax models p(y|x) ∝ exp(f(x)ᵀg(y)) under a diversity condition, the model distribution determines the embeddings and unembeddings up to an invertible linear map (f = Af′, g₀ = A⁻ᵀg′₀). The determination is not stable under KL divergence, though: two models can be ε-close in KL, even both near maximum likelihood, while their embeddings are far from linearly equivalent. Under a log-likelihood variance distance it is stable: d_LLV ≤ ε implies a representational dissimilarity of at most 2Mε. | Active — read 2026-10-14 (NOTE-738) in full; worth reading as the exact statement that softmax models are identified only up to the paired map f = Af′, g₀ = A⁻ᵀg′₀, and as the proof that this identification is not stable under KL divergence but is stable under a log-likelihood variance distance (d_LLV ≤ ε ⇒ d_f,g ≤ 2Mε). Read it knowing that the KL counterexample needs the unembedding norm to grow without bound, that how far apart its embeddings are is shown numerically rather than proved, that everything concerns the final embedding and unembedding only, and that the experiments use 2- to 5-dimensional representations. |
| LIT-254 v4 | Similarity of Neural Network Representations Revisited | Kornblith et al. (2019), ARXIV-1905.00414. Any similarity index invariant to invertible linear maps gives the same value for all representations of rank n, the number of examples (Theorem 1). The paper argues, without a theorem, for invariance to orthogonal maps and isotropic scaling only, and proposes centered kernel alignment, normalized HSIC of centred Gram matrices, which has exactly those invariances. R²_CCA and linear regression are CKA of re-weighted Gram matrices. Linear CKA alone matches corresponding layers across CNN initializations (99.3%, against at most 45.4% for the other linear indexes). | Active — read 2026-10-14 (NOTE-743) in full; worth reading as the source of CKA for comparing representations and of the argument that a similarity index should be invariant to orthogonal maps and isotropic scaling but not to all invertible linear maps. That argument rests on Theorem 1: any index with that wider invariance is constant once rank reaches the number of examples. It also shows R²_CCA, linear regression and linear CKA to be one family, differing in eigenvalue weighting, with canonical ridge interpolating between them. Read it knowing that the case for the orthogonal group is argued from training dynamics, not proved, that it does not state the converse that equal Gram matrices imply an orthogonal relation, and that RBF CKA is scale-invariant only because its bandwidth is set from the median distance. |
| LIT-256 v2 | Mutual Information Neural Estimation | Belghazi et al. (2018), ARXIV-1801.04062. Mutual information is stated measure-theoretically as the expectation under the joint of log dP_XZ/d(P_X⊗P_Z), and its Donsker–Varadhan dual is tight exactly at T* = log dP/dQ + C, so a neural critic trained on the dual is a log Radon–Nikodym-derivative estimator. | Active — read 2026-10-14 (NOTE-828) in full; worth reading as the paper that states mutual information as the expected log Radon–Nikodym derivative and proves the Donsker–Varadhan bound tight at that log derivative plus a constant. Read it knowing that its consistency theorem is for the exact empirical supremum over a compact network family, that two steps of its proof have gaps, that its gradient-bias fix is a heuristic, and that the finite-sample estimate is not a bound (LIT-246). |
| LIT-257 v2 | What Representational Similarity Measures Imply about Decodable Information | Harvey et al. (2024), ARXIV-2411.08197. For linear readouts regularised by w ↦ wᵀG(X)w, the optimal decoded signal is K_X z with K_X = X G(X)⁻¹ Xᵀ / M, and the expected alignment of two networks’ readouts is Tr(K_X K_z K_Y). With K_z = I that gives linear CKA and mean squared CCA as normalized averages, ENSD as the average, and the GULP distance as the average squared readout distance. The Procrustes distance of normalized representations and the average decoding distance bound each other with constants set by the participation ratio of K_X − K_Y. All at finite M. | Active — read 2026-10-14 (NOTE-834) in full; worth reading as the decoding reading of representational similarity: an optimal regularized linear readout outputs K_X z, so linear CKA, mean squared CCA, ENSD and the GULP distance are task-averaged readout statistics, and Procrustes distance and decoding distance bound each other through a participation ratio. Read it knowing that every identification fixes the target covariance to the identity at finite M, and that its population appendix is an outline that does not reach that case. |
| LIT-258 v2 | Contrastive Learning Can Find An Optimal Basis For Approximately View-Invariant Functions | Johnson et al. (2022), ARXIV-2210.01883. On a finite augmentation space, the NT-Logistic and spectral contrastive losses are minimized over unconstrained kernels by the positive-pair kernel K⁺(a₁,a₂) = p⁺(a₁,a₂)/(p(a₁)p(a₂)), and InfoNCE by K⁺ up to a constant on each communicating class. Kernel PCA under K⁺ returns exactly the eigenfunctions of the positive-pair Markov chain, and the top-d of those span the linear-predictor subspace that minimizes worst-case approximation error over approximately view-invariant targets. | Active — read 2026-10-14 (NOTE-751) in full; worth reading as the cleanest statement linking contrastive losses, one Mercer kernel and a minimax-optimal eigenbasis. The NT-Logistic and spectral losses are minimized over unconstrained kernels by K⁺ = p⁺/(p⊗p), and InfoNCE by K⁺ up to a constant per communicating class (over symmetric functions, with the minimizer form taken from Van den Oord et al.). Kernel PCA under K⁺ returns λ_i^(1/2) times the positive-pair chain’s eigenfunctions (Thm 3.2). The top-d span minimizes worst-case L²(p) error over approximately view-invariant targets, at ε/(2(1 − λ_(d+1))), provided d is at least the number of communicating classes (Thm 4.1). Read it knowing that everything is proved on finite spaces at the unconstrained population optimum. The supporting experiments are synthetic, and the authors report that they could not extract the eigenfunctions from real-data models. |
| LIT-260 v2 | Kernel Mean Embedding of Distributions: A Review and Beyond | Muandet et al. (2016), ARXIV-1605.09522. Point evaluation is bounded on an RKHS, so the Riesz theorem gives the reproducing kernel k_x. Likewise, when E√k(X,X) < ∞ the expectation functional f ↦ E_P f is bounded, so Riesz gives a unique mean embedding μ_P with E_P f = ⟨f, μ_P⟩. MMD is then the RKHS distance ‖μ_P − μ_Q‖, and it separates distributions exactly when the kernel is characteristic. | Active — read 2026-10-14 (NOTE-838) in full; worth reading as the one held text that runs the chain bounded evaluation → Riesz → k_x → Moore–Aronszajn → μ_P → MMD/HSIC in numbered statements, with Lemma 3.1 (μ_P exists and represents expectation when E_P√k(X,X) < ∞) proved in the text, and as a map of conditional mean embeddings and their limits. Read it knowing that it is a survey whose proofs are mostly sketched or cited, that every result assumes a fixed kernel, that it defines μ_P as a Bochner integral as well as by Riesz, so Riesz is one route to existence and not shown to be the only one, and that the unregularized conditional embedding C_YX C_XX⁻¹ often fails to exist. |
| LIT-305 v2 | On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups | Kondor & Trivedi (2018), ARXIV-1802.03690. Theorem 1 says that a feed-forward network whose layers are linear maps followed by pointwise nonlinearities, with every layer indexed by a homogeneous space G/H_ℓ of a compact group G, is G-equivariant (every layer transforming by the induced action) if and only if each linear map is a generalised convolution f ∗ χ with a filter on the double coset space H_{ℓ−1}\G/H_ℓ. The “if” direction is a short coset computation. The “only if” direction goes through the group Fourier transform: equivariant maps preserve isotypic components (Lemma 6) and act on each Fourier matrix by right multiplication, M ↦ MB (Lemma 10), which the convolution theorem (Prop. 2) turns back into a convolution. | Active — read in full 2026-09-29 (NOTE-282); worth reading as the standard ML statement of “equivariance ⇔ convolution” for compact groups and as a compact introduction to isotypic decomposition and noncommutative Fourier analysis for neural networks. Read the “only if” proof as an outline. It is written out for finite or countable groups and scalar channels only, and one step (removing the pointwise nonlinearity) does not hold as stated for ReLU. |
| LIT-314 v2 | Group Equivariant Convolutional Networks | Cohen & Welling (2016), ARXIV-1602.07576. Group convolution (eqs. 10–11) replaces translation by the elements of a discrete group G, so feature maps become functions on G and every layer is G-equivariant (eq. 12, with pooling and nonlinearities shown to commute with the action). With parameter counts held roughly fixed, p4-CNNs cut rotated-MNIST test error from 5.03% (planar baseline) to 2.28% (previous best 3.98%), and p4m convolutions cut plain-CIFAR10 error from 9.45% to 6.46% on a ResNet44 (Tables 1–2). | Active — read in full 2026-09-29 (NOTE-288); worth reading as the short, self-contained origin of the group-convolution layer and of the “equivariance as weight sharing over a symmetry group fixed in advance” design. Read the CIFAR “state of the art” claim as holding for un-augmented CIFAR10 only, and the paper as containing no representation theory (no irreps, characters or Fourier analysis on groups). |
| LIT-319 v2 | Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges | Bronstein et al. (2021), ARXIV-2104.13478. The authors cast CNNs, GNNs, Deep Sets, Transformers, spherical, mesh and gauge CNNs, and LSTMs as instances of one “blueprint” (p. 29). Each is a stack of linear G-equivariant layers, pointwise nonlinearities and local pooling, capped by a G-invariant global pooling, with the domain Ω and its symmetry group G given in advance (p. 27: “a symmetry group G”; p. 4: “exploiting the known symmetries”). The two priors are symmetry and scale separation. The Fourier transform is derived as the joint eigenbasis of shift-commuting (circulant) operators, which requires distinct eigenvalues (p. 37). Its group-theoretic generalisation via irreducible representations is announced and deferred to “future work” (p. 43). | Active — read in part 2026-09-29 (NOTE-274); worth reading as the standard systematisation of symmetry-as-architecture in deep learning and for its clean derivation of convolution and the Fourier basis from translation symmetry (§4.2). Read it as a proto-book that defers its representation theory (irreps, group Fourier transforms) and its learning theory to “future work”. It takes the symmetry group as known and never addresses discovering one. |
| LIT-322 v2 | Not All Language Model Features Are One-Dimensionally Linear | Engels et al. (2024), ARXIV-2405.14860. Defines a feature as irreducibly multi-dimensional when no rotation and translation makes its distribution either a product of independent parts (separable) or a mixture with a lower-dimensional component (Def. 2). Softened versions are the separability index S(f) and the ε-mixture index M_ε(f) (Def. 3). Clustering sparse-autoencoder dictionary elements by cosine similarity then finds circular representations of weekdays and months in GPT-2 (also 20th-century years) and in Mistral 7B. Replacing the circle coordinates at the day or month token causally changes the answer on “N days/months from X” prompts in Mistral 7B and Llama 3 8B, and in early layers the circle patch nearly matches patching the whole layer. The models’ task accuracy is modest on weekdays (29/49 Llama, 31/49 Mistral), high on months (143/144, 125/144), and near-trivial for GPT-2 despite its circles. | Active — read in full 2026-09-29 (NOTE-273); worth reading as the first causal demonstration that a pretrained language model computes with an irreducibly two-dimensional (circular) feature, and for a clean statistical definition of feature reducibility. Read the discovery as largely manual: circles for days, months and years, found by eye among about 2,500 inspected clusters, with the automatic score ranking them 9th, 28th and 15th of 1,000. Read “irreducible” as statistical, not representation-theoretic. |
| LIT-323 v3 | Toy Models of Superposition | Elhage et al. (2022), ARXIV-2209.10652. In a toy autoencoder x′ = ReLU(WᵀWx + b), with n sparse synthetic features compressed into m < n dimensions, a linear model keeps only the top-m features, but with the output ReLU and sparse enough features the model stores more features than dimensions (“superposition”), in non-orthogonal directions whose interference the ReLU and a negative bias filter out. Whether a feature is dropped, superposed or given a dedicated dimension changes discontinuously with sparsity and importance, and in the n = 2, m = 1 theory this is a first-order phase change. With uniform features the directions settle into uniform polytopes, with feature dimensionalities clustering at 3/4, 2/3, 1/2, 2/5 and 3/8. A ReLU hidden layer creates a privileged basis in which monosemantic and polysemantic neurons coexist, and a model can compute |x| for more features than it has neurons. Everything is shown in toy models only, and the authors expect the geometry and learning-dynamics results to generalise least. | Active — read in full 2026-09-29 (NOTE-278); worth reading as the founding statement of the superposition hypothesis and of the vocabulary the field now uses: features as directions, privileged versus non-privileged bases, polysemanticity as interference, and feature dimensionality. It is also the cleanest demonstration that a nonlinearity plus sparsity makes it optimal to store more features than dimensions. Its evidence is toy models alone, which the paper says plainly. Its phase-change and polytope results are exact only for n = 2, m = 1 and uniform features. |
| LIT-338 v2 | The information bottleneck method | Tishby et al. (1999), ARXIV-physics/0004057. The paper defines relevant information as what X tells about a second variable Y and poses the information-bottleneck problem, min over p(x̃|x) of L = I(X̃;X) − β·I(X̃;Y), under the Markov chain X̃ ← X ← Y (eq. 15). Theorem 4 gives the exact formal solution p(x̃|x) = p(x̃)/Z(x,β)·exp(−β·D_KL[p(y|x)‖p(y|x̃)]), so the KL divergence emerges as the effective distortion. Theorem 5 gives a Blahut–Arimoto-style alternating iteration that converges but, being non-convex jointly, not necessarily to a unique solution. | Active — read in full 2026-09-29 (NOTE-300); worth reading as the short, exact source of the IB functional, its self-consistent equations, the information plane and β as the one trade-off parameter. The phase-transition claim is cited out, not shown here. Nothing in it concerns gradients, SGD or batch size, so it is not the principle behind the gradient-as-channel narrative in the sense row 12 suggests. |
| LIT-347 v2 | Information-theoretic analysis of generalization capability of learning algorithms | Xu & Raginsky (2017), ARXIV-1705.07809. If the loss ℓ(w,Z) is σ-subgaussian for every w, then the expected generalisation error of any learning algorithm P_{W|S} obeys |E[L_μ(W) − L_S(W)]| ≤ √(2σ²·I(S;W)/n) (Theorem 1). This extends Russo–Zou to uncountable hypothesis spaces, and the paper adds a high-probability version, n = (8σ²/α²)(ε/β + log(2/β)) (Theorem 3). Minimising empirical risk plus I(S;W), relaxed to a KL term against a prior Q, yields exactly the Gibbs algorithm (Theorem 5). Noisy ERM and adaptive composition are analysed through the same mutual-information quantity. | Active — read in full 2026-09-29 (NOTE-306); worth reading as the short, clean source of the input–output mutual-information generalisation bound, and of “a learning algorithm is a channel P_{W|S}”, whose capacity is sup_μ I(S;W). The bound is in expectation and is vacuous for deterministic algorithms on continuous hypotheses, which is what the later compression and rate–distortion refinements (LIT-236, LIT-233) address. |
| LIT-354 v4 | Algebraic Geometry and Statistical Learning Theory | Watanabe (2009), DOI-10.1017/CBO9780511800474. Per publisher description: develops statistical learning theory for singular models (neural networks, mixtures, hidden Markov models) in which the Fisher information is degenerate, using resolution of singularities to show that the real log canonical threshold governs Bayesian free energy and generalization error. | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read; no lawful full text could be reached, and it stays Unavailable until a copy is supplied. Its core results are now read in a lawful substitute by the same author, Watanabe’s A Widely Applicable Bayesian Information Criterion (LIT-616; arXiv 1208.6338, JMLR 2013). That paper restates the book’s standard representation theorem, the real log canonical threshold and the free-energy asymptotics F = nL_n(w0) + λ log n − (m − 1) log log n, citing the book for each. Read it for the results; the book holds their proofs, the generalization-error theory and WAIC. |
| LIT-356 v2 | How much does your data exploration overfit? Controlling bias via information usage | Russo & Zou (2015), arXiv:1511.05219. Suppose an analyst reports statistic φ_T, chosen by any data-dependent rule T from m candidates each σ-sub-Gaussian. Then the selection bias obeys |E[φ_T − µ_T]| ≤ σ√(2·I(T;φ)) (Prop. 1), where “information usage” I(T;φ) ≤ H(T) ≤ log m. For argmax selection over φ ~ N(µ,I) the squared error is Θ(1 + H(T)) (Prop. 3). Adaptive analyses compose additively, with I(T_{k+1};φ) ≤ Σ_i I(Y_{T_i};φ_{T_i} | H_{i−1},T_i) (Lemma 1). Answering each query with Gaussian noise of variance σ²√j/n therefore keeps the k-th answer’s error at O(σk^{1/4}/√n) (Prop. 7). | Active — read in full 2026-09-29 (NOTE-308); Worth reading as the origin of the mutual-information bias and generalisation bound that LIT-347 (Xu & Raginsky) extends, and as the first place its adaptive-composition chain rule and its per-query “information budget” with Gaussian noise (½·log(1+SNR) per query) appear. Read it knowing that it is stated for a finite set of m candidate statistics, and that “the bound is tight” is shown only for Gaussian argmax and threshold selections, as squared-error (not bias) bounds except in the exponential case. |
| LIT-363 v2 | Understanding image representations by measuring their equivariance and equivalence | Lenc & Vedaldi (2014), arXiv:1411.5908. Given a fixed representation φ (HOG, or AlexNet Conv1–Conv5 before ReLU) and a transformation g of the input image chosen by the experimenter (flips, 90° and other rotations, rescaling by 2^{±1/2}, affine warps), it fits a linear map M_g on the representation space with φ(gx) ≈ M_g φ(x), by sparse or structured-sparse regression or by a “transformation layer” trained through the rest of the network, and fits “stitching layers” E with φ′(x) ≈ E φ(x) between different networks. In AlexNet, learned M_g recover most of the classifier’s accuracy for vertical flips and 90° rotations (top-1 error 0.75 uncompensated vs 0.43–0.51 and 0.44–0.53 across Conv1–Conv5, original 0.43), channels scored invariant rise to 99.61% for horizontal flips at Conv5, and Conv1–Conv2 are interchangeable between AlexNet and networks trained on ImageNet, Places or both. It detects no group and uses no spectrum. | Active — read in full 2026-09-29 (NOTE-309); worth reading as the origin of two standard probes of trained networks, fitted linear equivariance maps and model stitching, and as the closest prior art for “measuring symmetry in a trained network’s representation”; read it knowing that every transformation tested is named in advance and applied to the input, that each is fitted separately with no group structure, and that its deep-layer “equivariance” is judged by recovered task accuracy through a high-capacity learned layer. |
| LIT-369 v2 | Grokking and Generalization Collapse: Insights from HTSR theory | Prakash & Martin (2025), arXiv:2506.04434. A depth-3, width-200 ReLU MLP trained on 1,000 MNIST images (the Omnigrok recipe, with weights and biases scaled ×8, MSE loss, AdamW, WD = 0) for 10⁷ steps goes through three phases read off the accuracy curves. In pre-grokking (to ~10⁵ steps) training accuracy saturates while test accuracy stays low. In grokking (~10⁵–10⁶) test accuracy rises to ~0.85. In “anti-grokking” (~10⁶–10⁷) it falls to ~0.45–0.5 while training accuracy stays near 1. The average HTSR power-law exponent α of the two hidden weight matrices goes 4.0 → 2.9 → 1.1 at the phase ends (Table 1). In the collapse phase, correlation traps (outlier eigenvalues of the elementwise-shuffled weight matrix) appear. The paper’s own weight-decay control shows FC2’s α below 2, and traps in both layers, without a collapse (Fig. 6, Table 2). That undercuts its headline claim that α < 2 “invariably” heralds collapse. | Proposed — read in full 2026-09-30 (NOTE-317); worth knowing as the report of a late generalisation collapse after grokking at WD = 0, but it is one architecture on one 1k-sample MNIST subset, with the seed count unstated, and its central diagnostic (α < 2 ⇒ collapse) is contradicted by its own WD = 0.01 control. What would settle it: several seeds and architectures, with α and traps reported for runs that reach α < 2 without collapsing, and a comparison against the ℓ² norm under the Omnigrok LU account, which the paper does not cite. |
| LIT-370 v2 | New Evidence of the Two-Phase Learning Dynamics of Neural Networks | Zhou et al. (2025), arXiv:2505.13900. VGG-16 and ResNet-20 on CIFAR-10 (SGD with momentum, lr 0.1, 160 epochs) are compared across pairs of training times. Before an early “inflection point” (~2,500 iterations for VGG-16, ~100–500 for ResNet-20), a tiny parameter perturbation under identical SGD noise sends the run to a different basin by the end: loss barrier up to ~2, test disagreement ~0.15–0.5. After it, the same perturbation leaves the runs linearly connected. Also after it, the empirical NTK keeps changing step to step but stays within a bounded cosine distance of any later reference kernel (the “cone effect”, e.g. S ≈ 0.05–0.1 for VGG-16 vs ≈ 0.15–0.2 from initialisation). Switching to linearised training later gives better test accuracy (VGG-16 0.13 → 0.82 as the switch moves from ≤200 to 5,000 iterations). | Proposed — read in full 2026-09-30 (NOTE-318); a compact, readable demonstration that an early, short chaotic phase gives way to a confined, stable one in both basin and kernel terms, but it rests on two architectures on one dataset, one run each with no error bars. The chaos half largely re-measures Frankle et al.’s (2020) spawning / linear-mode-connectivity result with a parameter perturbation in place of fresh SGD noise, and the claimed role of the cone in generalisation is asserted. What would settle it: seeds and error bars, the batch size, a stated perturbation norm, other domains, and a quantitative test of confinement (e.g. an angular radius that stops growing) rather than a 3-D picture. |
| LIT-386 v2 | Homeostasis and soft robotics in the design of feeling machines | Man & Damasio (2019), Nature Machine Intelligence 1:446–452. A perspective proposing machines organized around homeostasis: a body that must be kept within viable states gives the machine a goal of its own, and something like feeling as a source of motivation and of evaluation. Soft robotics and multisensory abstraction are named as the means. The claimed payoffs are equivalents of feeling, better function across environments, and a platform for studying consciousness. Unread: placed from the abstract. | Unavailable — filed 2026-10-01 at the owner’s request; no full text could be reached, so it is placed from the abstract and figure captions only and no NOTE is filed. Nature’s page is paywalled, there is no arXiv version, Unpaywall, OpenAlex and Semantic Scholar list no open copy, and the ResearchGate, Scribd and ProQuest copies did not serve the text. By its abstract it is a design proposal for machines built around homeostasis, not a reported system or result. Flagged anthology-candidate because the anthology’s agents-and-environments topic could hold it; a reading should settle whether it carries an instruction for ML practice. |
| LIT-427 | Pandemonium: A Paradigm for Learning | Selfridge (1959), Mechanisation of Thought Processes Vol. I (HMSO), pp. 513–526 plus discussion to p. 531. Pattern recognition by a “pandemonium”: data demons hold the input, computational subdemons compute features, each cognitive demon “shrieks” a weighted sum of them, and a decision demon picks the loudest. The system learns by hill-climbing on the weights and by “natural selection” on the subdemons, culling those of low worth and breeding new ones by mutation and by “conjugation” of pairs. It is applied to telling dots from dashes in hand-keyed Morse. In the discussion McCarthy proposes reading the demons’ private computation as unconscious and their public shouting as conscious thought. | Active — read in full 2026-10-02 (NOTE-360); worth reading as the first published architecture of a mind-like process built from many quasi-independent “demons”, each doing one narrow job, with no part holding the whole’s competence. That is the decomposition template that Minsky’s agents inherit. Cite it as an engineering proposal with a preliminary result reported only in discussion, not as a theory of mind. Its one remark on consciousness is McCarthy’s, in the discussion, and it is an aside. |
| LIT-436 | Intelligence without representation | Brooks (1991), Artificial Intelligence 47:139–159, written in 1987. Argues that AI should be built incrementally from complete autonomous “Creatures” tested in the real world, and that the system should be decomposed by activity into parallel layers, each connecting sensing to action, not by function into perception, a central reasoner and action. His robots have no central representation and no central control: each layer is a fixed network of simple finite-state machines, and a higher layer subsumes a lower one by suppressing or inhibiting its wires. Coherent behaviour emerges, “in the eye of an observer”, from competing behaviours. Hypothesis: representation is the wrong unit of abstraction for the bulk of intelligence. | Active — read in full 2026-10-02 (NOTE-351) from the copy Brooks posts; worth reading as the decomposition side built and run: a robot whose purposive behaviour comes from parallel layers of simple finite-state machines with no central representation and no central control, which Brooks likens to Minsky’s account of human behaviour. The coherence of the whole, he says, exists “in the eye of an observer”. The evidence is three layers on one physical robot, described rather than measured. It makes no claim about minds or experience and disclaims interest in the philosophy. |
| LIT-577 | Sophisticated Inference | Friston, Da Costa, Hafner, Hesp & Parr (2021; arXiv 2020), Neural Computation 33(3):713–763. Ordinary active inference scores a fixed set of policies by expected free energy (risk plus ambiguity). Sophisticated inference instead evaluates the expected free energy of the next action recursively, averaging over the beliefs the agent would hold after each plausible outcome and action, which amounts to a tree search over belief states pruned below a probability of 1/16. In simulations it plans past short-term costs: a rat visits a mildly aversive cue, and a horizon of four escapes a local minimum in a maze. It recovers Bayes-adaptive RL in the zero-temperature limit. | Active — read in full 2026-10-03 (NOTE-454), from the arXiv preprint, whose displayed equations did not survive text extraction. Worth reading as the active-inference planning scheme the record’s free-energy holdings otherwise lack: expected free energy evaluated recursively over future belief states, a deep tree search pruned by probability, with the Bellman equation as its formal analogue. The evidence is small simulations (a T-maze rat and an 8 × 8 grid). Flagged anthology-candidate: it is a planning algorithm, compared with reinforcement learning, and the anthology’s agents-and-environments topic may hold it. |
| LIT-607 | Natural Gradient Works Efficiently in Learning | Amari (1998), Neural Computation 10(2):251–276. When the parameter space carries a Riemannian metric G, the steepest descent direction of a loss is the natural gradient G⁻¹∇L, not the ordinary gradient. For a statistical model the metric is the Fisher information, the only invariant one. Online natural gradient learning with step 1/t is Fisher efficient: it attains the Cramér–Rao bound asymptotically, as well as the best batch estimator. The paper computes the metric and its inverse for a simple perceptron, for the group of nonsingular matrices (blind source separation, giving the update ∇L·WᵀW) and for linear systems, proposes an adaptive learning rate with 1/t convergence, and conjectures that the natural gradient escapes the plateaus of backpropagation. | Active — read in full 2026-10-03 (NOTE-472); worth reading as the source of the natural gradient: steepest descent in a Riemannian parameter space is G⁻¹∇L (Theorem 1), the Fisher information is the metric on a statistical model, and online learning with the natural gradient and a 1/t rate is Fisher efficient (Theorem 2). Its efficiency proof needs the Fisher matrix at the optimum to be invertible, which is exactly what singular learning theory says fails for neural networks. |
| LIT-608 | Deep Equals Shallow for ReLU Networks in Kernel Regimes | Bietti & Bach (2021), ICLR 2021. Dot-product kernels on the sphere are diagonal in spherical harmonics, with one eigenvalue µ_k per degree k shared by N(d, k) ~ k^(d−2) harmonics. Theorem 1 reads the asymptotic decay of µ_k off the kernel function’s expansion at ±1: a non-integer exponent ν there gives µ_k ~ k^(−d−2ν+1). The arc-cosine kernels have ν = 1/2 and 3/2, and composing them preserves ν, so the NTK of a deep ReLU network decays as k^(−d) and its random-feature kernel as k^(−d−2), exactly as for two layers. Deep and shallow ReLU kernels therefore have the same RKHS, the same as the Laplace kernel’s for the NTK. | Active — read 2026-10-03 (NOTE-475): main text and Appendix A in full, the proofs in Appendices B and C followed at the level of their lemmas. Worth reading as the general rule behind the NTK eigenvalue decays the record holds: for a dot-product kernel on the sphere, the decay of its spherical-harmonic eigenvalues is fixed by the kernel function’s non-integer exponent at the endpoints ±1 (Theorem 1). Deep ReLU kernels keep the shallow exponent, so depth leaves the spectrum’s shape unchanged: µ_k ~ C(d, L) k^(−d) for the NTK and k^(−d−2) for the random-feature kernel, with only the constant growing in L (Corollaries 2–3). |
| LIT-611 | Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2 | Lieberum et al. (2024), Google DeepMind technical report. Gemma Scope releases JumpReLU sparse autoencoders trained on the attention, MLP and residual-stream activations of every layer of Gemma 2 2B and 9B, and of three layers of 27B: over 400 SAEs, over 2,000 counting sparsity variants, and over 30 million latents. Each latent fires only above a learned threshold θ_i, and training penalises the L0 count of active latents directly, through straight-through estimators. Evaluation is by sparsity–fidelity curves (delta LM loss and FVU against mean L0). Base-model SAEs transfer to the instruction-tuned model, wider SAEs split latents, and the authors state that no agreed metric of SAE quality exists. | Active — read in full 2026-10-03 (NOTE-482). Worth reading as the technical report for the open SAE suite that later SAE studies, including the anthology’s, are run on, and as the clearest statement of what a JumpReLU SAE is. A JumpReLU SAE is a learned hard threshold per latent, z ⊙ H(z − θ), trained against squared reconstruction error plus λ‖f(x)‖₀ (Eqs. 3–4). The report releases over 400 such SAEs (over 2,000 counting the sparsity variants) on every layer and sublayer of Gemma 2 2B and 9B and on three layers of 27B. Its evaluations are sparsity–fidelity curves, with no criterion offered for choosing a width or a sparsity; the authors say there is no consensus on what measures SAE quality. |
| LIT-612 | The Convergence Rate of Neural Networks for Learned Functions of Different Frequencies | Basri, Jacobs, Kasten & Kritchman (2019), NeurIPS 2019. In the linearised dynamics of a wide two-layer ReLU network, on uniform data on a sphere, the Gram matrix H∞ is a convolution, so its eigenvectors are Fourier modes or spherical harmonics. Its eigenvalues are computed in closed form per frequency. Without bias, every odd frequency k ≥ 3 has eigenvalue zero, so the network cannot represent or learn it (Theorems 2 and 4). With a zero-initialised bias all frequencies pass. The eigenvalues fall as 1/k² on S¹ and, numerically, about 1/k^(d+1) on S^d, so the time to learn frequency k grows as k² and k³ on the circle and the 2-sphere. Deep and residual networks show the same exponents. | Active — read 2026-10-03 (NOTE-481): main text in full, the supplementary derivations of the S^d eigenvalues skimmed. Worth reading for the closed-form NTK eigenvalues per frequency and the exact null space they reveal. On the circle the bias-free two-layer ReLU kernel has eigenvalue 2(k²+1)/(π²(k²−1)²) on even k ≥ 2 and exactly zero on odd k ≥ 3 (Eq. 12), so such a network cannot represent cos 3θ at all (Theorems 2 and 4). A zero-initialised bias fills the null space, with 1/(π²k²) on odd k (Eq. 13). Convergence time then grows as k² on S¹ and about k³ on S², and shallow, deep and residual networks match: fitted exponents 1.93–2.15 on S¹ and 2.74–3.13 on S² (Figs. 6–7). |
| LIT-613 | Traces of Class/Cross-Class Structure Pervade Deep Learning Spectra | Papyan (2020), JMLR 21(252). Cross-entropy indexes every deepnet quantity by example, class and would-be class, a two-way MANOVA layout. In the spectra of features, backpropagated errors, gradients, the Fisher, the Hessian and weights, knockouts attribute the top outlier to the global mean, C−1 outliers to between-class second moment, a mini-bulk of C(C−1) to between-cross-class covariance, and the main bulk to within-cross-class covariance. Feature and error class means separate from the bulk and become orthogonal with depth. Gradient class means are approximately feature class means ⊗ error class means, layer by layer; K-FAC’s single Kronecker product misaligns the Fisher’s outliers and the class-wise CFAC aligns them. For multinomial logistic regression on a Gaussian class model the expected Fisher spectrum is derived exactly. | Active — read 2026-10-03 (NOTE-474); worth reading as the general account of class/cross-class structure in deepnet spectra: features, backpropagated errors, gradients, Fisher, Hessian and weights all show one outlier from the global mean, C−1 from class means, a mini-bulk of about C(C−1) from cross-class means, and a main bulk from within-class variation. Per layer, gradient class means are approximately Kronecker products of feature class means and error class means, which is why K-FAC, pairing every class with every class, misaligns the Fisher’s outliers, and CFAC, pairing each class with itself, aligns them. Proved only for multinomial logistic regression on a Gaussian model. |
| LIT-616 | A Widely Applicable Bayesian Information Criterion | Watanabe (2012), arXiv:1208.6338; JMLR 14:867–897 (2013). A model is singular when the parameter-to-distribution map is not one-to-one or the Fisher information is not positive definite: neural networks, mixtures, reduced-rank regression, HMMs. Its Bayes free energy is nL_n(w0) + λ log n − (m − 1) log log n + O_p(1), with λ the real log canonical threshold (RLCT), not d/2. WBIC, the posterior average of nL_n(w) at inverse temperature 1/log n, has the same leading terms even for singular, unrealizable models, equals BIC for regular ones, and needs no knowledge of the true distribution. Two temperatures give an estimate of λ. In reduced-rank regression WBIC picked the true rank in 100 of 100 trials and recovered theoretical RLCTs closely. | Active — read in full 2026-10-03 (NOTE-473); worth reading as the record’s lawful, short statement of the core of singular learning theory, standing in for Watanabe’s unread book (LIT-354). It states the Bayes free energy asymptotics F = nL_n(w0) + λ log n − (m − 1) log log n + O_p(1), with λ the real log canonical threshold and m its multiplicity (Theorem 2), and defines λ through resolution of singularities (Theorem 1, Eq. 18). Its own result is WBIC, the posterior mean of nL_n(w) at inverse temperature 1/log n, which matches F to the λ log n term even for singular and unrealizable models (Theorem 4), equals BIC for regular ones (Theorem 5), and estimates λ from two temperatures (Corollary 3). The prior results are stated with citations to the book, not re-proved. |
| LIT-617 | The Full Spectrum of Deepnet Hessians at Scale: Dynamics with SGD Training and Sample Size | Papyan (2018), arXiv. Stochastic Lanczos without reorthogonalization (O(Mnp) time, O(p) memory) plus subspace-iteration deflation measures the full Hessian spectrum of VGG11 and ResNet18 on MNIST, Fashion-MNIST and CIFAR-10/100, over 16,000 trained models. It confirms a bulk with about C outliers, attributes the outliers to the Gauss–Newton term G and most of the bulk to the residual H, finds H’s spectrum symmetric with a power-law tail rather than Marchenko–Pastur, and finds H never negligible. Version 2 decomposes G into class means (outliers), cross-class variation and within-class variation (two bulks) and tracks them across epochs and sample sizes. | Active — read 2026-10-03 (NOTE-478); worth reading as the measurement that the bulk-and-C-outliers spectrum of the Hessian persists at full scale (VGG11, 28M parameters), that the outliers belong to the Gauss–Newton/Fisher term G and the bulk mostly to the residual H, and, in its second version, that G splits into class means (outliers) and two bulks (cross-class and within-class variation). Its tools are stochastic Lanczos spectral density estimation without reorthogonalization, plus subspace-iteration deflation. |
| LIT-618 | Prevalence of Neural Collapse during the terminal phase of deep learning training | Papyan, Han & Donoho (2020), PNAS 117(40). Training past zero error (the terminal phase, TPT) drives four linked phenomena in the last layer: within-class variability collapses (Tr(Σ_WΣ_B†)/C → 0), centred class means become equinorm and equiangular with cosines −1/(C−1), a simplex ETF; classifier rows become proportional to the means; and the decision becomes nearest class mean. Measured on MNIST, Fashion-MNIST, SVHN, CIFAR-10/100, STL10 and ImageNet with VGG, ResNet and DenseNet, 480 models. Test accuracy and DeepFool robustness keep improving during TPT. Theorems: NC1–NC2 imply NC3–NC4 for the Webb–Lowe and Soudry max-margin classifiers, and the simplex ETF uniquely maximizes the small-noise error exponent. | Active — read in full 2026-10-03 (NOTE-476); worth reading as the source of neural collapse: in the terminal phase of training, last-layer features collapse to their class means (NC1), the centred means form a simplex equiangular tight frame (NC2), the classifier aligns with them (NC3) and the decision becomes nearest class centre (NC4). Measured on 7 datasets × 3 architectures; NC1–NC2 imply NC3–NC4 for the Webb–Lowe and max-margin classifiers, and the simplex ETF is the unique optimal codebook in a vanishing-noise Gaussian channel. It explains Papyan’s Hessian outliers as class means outgrowing collapsing within-class variation. |
| LIT-619 | Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians | Papyan (2019), ICML. For a C-class network the Gauss–Newton term of the Hessian is G = (1/n)ΔΔᵀ, a second moment (not covariance) of logit derivatives δᵢ,c,c′ indexed by example, true class and logit. Grouping them into C² group means δc,c′ and C clusters with centres δc gives G = G₀ + G₁ + G₂ + G₃ (Eq. 28). The top-C outliers of G are those of G₁ = (C−1)Ave_c δcδcᵀ, because clusters are far apart and tight. Early in training the group means cluster by logit c′, later by true class c. Holds across MNIST, Fashion-MNIST, CIFAR-10 and ImageNet with VGG, ResNet and DenseNet; the outliers exceed their approximations slightly, as random matrix theory would predict. | Active — read in full 2026-10-03 (NOTE-479); worth reading as the paper that explains the C outliers of the deepnet Hessian: the Gauss–Newton/Fisher term G is a second moment, not a covariance, of logit derivatives indexed by (example, class, logit), and the C outliers are the eigenvalues of G₁ = (C−1)·Ave_c δ_c δ_cᵀ, the C class-cluster centres, computable from a C×C Gram matrix without high-dimensional eigenanalysis. The match is empirical, over 18,000 trained models. |
| LIT-621 | Shampoo: Preconditioned Stochastic Tensor Optimization | Gupta, Koren & Singer (2018), ICML. For a parameter tensor of order k, Shampoo keeps one preconditioner per dimension, each the gradient contracted with itself over all other dimensions, and multiplies the gradient along dimension i by (Hⁱ)^{-1/2k}. In the matrix case this is W ← W − η L^{-1/4} G R^{-1/4}, equivalent to preconditioning the flattened gradient by (L ⊗ R)^{1/4}, which upper-bounds the full-matrix AdaGrad preconditioner up to the gradient rank (Lemma 8). Regret is O(√T) in online convex optimization (Theorems 7, 10). On CIFAR-10/100 and LM1B it converges faster than SGD with momentum, Adam and AdaGrad at similar step time. | Active — read 2026-10-03 (NOTE-480); worth reading as the source of per-dimension Kronecker preconditioning: for a matrix parameter, left and right statistics L = Σ GGᵀ and R = Σ GᵀG, update W ← W − η L^{-1/4} G R^{-1/4}, i.e. a step preconditioned by (L ⊗ R)^{1/4}, which by Lemma 8 dominates the full AdaGrad matrix. Its statistic is the accumulated gradient outer product, not the Fisher; its guarantee is an O(√T) regret in online convex optimization; its deep-learning evidence is four training curves. |
| LIT-622 | Optimizing Neural Networks with Kronecker-factored Approximate Curvature | Martens & Grosse (2015), ICML. K-FAC approximates natural gradient descent in neural networks. Each Fisher block between layers i and j, E[āᵢ₋₁āⱼ₋₁ᵀ ⊗ gᵢgⱼᵀ], is replaced by E[āᵢ₋₁āⱼ₋₁ᵀ] ⊗ E[gᵢgⱼᵀ] (activations times backpropagated derivatives), whose error is a sum of third- and fourth-order cumulants. The inverse is then taken as block-diagonal or block-tridiagonal across layers. With factored Tikhonov damping, re-scaling by the exact Fisher, an adaptive momentum and a growing mini-batch, it needs orders of magnitude fewer iterations than tuned SGD with Nesterov momentum on three deep autoencoder benchmarks, and it is invariant to affine reparameterizations of each layer’s inputs and pre-activations. | Active — read in full 2026-10-03 (NOTE-486); worth reading as the source of the Kronecker-factored Fisher: each layer’s Fisher block E[āāᵀ ⊗ ggᵀ] is approximated by E[āāᵀ] ⊗ E[ggᵀ], so it inverts as two small matrices, and the inverse is further taken as block-diagonal or block-tridiagonal across layers. The factorization is justified by small higher-order cumulants, checked on one small tanh network; the speedups are on three deep autoencoders. |
| LIT-623 v2 | Bayesian Interpolation | MacKay (1992), Neural Computation 4(3):415–447. Two levels of inference: fit each model, then rank models by their evidence P(D|H). Under a Gaussian approximation the evidence is the best-fit likelihood times an Occam factor, the ratio of posterior to prior accessible parameter volume, so complex or finely tuned models penalise themselves (Eqs. 5–6). For linear interpolation with quadratic regularisers the evidence sets the regularising constant and the noise level through γ, the number of well-determined parameters (2αE_W = γ, 2βE_D = N − γ). It ranks polynomial, radial-basis, spline, Hermite and neural-network models on two data sets. In expectation, no fixed model’s log evidence beats the true model’s. | Active — read in full 2026-10-03 (NOTE-471); worth reading as the source of the evidence framework and the Occam factor in the form the field uses: evidence ≈ best-fit likelihood × Occam factor, the Occam factor being the ratio of posterior to prior accessible parameter volume, (2π)^{k/2} det^{-1/2}A · P(w_MP|H) under a Gaussian approximation (Eqs. 5–6). It also gives the “number of good parameter measurements” γ = Σ λ_a/(λ_a + α), which sets the regularising constant (2αE_W = γ) and the noise level (2βE_D = N − γ) at the evidence maximum. The framework is Gull and Skilling’s, as MacKay says, and every demonstration is one-dimensional interpolation. Flagged anthology-candidate for its regularisation and early-stopping arguments. |
| LIT-624 | Towards Understanding the Spectral Bias of Deep Learning | Cao, Fang, Wu, Zhou & Gu (2019). For a two-layer ReLU network trained by gradient descent on squared loss, the training residual projected onto the eigenfunctions of the NTK integral operator with the r_k largest eigenvalues decays like (1 − λ_(r_k))^T (Theorem 4.2). This holds for any labelling, provided the sample size beats the eigengap and the width is polynomial in λ_(r_k)^(−1). On uniform data on S^d the eigenfunctions are spherical harmonics, with µ_k = Ω(k^(−d−1)) for k ≫ d and Ω(d^(−k+1)) for d ≫ k on even degrees, and µ_k = 0 on odd k ≥ 3 (Theorem 4.3). Lower degrees are learned first, faster, and with fewer samples and narrower networks. | Active — read 2026-10-03 (NOTE-483): main text and Appendix A in full, the proofs in Appendix B followed at the level of their lemmas. Worth reading as the theorem that makes “spectral bias” a statement about NTK eigendirections: for a two-layer ReLU network trained by gradient descent, the residual projected onto the top r_k eigenfunctions of the NTK integral operator shrinks like (1 − λ_(r_k))^T (Theorem 4.2), for any target, with sample size governed by the eigengap λ_(r_k) − λ_(r_k+1). On uniform spherical data the eigenfunctions are spherical harmonics, the odd degrees above 1 have eigenvalue zero, and µ_k = Ω(max(k^(−d−1), d^(−k+1))) for even k (Theorem 4.3). |
| LIT-642 | Psychologism and Behaviorism | Block (1981), The Philosophical Review 90(1):5–43. Psychologism, the doctrine that whether behaviour is intelligent depends on the information processing that produces it, is true. A machine that stores all sensible conversations of a test’s length and looks up its replies has the capacity to respond sensibly to any sequence of inputs, so it satisfies the strongest behaviourist conception of intelligence, yet all its intelligence is its programmers’. The standard objections to behaviourism (Chisholm–Geach, the perfect actor, paralytics) defeat behaviourist necessary conditions but not sufficient ones; this machine defeats the sufficient condition. Psychologism is not chauvinism: it excludes one kind of processing without requiring ours. | Active — read in full 2026-10-03 (NOTE-506); worth reading as the source of the “Blockhead”: a machine that stores every sensible hour-long conversation and so has the capacity to pass any Turing test of that length, yet “has the intelligence of a toaster”. The argument refutes behaviourist sufficient conditions for intelligence, including a capacity version that survives the standard objections, and establishes psychologism: whether behaviour is intelligent depends partly on the processing that produces it. It does not say which processing is required, only that it needs some “richness”, and the closing case against simulators of physics, neurons or psychology is admitted to be a sketch. |
| LIT-651 | Linear Mode Connectivity in Multitask and Continual Learning | Mirzadeh, Farajtabar, Gorur, Pascanu & Ghasemzadeh (2020). When a multitask solution and a continual solution start from the same point, the multitask solution lies on a low-loss line to every earlier continual solution, while consecutive continual solutions are separated by rising loss. Euclidean distance and CKA do not detect the difference. The multitask direction lies outside the top Hessian eigenvectors of the earlier task. MC-SGD trains towards a point linearly connected to both the previous solution and the current task’s solution, using a small replay buffer, and leads five baselines on three 20-task benchmarks. | Active — read in full 2026-10-03 (NOTE-520); worth reading as the first study of where continual and multitask solutions lie relative to each other. Started from the same point (the previous task’s solution), the multitask solution is linearly connected with low loss to every earlier continual solution, while consecutive continual solutions are not. The authors explain it by curvature: the multitask path moves in directions the earlier task’s Hessian sees as flat, within a region where a second-order expansion holds. MC-SGD turns this into a regulariser and beats EWC, A-GEM, ER-Reservoir and Stable SGD on 20-task Permuted MNIST, Rotated MNIST and Split CIFAR-100. The connectivity breaks with a different start, input noise, label corruption or removed classes. |
| LIT-652 | The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks | Entezari, Sedghi, Saukh & Neyshabur (2022), ICLR 2022. Conjecture: most SGD solutions lie in a set whose members can be permuted so that no barrier remains on the line between any two. Barriers between networks from different initializations first rise then fall with width, rise with depth, and saturate high for VGGs and ResNets. A proof covers a wide one-hidden-layer net at uniform random initialization. The evidence is a model: random permutations of one SGD solution show the same barriers as independent solutions across width, depth and dataset. A simulated annealing search lowers barriers only for shallow networks. | Active — read in full 2026-10-03 (NOTE-537); the source of the conjecture that SGD solutions, once their hidden units are suitably permuted, have no barrier on the straight line between them: one basin modulo permutation. Its proof covers only a one-hidden-layer network at uniform random initialization, and its main evidence is indirect: barriers between independently trained networks behave like barriers between random permutations of a single trained network, across width, depth and dataset, in more than 3,000 networks. Its own permutation search lowers barriers only for shallow nets on easy data and fails for VGGs and ResNets. Second reading under ADR-013; the anthology holds it as ANTH-LIT-251, and that entry misstates the theorem. |
| LIT-653 | Essentially No Barriers in Neural Network Energy Landscape | Draxler, Veschgini, Salmhofer & Hamprecht (2018), ICML 2018. Using AutoNEB, a minimum-energy-path method from molecular statistical mechanics, the paper connects ten minima per architecture for CNNs, ResNets and DenseNets on CIFAR-10 and CIFAR-100. The saddle losses on the best paths are almost those of the minima for deep architectures, and test error rises by at most a few tenths of a percent on CIFAR-10. The conclusion is that minima are points on one connected low-loss manifold, not bottoms of separate valleys, and that the effect grows with width and especially depth. | Active — read in full 2026-10-03 (NOTE-531); the other founding demonstration of mode connectivity, reached independently of Garipov et al. by borrowing the nudged elastic band from chemical physics. Over ten minima per architecture, the highest training loss on the best paths found is almost the minima’s own for deep ResNets and DenseNets on CIFAR-10, with a small residual gap on CIFAR-100, and barriers fall as networks get wider and especially deeper. Its two explanations, resilience and redundancy, are qualitative; the redundancy toy (swapping two hidden units needs a third) is the first statement in this batch of why a permutation costs a barrier without spare capacity. |
| LIT-654 | Linear Mode Connectivity and the Lottery Ticket Hypothesis | Frankle, Dziugaite, Roy & Carbin (2020), ICML 2020. Instability analysis trains two copies of a network from the same state under different SGD noise and measures the error barrier on the straight line between the results. Standard vision networks are unstable at initialization but become stable early in training, after which the outcome is fixed to one linearly connected minimum. Sparse subnetworks found by iterative magnitude pruning reach full accuracy only when they are stable, which explains where lottery tickets were and were not found, and rewinding to an early iterate rather than to initialization finds matching subnetworks at ImageNet scale. | Active — read in full 2026-10-03 (NOTE-514); the source of linear mode connectivity as a measurement and of instability analysis. Two copies of a network trained from the same state under different SGD noise end up linearly connected once that state is a little way into training (3% for ResNet-20, 20% for ResNet-50 on ImageNet), and iterative-magnitude-pruning subnetworks are matching only when they are stable in this sense. The evidence is correlational: stability and matching coincide across the settings tested, and nothing shows that one causes the other. |
| LIT-655 | Understanding Mode Connectivity via Parameter Space Symmetry | Zhao, Dehmamy, Walters & Yu (2025), ICML 2025. Relates the topology of a loss’s symmetry group to that of its minimum. The minimum of a full-rank l-layer linear regression is homeomorphic to GL_h(ℝ)^{l−1}, so it has 2^{l−1} components; a skip connection reduces 4 to 3 in a 1-D case, and permutations connect all components when h ≥ 2. Rescaling symmetry yields minima in one component with arbitrarily large barriers on the straight line. Group actions give explicit constant-loss curves, and a curvature bound κ on such a curve bounds the chord’s distance from the minimum by about κ‖w₂ − w₁‖²/8. | Active — read 2026-10-03 (NOTE-528); worth reading as the account of mode connectivity from continuous parameter symmetries. The minimum of a full-rank l-layer linear network is homeomorphic to GL_h(ℝ)^{l−1} and so has 2^{l−1} connected components, a skip connection merges some of them, and a permutation with negative determinant joins the rest. Rescaling symmetry gives minima in the same component with an arbitrarily large loss barrier on the straight line between them, even after a restricted permutation. A group element g gives an explicit constant-loss curve exp(t log g)·w, and bounded curvature of such curves bounds the barrier on the chord. The theory is for linear networks and homogeneous last layers; the experiments are small synthetic regressions. |
| LIT-656 | Loss Surface Simplexes for Mode Connecting Volumes and Fast Ensembling | Benton, Maddox, Lotfi & Wilson (2021), ICML 2021. SPRO trains the connecting vertices of simplexes so that loss is low at points sampled uniformly inside them, with a log-volume bonus. It finds simplicial complexes of low loss joining up to seven independently trained networks through shared vertices, and a low-loss region around VGG-16 on CIFAR-10 of at least 10 dimensions. Started from a single trained network, a few vertices cost about a tenth of a training run each, and ensembling inside the simplexes (ESPRO) beats deep ensembles in accuracy, calibration and robustness to corruption. | Active — read in full 2026-10-03 (NOTE-539); mode connectivity raised from curves to volumes. Training the vertices of simplexes so that loss stays low at random points inside them finds simplicial complexes of low loss joining several independently trained networks at once, and a low-loss region around a VGG-16 on CIFAR-10 of at least 10 dimensions. The same trick, run from one trained network, gives cheap ensembles (ESPRO) that beat deep ensembles at equal size and budget. The dimension estimate is a lower bound from one architecture and dataset, and the paper’s picture of one connected volume containing all SGD solutions is drawn from complexes of up to 7 modes. |
| LIT-657 | The Generalization Ridge: Information Flow in Natural Language Generation | Chang, Deng & Chen (2025), COLM 2026. InfoRidge estimates, layer by layer and through fine-tuning, the matrix-based mutual information between a transformer’s last-token hidden state and the embedding of the target token, and between each block’s residual update and that target. In GPT-2 Small and Medium, Qwen-2.5 0.5B and Llama 3.1 8B on CLUTRR, ECQA and a synthetic arithmetic task, it rises, peaks in the upper-middle layers and falls. On the synthetic task the peak coincides with the best OOD early-exit accuracy while ID accuracy keeps rising. Middle blocks add the most information, attention to task tokens peaks there, and residual scales fitted on OOD data downweight the last layers. The profile persists over 50 decoding steps on CNN/DailyMail. | Active — read 2026-10-03 (NOTE-536); worth reading as a layer-wise information profile of fine-tuned language models: a matrix-based Rényi estimate of I(Zℓ; Y), between the last-token hidden state at layer ℓ and the input embedding of the true next token, rises through the early layers, peaks in the upper-middle layers and falls in the last few, the “generalization ridge”. On a synthetic arithmetic task with GPT-2 Small the peak (0.27 at layer 10 of 12) is where OOD early-exit accuracy is best (56.9%) while ID accuracy keeps rising to 93.2% at layer 12. Learned residual scales put less weight on the final layers when fitted on OOD data. The estimator is a kernel-alignment quantity, not Shannon mutual information: at layer 12 it falls to 0.019, near its layer-6 value, where the same layer decodes the target with 93% ID accuracy. |
| LIT-658 | Knowledge is a Region in Weight Space for Fine-tuned Language Models | Gueta, Venezian, Raffel, Slonim, Katz & Choshen (2023). RoBERTa-base models fine-tuned on similar data end up close in weight space: task vectors cluster by dataset (98% clustering accuracy over 280 models) and by task family (90%), and the grouping follows data type, not data size. Points between the fine-tuned models, along lines and across their convex hull, perform as well or better than the fine-tuned models themselves, and linear extrapolation beyond them quickly fails. The region’s centroid, re-fine-tuned with BitFit, beats starting from the pretrained model on 9 of 12 target datasets. | Active — read in full 2026-10-03 (NOTE-515); worth reading as the paper that turns linear connectivity between fine-tuned models into a claim about regions. RoBERTa-base models fine-tuned on the same dataset, the same task, or any of 12 GLUE/SuperGLUE datasets cluster in weight space by cosine of their task vectors, and the convex hull of each group performs as well as or better than the fine-tuned members: sampled hull points beat the members in 88% of comparisons on MNLI. Linear extrapolation past the members fails quickly, so the fine-tuned models sit near the edge of a small basin. Averaging the region (its centroid) is a better start for further fine-tuning than the pretrained model on 9 of 12 datasets. One pretrained model, classification only, English only. |
| LIT-659 | Do Deep Neural Network Solutions Form a Star Domain? | Sonthalia, Rubinstein, Abbasnejad & Oh (2024), preprint. Conjectures that SGD solution sets, modulo permutation, are star domains, which needs less width than convexity does. Starlight trains a star model by sampling points on the lines to permuted source models. The star model has much lower loss barriers to held-out solutions than those solutions have to each other, across ResNet-18, VGG, DenseNet, CIFAR and ImageNet-1k (e.g. 0.078 against 0.383 on CIFAR-10). It gains with more source models, ranks uncertainty better in BMA than a deep ensemble, and is about 1 point more accurate than a single model. | Active — read 2026-10-03 (NOTE-533); worth reading as the star domain conjecture: where SGD solutions are not convex modulo permutation, they still form a star domain, with a star model linearly connected, after weight matching, to every other solution. Starlight trains a candidate by sampling points on the lines to 50 permuted source models, and the candidate has lower barriers to held-out models than those models have to each other (ResNet-18 CIFAR-10 0.078 against 0.383; ImageNet-1k 2.794 against 5.948). Those barriers are lower, not zero, and the authors say the conjecture is unproved and the barriers are often significantly above zero. Preprint. |
| LIT-660 | Mode Connectivity Beyond Classifiers: Evidence from Generative and Contrastive Models | Yao, Zhang & Tian (2026), preprint. Extends layer-wise path construction (LLPF, Tian et al. 2026) to a DDPM on Flowers102 and a NanoCLIP model on Flickr30k. Layers move towards the target mode in a dataflow order (symmetric encoder–decoder pairs for the U-Net, alternating vision and text blocks for CLIP); each layer is rescaled to a fixed variance; the point is retrained with AdamW until its loss is below a threshold. The resulting chains keep training loss below 0.04 (DDPM) and 0.09 (NanoCLIP), where AutoNEB’s maximum NanoCLIP loss is 6.69. Intermediate DDPMs have worse FID than the endpoints at equal training loss. | Proposed — read in full 2026-10-03 (NOTE-534); watching, as the first claim of mode connectivity between independently trained generative and contrastive models: one pair of DDPMs on Flowers102 and one pair of NanoCLIP models on Flickr30k, joined by a chain of checkpoints along which the training loss stays below a threshold, where AutoNEB’s paths rise steeply on NanoCLIP. The path is built by moving layer groups towards the far end, rescaling each layer’s variance, and retraining with AdamW until the loss is under threshold, so every anchor is a retrained model. What would settle it: more than one pair per model, a barrier measure along the whole path rather than at the anchors, and a reason to think a path made of retrained anchors says more about the landscape than that retraining works. It is a preprint of 31 August 2026. |
| LIT-661 | Git Re-Basin: Merging Models modulo Permutation Symmetries | Ainsworth, Hayase & Srinivasa (2023), ICLR 2023. Three ways to permute one network’s hidden units into alignment with another: activation matching (a linear assignment per layer), weight matching (an NP-hard sum of bilinear assignments, solved by coordinate descent over layers), and a straight-through estimator. After alignment the straight line between independently trained networks has zero loss barrier for MLPs on MNIST and wide ResNet-20s on CIFAR-10. Connectivity grows with width and with training time, fails for thin models, and a counterexample shows it is a property of SGD solutions. Merging models trained on disjoint data improves test loss and calibration but not accuracy. | Active — read in full 2026-10-03 (NOTE-532); the strongest evidence for one basin modulo permutation. A weight-matching algorithm that needs no data and runs in seconds aligns independently trained networks so that the straight line between them has no loss barrier, for MLPs on MNIST and for ResNet-20 on CIFAR-10 at large width, the first such result for ResNets. It also finds where the picture stops: thin models are not linearly connected after any permutation it finds, connectivity appears only after some training, ImageNet ResNet-50 keeps a barrier, and a constructed counterexample shows that the property belongs to SGD solutions, not to architectures. Second reading under ADR-013; the anthology holds it as ANTH-LIT-333, and that entry misdescribes the algorithm and overstates the results. |
| LIT-662 | Geodesic Mode Connectivity | Tan, Long, Zhao & Laine (2023), ICLR 2023 Tiny Paper. Treats a network as a distribution p(x, ŷ; θ) on a Fisher–Rao manifold and hypothesises that geodesics between SGD solutions are mode-connecting paths. A chain of 25 models between two ResNet-20s (4× width, LayerNorm), initialised on the straight line after Git Re-Basin weight matching, is optimised to minimise summed Jensen–Shannon divergence between neighbours. The curve is longer in Euclidean terms but shorter in distribution space, and keeps train and test loss low where linear interpolation shows a large barrier. | Active — read in full 2026-10-03 (NOTE-521); worth reading as the short proposal that mode-connecting paths are geodesics of the Fisher–Rao metric on the space of the networks’ output distributions. It approximates a geodesic between two trained ResNet-20s (4× width, LayerNorm), after Git Re-Basin weight matching, by optimising 25 models on a chain to minimise the summed Jensen–Shannon divergence between neighbours, using unlabelled training images only, and the resulting curve has low train and test loss where the straight line does not. One architecture, one pair, one figure; the hypothesis that all geodesics between SGD solutions are mode-connecting is stated, not tested. |
| LIT-663 | Taxonomizing local versus global structure in neural network loss landscapes | Yang, Hodgkinson, Theisen, Zou, Gonzalez, Ramchandran & Mahoney (2021), NeurIPS. Thousands of trained networks are placed on a load-like by temperature-like grid and measured three ways: Hessian top eigenvalue and trace (local sharpness), Bezier-curve mode connectivity between two independently trained models (global connectivity), and CKA similarity of their outputs on mixup points. Two transitions give four phases: I sharp and poorly connected, II sharp and well connected, III flat and poorly connected, IV flat and well connected, with IV split by CKA into IV-A and IV-B. Test accuracy is best in IV-B. Sharpness alone mispredicts accuracy across phases, and in phase III training to zero loss does worse than stopping in the sharp phases. Double descent appears as a dark band at the phase boundary under label noise. | Active — read 2026-10-03 (NOTE-516); worth reading as the empirical taxonomy of loss landscapes by local and global structure: Hessian top eigenvalue and trace separate locally sharp from locally flat, Bezier-curve mode connectivity between independently trained models separates globally poorly-connected from well-connected, and CKA similarity of their outputs splits the well-connected, flat phase in two. Across a load-like by temperature-like grid (width, data amount or label noise against batch size, learning rate or weight decay), the best test accuracy sits in the flat, well-connected, high-similarity phase IV-B; the Hessian alone mispredicts, and where connectivity is poor, training to zero loss can lower test accuracy. Empirical only, mostly ResNet18 on CIFAR-10. |
| LIT-664 | Generalized Linear Mode Connectivity for Transformers | Theus, Cabodi, Anagnostidis, Orvieto, Singh & Boeva (2025), NeurIPS 2025. A hierarchy of function-preserving maps (permutation ⊂ semi-permutation ⊂ orthogonal ⊂ invertible) fits transformer components: permutations for feed-forward units, semi-permutations for attention heads, an orthogonal map on the RMSNorm residual stream. Aligning with them gives, for the first time, low and zero test-loss barriers between independently trained ViTs and GPT-2s (learned matching 0.00 on CIFAR-10/100 and Tiny ImageNet, 0.02 Tiny Shakespeare, 0.42 BookCorpus), for three-way merges and across embedding widths. Permutation-only alignment does not get there. | Active — read 2026-10-03 (NOTE-540); worth reading as the result that independently trained transformers become linearly mode connected once alignment uses their real symmetries: permutations of feed-forward units, semi-permutations of attention heads, and an orthogonal map on the residual stream after RMSNorm reparameterisation. Data-free weight matching cuts test-loss barriers from 1.69–4.34 to 0.34–1.56, and learned matching, which optimises the alignment against task loss at the midpoint, reaches 0.00 on three vision datasets and 0.02 and 0.42 on two language ones. Most of the gap is in the orthogonal map. Small ViTs and GPT-2s only. |
| LIT-665 | ZipIt! Merging Models from Different Tasks without Training | Stoica, Bolya, Bjorner, Ramesh, Hearn & Hoffman (2023), ICLR 2024. Merging differently initialised models trained on disjoint tasks fails under permutation, because the permuted model sits in a basin for its own task and not the other’s. ZipIt! instead merges correlated features within and across models through merge and unmerge matrices, and can stop merging part-way, giving a shared trunk with task heads. The abstract reports a 20–60% improvement over prior work on CIFAR and ImageNet splits; it comes near the ensemble with partial zipping, and proves a tighter LMC barrier bound than Entezari et al. when units are redundant. | Active — read in full 2026-10-03 (NOTE-523), main text and Appendices A–F, the proof in Appendix G followed to its reduction to Entezari et al.; worth reading as the case that permutation is the wrong symmetry for merging models trained on different tasks. ZipIt! matches features by activation correlation within each model as well as across the two, and can stop merging at a chosen layer, leaving a multi-head model. On disjoint halves of CIFAR-10 it reaches 79.1% joint accuracy where Git Re-Basin reaches 46.2%, and partial zipping comes within a few points of the ensemble. Its theorem tightens Entezari et al.’s barrier bound when a model has redundant units. With full merging on ImageNet-1k it is no better than a permutation baseline (8.6%), so the gains there come from leaving later layers unmerged. |
| LIT-666 | Mode Connectivity and Data Heterogeneity of Federated Learning | Zhou, Zhang & Tsang (2023), preprint. Under Dirichlet label skew and PACS domain shift, low-loss overlaps between FedAvg client models and the global model shrink as heterogeneity grows; client models stay close to the global one yet move in different directions within a round. Global models from one initialisation under different heterogeneity have function dissimilarity above 0.2, show an error barrier on the line, and are joined without barrier by a polygonal chain. A mean-field analysis of two-layer networks bounds dropout and connectivity errors by terms proportional to √h(α) and decreasing in width N. | Active — read in full 2026-10-03 (NOTE-517); worth reading as the only study in the batch of connectivity under federated averaging. Client models, though close in weight space, leave the low-loss region of the global model more as client data grow more heterogeneous, and FedAvg global models trained from one initialisation under different heterogeneity are separated on the straight line (accuracy drops of about 10%) but joined by a one-bend polygonal chain. A mean-field bound, extending the dropout-stability route of Kuditipudi et al. and Shevchenko and Mondelli, makes the connectivity error grow with √h(α), a heterogeneity factor, and shrink with width; width and heterogeneity trends in MNIST and CIFAR-10 runs agree. The theory treats heterogeneity as isotropic Gaussian noise on each neuron’s update, an assumption the paper checks only by plotting noise norms. |
| LIT-667 | Does Data Scaling Lead to Visual Compositional Generalization? | Uselis, Dittadi & Oh (2025), ICML 2025. Two labelled concepts with n values each form an n × n grid; training shows k combinations per value and testing uses the rest. On dSprites, 3DShapes, PUG, colored MNIST and a new FSprites, ResNet-50s trained from scratch reach near 100% on seen pairs and fall sharply on unseen ones; 4× more in-distribution data does not close the gap, while more values and more combinations do. Features pass from spurious, to discriminative but entangled, to linearly factored and near-orthogonal as combinatorial coverage grows. Exact linear factorization makes k = 2 sufficient for perfect generalization. Frozen DINO, DINOv2 and CLIP features are above chance but far from that ideal. | Active — read 2026-10-03 (NOTE-525); worth reading as a controlled test of whether more data buys visual compositional generalization: in an (n, k) grid where each of n values of two concepts is seen in only k combinations, ResNet-50s trained from scratch fail on unseen pairs whatever the data volume (4× more in-distribution data leaves 60–80% drops), and succeed only as the fraction of combinations seen rises, when their features become linearly factored (a pair’s embedding is the sum of per-concept vectors; R² > 0.8). Under exact linear factorization, k = 2 combinations per value suffice for a linear classifier to get all unseen pairs right (Proposition 4.1). DINO, DINOv2 and CLIP features are partly factored. Model selection is oracle, on the test set, by design. |
| LIT-668 | Bridging Mode Connectivity in Loss Landscapes and Adversarial Robustness | Zhao, Chen, Das, Ramamurthy & Lin (2020), ICLR 2020. A Garipov-style quadratic Bezier path between two tampered models, trained on a small clean set, contains models that keep clean accuracy and shed backdoor or injected-error attacks, outperforming fine-tuning, retraining, pruning and weight noise on the clean–attack trade-off in most settings. On a standard-loss path between a regular and an adversarially trained model, standard loss stays flat but robustness loss rises in the middle, and it correlates with the largest input-Hessian eigenvalue (Pearson 0.71–0.88). | Active — read 2026-10-03 (NOTE-530); worth reading as the first use of mode connectivity for adversarial robustness. A quadratic Bezier path trained between two backdoored or error-injected models on as few as 50–2,500 clean samples contains models near its ends that keep most clean accuracy and lose the attack (CIFAR-10 VGG with 2,500 samples: 88% clean, 1.1% backdoor success at t = 0.1). Between a regular and an adversarially trained model, a path trained with the standard loss has no standard-loss barrier but a robustness-loss barrier, which tracks the largest eigenvalue of the input Hessian. The proposition offered for that correlation rests on an assumption that, taken literally, would rule out the barrier it explains. |
| LIT-669 | Mechanistic Mode Connectivity | Lubana, Bigelow, Dick, Krueger & Tanaka (2022), ICML 2023. Two models are mechanistically similar if they are invariant to the same interventions on the data-generating process. Models that use a spurious cue and models that ignore it are mode connected along quadratic paths but not along linear ones, even after permutation matching, and the paper conjectures that a linear barrier implies mechanistic dissimilarity. For a one-hidden-layer ReLU network it proves only the converse: mechanisms of different complexity cannot be linearly connected. Naive fine-tuning on clean data stays linearly connected to the pretrained model and keeps its reliance on the cue; a fine-tune that forces a barrier removes it. | Active — read 2026-10-03 (NOTE-538); worth reading as the paper that ties a loss barrier to a difference in what two models compute. It defines two models as mechanistically similar when they are invariant to the same interventions on the data-generating process, finds that models relying on a spurious cue and models ignoring it are connected by quadratic paths but not by linear ones, even after permutation, and conjectures that a linear barrier always signals mechanistic dissimilarity. The conjecture is proved only for a one-hidden-layer ReLU network on a synthetic process; the evidence is synthetic cues on CIFAR-10, CIFAR-100 and Dominoes. |
| LIT-670 | On Linear Mode Connectivity of Mixture-of-Experts Architectures | Tran, Trinh, Bui & Nguyen (2025), NeurIPS 2025. The group generated by permuting experts with their gates and translating all gating weights by a common vector preserves an MoE’s function. Under genericity conditions it is the whole symmetry of the dense gate and of the top-k gate for k ≥ 2 (Theorems 4.1–4.2). Translation does not change barriers, so only the permutation matters. Matching expert order by centred gate weights or by Gram matrices, then weight-matching inside each expert, brings barriers between MoE layers fine-tuned from different seeds near zero, across dense, top-2 and DeepSeekMoE variants, on ViT and GPT-2 backbones that are shared and frozen. | Active — read 2026-10-03 (NOTE-513); worth reading as the characterisation of the symmetries of a mixture-of-experts gate. Under genericity conditions, two dense or top-k (k ≥ 2) MoEs compute the same function only if they differ by a permutation of experts together with their gates and a common translation of the gating weights (Theorems 4.1–4.2), and the translation leaves every barrier unchanged. Expert-order matching followed by per-expert weight matching gives near-zero barriers between MoE layers fine-tuned from different seeds. But the backbone around the MoE layer is shared and frozen, so what is connected is one inserted layer, not two independently trained networks. |
| LIT-671 | Explaining Landscape Connectivity of Low-cost Solutions for Multilayer Nets | Kuditipudi, Wang, Lee, Zhang, Li, Hu, Arora & Ge (2019), NeurIPS 2019. Mode connectivity follows from generic robustness properties of trained nets. Two ε-dropout-stable ReLU networks, each able to zero half its units per layer at loss cost ε, are connected by a piecewise-linear path of loss at most max endpoint + ε; ε-noise-stable networks are connected by 10 segments with barrier Õ(ε). Conversely, for any width there is a dataset on which global minima of a two-layer student are disconnected, so connectivity is a property of the solutions training finds. Checked on MNIST convnets and VGG-11 on CIFAR-10. | Active — read in full 2026-10-03 (NOTE-544); the first explanation of mode connectivity that applies to realistic multilayer nets. If two solutions can each lose half their units per layer at a cost of at most ε in loss (dropout stability), they are joined by a piecewise-linear path whose loss never exceeds the worse endpoint by more than ε; a stronger noise-stability condition gives 10 segments. A counterexample shows that connectivity fails for some global minima of arbitrarily overparametrized two-layer nets, so the property belongs to the solutions training finds, not to the landscape. The conditions are checked on small MNIST convnets and one VGG-11 trained with channel-wise dropout. |
| LIT-672 | Rethinking generalization requires revisiting old ideas: statistical mechanics approaches and complex learning behavior | Martin & Mahoney (2017), arXiv preprint. Zhang et al.’s observations, that deep networks fit noisy labels and that only early stopping regularizes them, are read through the statistical mechanics of learning. A Very Simple Deep Learning model has two control parameters: a load α = m_eff/N, lowered by randomizing labels, and a temperature τ, raised by stopping early. In the thermodynamic limit (m, N → ∞, α fixed), generalization error can drop discontinuously at a critical αc, and a two-dimensional (α, τ) phase diagram has a spin-glass-like overtraining phase. The mechanism is entropy–energy competition, shown on the continuous and Ising perceptrons: the Ising entropy vanishes near zero error, so the first-order condition loses its interior solution. | Active — read in full 2026-10-03 (NOTE-542); worth reading as the position paper that brought the statistical-mechanics theory of generalization back to deep learning: a network is idealized as a black box governed by a load-like parameter α (effective data per capacity, lowered by label noise) and a temperature-like parameter τ (raised by early stopping), and in the thermodynamic limit m, N → ∞ with α = m/N fixed, generalization has phases with sharp transitions. The mechanism is shown on the continuous versus Ising perceptron: where the entropy of near-optimal hypotheses vanishes (s(ε) ~ −(π²/2)ε² ln ε), the error drops discontinuously at αc, which PAC/VC bounds do not even qualitatively describe. A qualitative argument with no new theorem and no experiment. |
| LIT-673 | Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs | Garipov, Izmailov, Podoprikhin, Vetrov & Wilson (2018), NeurIPS 2018. Optima of deep networks trained from different initializations are joined by simple learned curves (a polygonal chain with one bend, or a quadratic Bezier curve) along which train loss and test error stay nearly constant, although the straight line between them rises to near-chance error. The curve is found by minimizing the loss averaged over a uniform t on the curve. Points along it are functionally diverse, which motivates Fast Geometric Ensembling: short cyclical learning-rate cycles late in training, collecting a checkpoint per cycle, beating Snapshot Ensembles on CIFAR-10, CIFAR-100 and ImageNet at equal budget. | Active — read in full 2026-10-03 (NOTE-527); one of the two founding demonstrations of mode connectivity. Two independently trained networks are joined by a learned one-bend polygonal chain or quadratic Bezier curve along which train loss and test error stay near the endpoints, while the straight segment between them climbs to near chance (VGG-16 on CIFAR-10: 90% test error at the midpoint). Points a short way along a curve already make useful ensemble partners, which motivates Fast Geometric Ensembling. The connectivity evidence is a table of single runs per architecture with no error bars, and the paper does not claim the curves are anything more than found. |
| LIT-674 | Going Beyond Linear Mode Connectivity: The Layerwise Linear Feature Connectivity | Zhou, Yang, Yang, Yan & Hu (2023), NeurIPS 2023. When two trained networks are linearly mode connected, whether by spawning from a shared early checkpoint or by permutation alignment, the features of nearly every layer of the weight-interpolated network are proportional to the interpolation of the two networks’ features (LLFC). LLFC on the output layer implies LMC. Weak additivity of ReLU and a commutativity condition between layers’ weights and features together imply LLFC, and Git Re-Basin’s two matching objectives are each one factor of the commutativity residual. | Active — read 2026-10-03 (NOTE-512); worth reading as the finding that linear mode connectivity reaches inside the network: whenever two trained networks are linearly connected, by spawning or by permutation, the feature map of almost every layer of the interpolated network points in the direction of the interpolated feature maps (LLFC). Two conditions, weak additivity of ReLU on the two networks’ pre-activations and a commutativity of each layer’s weights with the other network’s features, imply exact LLFC (Theorem 1), and Git Re-Basin’s weight and activation matching each minimise one factor of the commutativity residual. Image classification only. |
| LIT-675 | TIES-Merging: Resolving Interference When Merging Models | Yadav, Tam, Choshen, Raffel & Bansal (2023), NeurIPS 2023. Merging fine-tuned models by averaging or adding task vectors suffers from two kinds of interference: redundant small updates dilute influential ones, and influential updates of different tasks disagree in sign. TIES-Merging trims each task vector to its top-k% magnitudes, elects a sign per parameter by total mass, and averages only values agreeing with it. It beats averaging, task arithmetic, Fisher merging and RegMean across (IA)³, T5 and ViT settings when a validation set tunes it, degrades more slowly as tasks are added, and with oracle signs nearly matches multitask training. | Active — read in full 2026-10-03 (NOTE-526); worth reading as the account of why averaging task vectors loses information: most of a fine-tuning update is small and redundant, and the influential coordinates of different tasks often disagree in sign, so a plain mean shrinks them. TIES-Merging keeps the top 20% of each task vector by magnitude, elects a sign per coordinate by total mass, and averages only the agreeing values. With a validation set it beats task arithmetic, Fisher merging and RegMean on all five settings (by 0.7 to 3.6 points); without one it falls below task arithmetic on T5-base (69.7 against 73.2 in Table 1), which the introduction’s summary of results leaves out. With the multitask model’s signs it reaches 72.0 against 73.1 for multitask training, which is the paper’s strongest evidence that sign is the bottleneck. |
| LIT-676 | Unveiling Mode Connectivity in Graph Neural Networks | Li, Chen, Han, Zeng, Liu & Tang (2025), preprint. GCN solutions for node classification from different seeds are connected by quadratic Bezier curves but often separated on the straight line, more so on co-authorship and heterophilous graphs than on citation graphs. GNN variants (GCN, GraphSAGE, GAT) give similar barriers, MLPs different ones. On CSBM graphs, barriers fall with density and feature separability and are highest at middling homophily. The authors bound the barrier through the spectral gap, bound the generalisation gap by the barrier, and propose a Wasserstein distance between graphs’ interpolation-loss curves that correlates with transfer gaps. | Proposed — read in full 2026-10-03 (NOTE-529); watching, as the first study of mode connectivity in graph neural networks. Two GCNs trained on the same graph with different seeds are usually joined by a low-loss quadratic Bezier curve but often not by a straight line, and the barrier depends on the graph more than on the GNN variant: it falls with density, with feature separability, and with homophily far from one half, in synthetic CSBM graphs and, by the authors’ reading, in 12 real ones. A loss-barrier bound, a generalisation bound in terms of the barrier, and a barrier-based distance between graphs are stated, but the appendix that should hold the proofs does not; the v1 text still carries an editor’s placeholder. What would settle it: the proofs, and numbers for the barriers the figures show. |
| LIT-677 | Unveiling mode-connectivity of the ELBO landscape | Zhang & Blei (2021), NeurIPS 2021 Bayesian Deep Learning workshop. Two SVI runs of LDA on the New York Times corpus give distinct local maxima of the ELBO; the simplified string method, with natural-gradient bead updates, is said to find an essentially flat maximum-energy path between them. Paths are said to flatten as data and the number of topics grow. The authors explain connectivity by over-parameterisation (more topics than the corpus holds) and by continuous statistical degeneracy, offering a landscape account of topic modelling’s “no best answer”. No results are reported in the paper. | Proposed — read in full 2026-10-03 (NOTE-522); watching, as the only work in the record to carry mode connectivity from neural networks to variational inference. It claims that local maxima of the LDA ELBO, from two SVI runs on the New York Times corpus, are joined by essentially flat maximum-energy paths found with the simplified string method, that paths flatten as the number of topics grows, and that statistical degeneracy explains this. The three-page workshop paper reports none of the results it describes: no path, no ELBO values, no synthetic-data experiment; its one figure is a schematic. What would settle it: the experiments themselves, which are not in this version. |
| LIT-678 | Intermediate Layer Classifiers for OOD generalization | Uselis & Oh (2025), ICLR 2025. Linear probes on the frozen features of an intermediate layer (ILCs) are compared with last-layer retraining for out-of-distribution classification on CMNIST, Waterbirds, CelebA, MultiCelebA, CIFAR-10C, CIFAR-100C and ImageNet variants, with ResNets and ViTs. Few-shot (probe trained on OOD data), the best intermediate layer usually wins, by more when OOD data are scarce. Zero-shot (probe trained on ID data, OOD only for selecting the layer), it beats retraining the last layer, sharply under subpopulation shift. Layer L − 2 beats L − 1 in every dataset shown. A distance-ratio sensitivity score finds intermediate features less displaced by the shift than penultimate ones, most of all for minority groups. | Active — read 2026-10-03 (NOTE-541); worth reading as the measurement that, under distribution shift, a trained classifier’s best linear probe is usually not on the penultimate layer: Intermediate Layer Classifiers (linear probes on frozen layer-l features, l ≤ L − 2) beat last-layer retraining in the few-shot setting on most of nine shifted datasets, and, trained only on in-distribution data, raise worst-group accuracy from 79.4% to 87.1% on Waterbirds and 56.0% to 82.0% on CelebA. The proposed reason is that penultimate features move further under shift, especially for minority groups, while intermediate ones stay put. Selection of the layer uses OOD validation data, and the gains for ViTs are small. |
| LIT-679 | A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent | Liao, Couillet & Mahoney (2020), NeurIPS 2020 and J. Stat. Mech. (2021). For random Fourier features ΣX = [cos(WX); sin(WX)] with n, p, N large and comparable, the expected resolvent of ΣXᵀΣX/n is asymptotically Q̄ = ((N/n)(Kcos/(1+δcos) + Ksin/(1+δsin)) + λI)⁻¹, with δσ = (1/n)tr KσQ̄, not the Gaussian-kernel prediction. Closed forms for training and test MSE follow and match MNIST, Fashion-MNIST and Kannada-MNIST. In the ridgeless limit the δ jump at 2N = n, separating an under- and an over-parameterized phase; the test MSE diverges there unless test data sit within about λ of the training data, which is the double-descent peak. A tuned λ removes it. | Active — read 2026-10-03 (NOTE-519); worth reading as an exact random-matrix account of double descent in random Fourier feature ridge regression when n, p and N are large together: the RFF Gram matrix does not converge to the Gaussian kernel in spectral norm, but its resolvent has a deterministic equivalent built from the cos and sin kernels with corrections δcos, δsin solving a fixed point (Theorem 1). Training and test MSE follow in closed form (Theorems 2–3). As λ → 0 the δ jump from O(1) to order 1/λ across 2N = n, so the model has two phases, and the test error diverges at the boundary when test data differ from training data; that singularity is the double-descent peak. Data need only be concentrated random vectors, not Gaussian. |
| LIT-680 | Proving Linear Mode Connectivity of Neural Networks via Optimal Transport | Ferbach, Goujaud, Gidel & Dieuleveut (2024), AISTATS 2024. Finding the permutation that aligns two networks’ neurons layer by layer is an optimal-transport problem between empirical weight distributions, so Wasserstein convergence rates bound when the permuted straight line stays low-loss. Wide two-layer nets trained by SGD in the mean-field regime are linearly connected modulo permutation with high probability. Deep nets with i.i.d. neuron weights (as at initialization) are too, once each layer is wider than about (1/ε) to the power of the previous one, a rate shown tight. Low-dimensional weight distributions need less width, and a covariance-weighted weight matching beats plain weight matching. | Active — read in full 2026-10-03 (NOTE-543); the first proof of linear mode connectivity modulo permutation beyond a single hidden layer. Aligning neurons is an optimal-transport problem between the empirical distributions of their weights, so convergence rates of empirical measures in Wasserstein distance bound how wide a network must be for the permuted straight line to stay low-loss. It proves the result for wide two-layer nets trained by SGD in the mean-field regime, and for deep nets whenever neuron weights within a layer are i.i.d., which holds at initialization. The width needed grows like (1/ε) to the power of the previous layer’s width, a rate shown to be tight, and falls if weights concentrate near a low-dimensional subspace. The proofs rest on independence of neurons, which the authors call the central and least realistic assumption. |
| LIT-681 | Optimizing Mode Connectivity for Class Incremental Learning | Wen, Cheng, Qiu, Wang, Pan & Li (2023), ICML 2023. In class-incremental learning the linear path between adjacent continual minima crosses a high-loss ridge for earlier tasks. OPC finds a curved low-loss path, modelled as the line plus a layer-wise Fourier-series perturbation, with the path loss split at a switching point between old-task and new-task memory losses. EOPC averages points in a cylinder around the switching point and is added as post-processing to existing methods, improving PODNet and AANet on CIFAR-100, ImageNet-100 and ImageNet-1K and sharply reducing forgetting. | Active — read in full 2026-10-03 (NOTE-524); worth reading as the class-incremental counterpart of Mirzadeh et al.: between adjacent continual minima in class-incremental learning, the straight line crosses a high-loss ridge for the earlier tasks, and a curved path avoids it. OPC parameterises that path as the line plus a layer-wise Fourier perturbation that vanishes at both ends (2 × 96 × 4 learned numbers for a 96-layer ResNet), splits the path loss at a switching point into old-task and new-task halves, and pushes it to flat regions by sampling a cylinder around it. Averaging points near the switching point (EOPC) adds 0.3 to 5.1 points of average incremental accuracy to PODNet and AANet on CIFAR-100, ImageNet-100 and ImageNet-1K. In Table 4 it beats Bezier and simplex paths only on the longer task sequences. |
| LIT-682 | Exploring Neural Network Landscapes: Star-Shaped and Geodesic Connectivity | Lin, Li & Wu (2024), preprint. For a two-layer ReLU student of an orthonormal teacher, and for deep linear networks, over-parameterisation makes any two global minima two-piece linearly connected and gives any finite set of minima a common centre linearly connected to each (star- shaped connectivity). Under a sparsity bias like SGD’s, the shortest path on the minimum manifold is nearly as short as the straight line, with normalised geodesic distance ≤ 1 + c₂/(r√m). A centre-finding objective locates such centres for independently trained FNNs, VGG16 and ResNets on MNIST and CIFAR-10, with no permutation alignment. | Active — read 2026-10-03 (NOTE-535); worth reading as the theory of star-shaped connectivity without permutations. For a two-layer ReLU student of an orthonormal teacher, and for deep linear networks, sufficiently wide networks have any two minima joined by a two-piece linear path and any finite set of minima linearly joined to a common centre on the minimum manifold. SGD’s bias toward sparse solutions explains why the shortest path on the manifold is nearly as short as the straight line (normalised geodesic distance ≤ 1 + c₂/(r√m)). Experiments find centres for three to five independently trained FNNs, VGG16s and ResNets on MNIST and CIFAR-10 with almost no barrier. Preprint. |
| LIT-683 | Exact expressions for double descent and implicit regularization via surrogate random design | Dereziński, Liang & Mahoney (2019), NeurIPS 2020. For the Moore–Penrose estimator X⁺y in linear regression, a determinantal surrogate design (density ∝ pdet(XXᵀ), Poisson sample size with mean n) gives exact MSE: σ² tr((Σ + λnI)⁻¹)(1 − αn)/(d − n) + wᵀ(Σ + λnI)⁻¹w·(d − n)/tr((Σ + λnI)⁻¹) for n < d, σ² tr(Σ⁻¹) at n = d, σ² tr(Σ⁻¹)(1 − βn)/(n − d) for n > d, where n = tr(Σ(Σ + λnI)⁻¹). For n < d, E[X⁺y] equals the population ridge solution with that λn: over-parameterization is implicit ridge regularization whose strength makes n the effective dimension. The surrogate agrees with the i.i.d. design as d, n → ∞ for sub-Gaussian data. | Active — read 2026-10-03 (NOTE-518); worth reading as the exact, non-asymptotic account of double descent and implicit regularization for minimum-norm least squares: replacing the i.i.d. design by a determinantal “surrogate” design gives closed-form MSE in every regime (Theorem 1) and shows that, below the interpolation threshold n < d, the expected minimum-norm solution equals the population ridge solution with the λn for which the ridge effective dimension tr(Σ(Σ + λnI)⁻¹) equals n (Theorem 2). The surrogate matches the i.i.d. design asymptotically for sub-Gaussian data (Theorem 3), at an empirically O(1/d) rate. The tool is a class of random matrices for which determinant and expectation commute. |
| LIT-770 | Compositional Visual Generation with Composable Diffusion Models | Liu, Li, Du, Torralba & Tenenbaum (2022), ECCV. Reading diffusion models as energy-based models lets several of them be combined at sampling time, as conjunctions and negations of concepts, generating scenes more complex than any seen in training and binding attributes that a single text-conditioned model confuses. Read: strong on object positions in CLEVR, weak on relations, qualitative on text models. | Active — read 2026-10-09 (NOTE-597); worth reading as the paper that moved compositional energy-based generation onto diffusion models: reading a diffusion model’s noise prediction as an energy gradient, concepts are composed at sampling time by adding one classifier-free-guidance term per concept (AND) or subtracting a concept’s prediction (NOT), with no retraining. The product-of-experts reading is exact only for conditionally independent concepts, unit weights and a conservative score, and is derived at zero noise but applied at every noise level. It wins clearly on composing CLEVR object positions (31.36% against an EBM’s 7.34% at three objects), not on relations (2.80% against 4.26%), and is shown on GLIDE and Stable Diffusion only qualitatively. |
| LIT-776 | Conditional Simulation Using Diffusion Schrödinger Bridges | Shi, De Bortoli, Deligiannidis & Doucet (2022), UAI, PMLR 180. Extends the Schrödinger-bridge formulation of diffusion generative modelling from unconditional to conditional simulation, applied to super- resolution, filtering in state-space models and refining pre-trained networks. Read: the extended-space bridge is proved; the empirical gains are at small step counts and mixed across metrics. | Active — read 2026-10-09 (NOTE-589); worth reading for the result that posterior sampling from simulation alone, when (X, Y) can be drawn but the likelihood cannot be evaluated, is a Schrödinger bridge between the joint p(x, y) and a reference p_ref(x|y)p(y) with y held fixed along the path (Proposition 1), solvable by iterative proportional fitting with learned drifts (CDSB). Its first iteration is the ordinary conditional score-based model, so later ones refine it, and the reference end can be a cheap posterior approximation rather than noise. Gains over the conditional score model are at equal small step counts and are not uniform across metrics; it filters the Lorenz-63 model with 20 steps where the score model diverges. |
| LIT-782 | GenEval: An Object-Focused Framework for Evaluating Text-to-Image Alignment | Ghosh, Hajishirzi & Schmidt (2023), NeurIPS 2023 Datasets and Benchmarks. An automated benchmark that uses object detectors to score text-to-image outputs on object co- occurrence, position, count and colour, agreeing well with human judgement; models of 2023 still fail at spatial relations and attribute binding. Detector verdicts agree with annotators 83% of the time (annotators with each other 88%); the best open model, IF-XL, scores 0.61 overall, 0.13 on position and 0.35 on binding. | Active — read 2026-10-09 (NOTE-593); worth reading as a measurement of where text-to-image models of 2023 fail at composition, and of how badly embedding similarity sees it: a COCO-trained detector plus masked-crop CLIP colour classification judges each image correct or not on presence, count, position and colour, agreeing with crowd annotators 83% of the time against 88% between annotators, and beating a per-task tuned CLIPScore on counting (by 22 points), position and attribute binding. The best open model (IF-XL) gets 0.61 overall but at most 0.15 on relative position and 0.35 on binding; scaling IF helps binding but not position, and continued training of Stable Diffusion v1 leaves scores flat. |
| LIT-783 | T2I-CompBench: A Comprehensive Benchmark for Open-world Compositional Text-to-image Generation | Huang et al. (2023). A benchmark of compositional text-to-image prompts in categories of attribute binding, object relationships and complex compositions, with category-specific automatic metrics, in the conference version (NeurIPS 2023 Datasets and Benchmarks). Disentangled BLIP-VQA and a UniDet box rule agree with human rankings far better than CLIPScore; spatial relations are hardest; a reward-weighted finetuning baseline, GORS, is rated best by humans. | Active — read 2026-10-09 (NOTE-592), in the conference version (arXiv v2); worth reading as a map of which compositions text-to-image models of 2023 get wrong and which automatic judges can tell: 6,000 prompts in six sub-categories (colour, shape and texture binding; spatial and non-spatial relations; complex), with spatial relations hardest and interactions easiest by human rating. Asking a VQA model one question per object–attribute pair ranks images far more like humans than CLIPScore does (Kendall τ 0.63 against 0.19 on colour), and detector boxes do so for spatial relations; nothing beat CLIPScore on interactions. A reward-weighted finetuning baseline (GORS) is rated best by humans, but its binding score falls sharply on attribute–noun pairs absent from its finetuning prompts (BLIP-VQA 0.55 to 0.34 on shape, 0.76 to 0.36 on texture), a gap the paper calls slight. |
| LIT-797 | An Explanation of In-context Learning as Implicit Bayesian Inference | Xie, Raghunathan, Liang and Ma (2021), ICLR 2022. Models pretraining text as a mixture of HMMs indexed by a latent concept θ and assumes the LM fits it exactly; in-context prediction is then the posterior predictive, and Theorem 1 shows the in-context predictor reaches the Bayes-optimal 0-1 risk as the number of examples grows, when each example’s KL signal about θ exceeds the error from the low-probability transitions between examples. Without that distinguishability, excess risk falls roughly as 1/k in the example length. Small Transformers and LSTMs trained on the synthetic GINC dataset reproduce it, and also show order sensitivity and zero-shot beating few-shot. | Active — read 2026-10-09 (NOTE-633); worth reading as the cleanest formal statement of in-context learning as posterior inference over a latent concept: if pretraining documents are a mixture of HMMs and the model fits that mixture exactly, then conditioning on a prompt of concatenated examples selects the prompt’s concept by posterior concentration, despite the prompt being low-probability under pretraining, provided each example carries more KL signal about the concept than the error from its unnatural boundaries (Theorem 1). The guarantees are asymptotic in the number of examples and say nothing about how a trained network computes the posterior; the experiments are on a synthetic dataset (GINC) built to satisfy the assumptions. |
| LIT-798 | T2I-CompBench++: An Enhanced and Comprehensive Benchmark for Compositional Text-to-Image Generation | Huang, Duan, Sun, Xie, Li & Liu (2025), IEEE TPAMI 47(5). The journal version of T2I-CompBench: 8,000 prompts in attribute binding, 2D/3D spatial and non-spatial relations, numeracy and complex compositions, with detector, depth, VQA and multimodal-LLM judges validated against human ratings, and eleven models benchmarked. GPT-4V is the best judge of interactions and complex prompts; spatial relations remain hardest and interactions easiest; DALL·E 3, SD3 and FLUX.1 lead. | Active — read 2026-10-09 (NOTE-610); worth reading as the 2025 journal extension of T2I-CompBench (LIT-783): 8,000 prompts in eight sub-categories, adding numeracy and 3D-spatial relations, eleven models up to SD3, DALL·E 3 and FLUX.1, and multimodal LLMs as judges. Two things change the conference picture. GPT-4V now agrees with human rankings better than CLIPScore on interactions (Kendall τ 0.48 against 0.25) and complex prompts (0.51 against 0.07), so interactions are no longer scored only by CLIPScore; but human scores on interactions are still near ceiling (0.95–0.99 for five of six rated models), and no new category asks about stance, purpose or social relation. Human ratings cover only the six 2023-era models; the 2024–25 models are ranked by automatic metrics alone. |
| LIT-801 | Learning to (Learn at Test Time): RNNs with Expressive Hidden States | Sun, Li, Dalal, Xu, Vikram, Zhang, Dubois, Chen, Wang, Koyejo, Hashimoto and Guestrin (2024), ICML 2025. Makes a sequence layer’s hidden state the weights W of an inner model f, updated per token by a gradient step on a learned self-supervised reconstruction loss, with output f(θ_Q x_t; W_t). With a linear f and batch gradient descent this is exactly linear attention (Theorem 1); with the Nadaraya–Watson estimator as inner learner it is exactly softmax self-attention (Theorem 2). TTT-Linear and TTT-MLP match or beat Mamba at 125M–1.3B and keep lowering perplexity to 32k tokens of context, where Mamba plateaus after 16k. | Active — read 2026-10-09 (NOTE-620); worth reading for its reframing of a sequence layer as a learner trained on its own context: the hidden state is the weights of an inner model, the update rule a gradient step on a self-supervised loss, the output rule the inner model’s prediction. Two exact identities follow: a linear inner model with batch gradient descent is linear attention (Theorem 1), and the Nadaraya–Watson estimator as inner learner is softmax self-attention (Theorem 2). The empirical part, TTT-Linear and TTT-MLP against Mamba and a Transformer at 125M–1.3B, is an architecture result that belongs to the anthology’s question. |
| LIT-806 v2 | Attend-and-Excite: Attention-Based Semantic Guidance for Text-to-Image Diffusion Models | Chefer et al. (2023), ACM TOG 42(4) (SIGGRAPH 2023). Stable Diffusion often omits one of two named subjects (“catastrophic neglect”) and misbinds attributes. Shifting the latent at each early denoising step to raise the peak cross-attention of the most neglected subject token mitigates neglect, by CLIP and caption similarity and by human preference, with no training. Binding gains are shown in figures, not measured. | Active — read 2026-10-09 (NOTE-639); worth reading as the paper that names “catastrophic neglect” (a text-to-image model omits one of the subjects a prompt names) and links it to attribute binding: the subject token that no image patch attends to is the one that goes missing. Its fix is an inference-time correction of the latent: push up the maximum cross-attention of the most neglected subject token, smoothed over neighbouring patches, during the first half of sampling. Evidence is on 276 two-subject Stable Diffusion prompts, by CLIP similarities, BLIP captions and a 65-person preference study (77–91% preferred it). That it improves attribute binding is shown only in figures; no binding accuracy is measured, and relations are left out of scope. |
| LIT-814 | Rethinking the Role of Demonstrations: What Makes In-Context Learning Work? | Min, Lyu, Holtzman, Artetxe, Lewis, Hajishirzi and Zettlemoyer (2022), EMNLP 2022. Replacing the gold labels of in-context demonstrations with random labels from the label set barely lowers accuracy (0–5 points absolute, over 12 model–method pairs including GPT-3 and 26 datasets). What the demonstrations do supply is the input distribution, the label space and the input-label pairing as a format; keeping the format with only inputs or only labels retains up to 95% of the gain. Meta-training for in-context learning sharpens the effect. | Active — read 2026-10-09 (NOTE-607); worth reading as the measurement that separates what a demonstration supplies from whether its labels are right: across 12 model–method pairs up to GPT-3 and 26 classification and multi-choice datasets, replacing gold labels with random ones costs 0–5 points, while removing the input distribution, the label space or the paired format costs much more. The finding is empirical and aggregate; some dataset–model pairs lose up to 14 points, and generation tasks are not tested. |
| LIT-827 | Backprop as Functor: A compositional perspective on supervised learning | Fong, Spivak & Tuyéras (2017; LICS 2019), ARXIV-1711.10455. Supervised learners compose only when each carries a request function that passes a corrected input upstream; learners (P, I, U, r) then form a symmetric monoidal category Learn, and gradient descent with backpropagation, for a fixed step size ε and an error function e with ∂e/∂x(x₀, −) invertible, is a faithful, injective-on-objects strong symmetric monoidal functor Para → Learn. Functoriality is the chain rule. | Active — read 2026-10-09 (NOTE-625); worth reading as the paper that makes supervised learners composable: a learner A → B is a parameter set with implement, update and request functions, learners form a symmetric monoidal category Learn, and for a fixed step size and an error function whose derivative in its first argument is invertible, gradient descent with backpropagation is a faithful strong symmetric monoidal functor from differentiable parametrised functions to Learn. The backward message is a requested input, not a gradient step; cross-entropy falls outside the theorem. |
| LIT-833 | Fantastically Ordered Prompts and Where to Find Them: Overcoming Few-Shot Prompt Order Sensitivity | Lu, Bartolo, Moore, Riedel and Stenetorp (2021), ACL 2022. With the same four SST-2 examples, the 24 orders range from near chance (~50%) to above 85% accuracy, in GPT-2 and GPT-3 from 0.1B to 175B. Good orders are model-specific (pairwise Spearman correlations between sizes all between −0.35 and 0.27, on one example set), the variance persists with up to 32 examples and after calibration, and failing orders mostly predict one label. Ranking orders by the label entropy they induce on a self-generated probing set recovers an average 13% relative gain across eleven classification tasks. | Active — read 2026-10-09 (NOTE-626); worth reading as the measurement that the order of the same few-shot examples, with content held fixed, can move a language model from near chance to near the fine-tuned state of the art, at every size from GPT-2 Small to GPT-3 175B. Good orders do not transfer between model sizes (rank correlations near zero), more examples do not remove the variance, and failing orders mostly collapse the predicted label distribution. The rest of the paper is a selection method (entropy over a self-generated probing set) whose interest here is secondary. |
| LIT-844 | Quantum-Like Contextuality in Large Language Models | Lo, Sadrzadeh & Mansfield (2024), ARXIV-2412.16806, Proc. R. Soc. A 481:20240399 (2025). Instantiates a three-sentence anaphora schema with the support of a PR prism 51,966,480 times from Simple English Wikipedia and takes referent probabilities from BERT: 0.148% of the models are sheaf-contextual by a signalling-corrected test and 71.1% are CbD-contextual. The support is fixed by the schema, so each verdict turns on how uncertain BERT is in three sentences. | Active — read in full 2026-10-09 (NOTE-648) from the arXiv preprint (the journal version could not be reached); worth reading as the large-scale test of language-model output for contextuality, and for what its numbers rest on. A PR-prism schema of three sentences (“It is X1 and the same one is X2”, ”… X2 … same … X3”, ”… X3 and the other one is X1”) is instantiated with 866,108 Simple English Wikipedia noun pairs and their adjectives; BERT gives one referent probability per sentence, and the support is fixed by the schema. Of 51,966,480 models, 77,118 (0.148%) pass the signalling-corrected sheaf test (SF < 1/6) and 36,938,948 (71.1%) the CbD test (Δ < 2). Both verdicts are fixed by three numbers per instance; the CbD one holds whenever BERT is uncertain and roughly consistent across the three sentences. The link to embedding distance is a softmax identity plus weak correlations (R² ≤ 0.08). |
| LIT-848 | Large AI models are cultural and social technologies | Farrell, Gopnik, Shalizi & Evans (2025), Science 387(6739):1153–1156, Policy Forum. Large models are best understood not as intelligent agents but as a new cultural and social technology: like writing, print and search they give access to what other people have produced, and like markets, bureaucracies and elections they make lossy, uninvertible summaries of it that can be operated on at scale. The questions that follow are the old ones about such technologies: homogenization or diversity, who gains and who loses, and which countervailing institutions emerge. Argued by historical analogy, without data. | Active — read in full 2026-10-09 (NOTE-649) from the authors’ accepted version (Science galley of 25 February 2025, posted on Farrell’s site); worth reading as the compact statement of the view that large models are not agents but a new cultural and social technology, of a kind with writing, print, markets, bureaucracies and elections: lossy summaries of what other humans have produced that let people reorganize it at scale. It is a four-page Policy Forum argued by analogy, with no data. Its one mechanism with consequences is that models fit their training distribution on average, so are least accurate where the data are rare, which may homogenize culture; it sets this against the possibility of “society-like” ecologies of models that keep diversity. |
| LIT-849 | Relative representations enable zero-shot latent space communication | Moschella, Maiorca, Fumero, Norelli, Locatello & Rodolà (2022; ICLR 2023), arXiv:2209.15430. Re-encodes each sample as its cosine similarities to a fixed set of anchor samples, which is invariant by construction to every cosine-preserving map of the latent space (rotations, reflections, rescaling), not to translations or general linear maps. With anchors matched across two encoders (parallel anchors), a decoder trained on one encoder’s relative vectors runs on another’s with no training: across seeds, architectures, latent sizes and languages. Stitching works far better than with raw latents but is lossy and uneven, and the assumption that independently trained spaces differ by an angle-preserving map is supported only approximately. | Active — read in full 2026-10-09 (NOTE-652); worth reading as the source of “relative representations”: a sample is re-encoded as its vector of cosine similarities to a fixed set of anchor samples, and a decoder trained on that vector runs, without any training, on a different encoder that sees corresponding anchors. Read it knowing what is guaranteed and what is observed. The invariance is by construction and one line long: the vector is unchanged by any map that preserves cosines, that is rotations, reflections and rescaling (and, though the paper does not say so, any per-sample positive rescaling), but not by translations, which it assumes normalisation removes, nor by general linear maps. That training noise, a change of architecture or a change of language moves a latent space only by such a map is an assumption, supported indirectly and only approximately: across FastText and Word2Vec the relative vectors agree at cosine 0.86 but share only 34–39% of their ten nearest neighbours, and between two ViTs on CIFAR-10 11–12%. Zero-shot stitching is not correspondence-free: it needs the anchors matched across the two sides (parallel anchors), which here means 300–768 translated or aligned sentence pairs or the same images. It beats absolute stitching everywhere absolute stitching can be run, but loses a great deal against the unstitched model in places: 98.6 to 80.5 ± 21.1 F1 on DBpedia across architectures, 90.1 to 65.7 for a Japanese encoder under an English decoder, 72.6 to 37.4 on ImageNet when ViT features feed a RexNet decoder. On a model that is already multilingual (XLM-R) it is worse than doing nothing. |
| LIT-850 | Revisiting Model Stitching to Compare Neural Representations | Bansal, Nakkiran & Barak (2021), NeurIPS 34. Revives Lenc and Vedaldi’s model stitching (LIT-363) as a measure of representations: the bottom of one frozen network feeds the top of another through a trained 1×1 convolution, and the change in test error is the “stitching penalty”. Independently seeded networks, and supervised and self-supervised ResNet-50s, stitch at all layers within a few points, where CKA says they differ. Representations trained on more data plug into a weaker network and lower its error; for width and training time this happens only near the top. All empirical, single runs, mostly CIFAR-10. | Active — read in full 2026-10-09 (NOTE-653); worth reading as the paper that made model stitching a standard way of comparing trained networks, and as the source of “stitching connectivity”: two networks of one architecture, trained from different seeds, can be joined at any layer by a learned 1×1 convolution with little loss of test accuracy. Read it knowing that it proves nothing. Its definitions are informal (“low penalty”, “comparable”), stitching connectivity is an informal conjecture, and every result is a single run, mostly ResNet-18 on CIFAR-10, read off plots with no error bars. Its “more is better” ordering is an ordering by one task’s loss through a fixed top network and a restricted map class, not an ordering by information: the paper itself says a pixel shuffle, which loses nothing, can stitch worse, and a negative penalty means the fitted map did not reproduce the layers it replaced. The width and training-time improvements appear only near the top of the network; at mid layers the wider and longer-trained representations stitch slightly worse. |
| LIT-855 | Closed-Form Training Dynamics Reveal Learned Features and Linear Structure in Word2Vec-like Models | Karkada, Simon, Bahri and DeWeese (2025), ARXIV-2502.09863, NeurIPS 2025. The quartic approximation of word2vec’s loss, under symmetric reweighting with constant aggregate weight, is exactly unweighted factorisation of M*, the co-occurrence matrix’s relative deviation from independence, and from small initialisation gradient flow learns M*‘s eigenvectors one at a time in sigmoidal steps. The embeddings this predicts match word2vec’s features and benchmark scores far better than truncated PMI or PPMI factorisations do. | Active — read 2026-10-09 (NOTE-660); worth reading as the first closed-form account of what a contrastive word-embedding algorithm learns under a rank constraint, and in what order. Replacing word2vec’s logistic loss by its quartic Taylor approximation, and choosing symmetric reweightings whose aggregate weight is constant, turns the objective into unweighted symmetric matrix factorisation of M*, the matrix of relative deviations of word co-occurrence from independence (Theorem 1, proved). From vanishing random initialisation, gradient flow then learns the eigenvectors of M* one at a time, each by a sigmoidal step in time (ln(λ_k/σ²))/λ_k (Result 3, a derivation that drops terms, not a proof). The resulting embeddings track word2vec’s features and benchmark scores closely, and much more closely than truncated factorisations of PMI or positive PMI. The analogy part is empirical: difference vectors of semantic pairs look like a spiked random matrix, and the spike’s strength tracks analogy accuracy. |
| LIT-857 | Single-Head Attention in High Dimensions: A Theory of Generalization, Weights Spectra, and Scaling Laws | Boncoraglio, Erba, Troiani, Xu, Krzakala and Zdeborová (2025), arXiv. For a tied single-head softmax attention layer trained by ridge-penalized ERM on Gaussian token sequences labelled by an attention-indexed target, in the limit n/d², p/d fixed, a six-parameter variational problem gives the exact train and test error, interpolation and perfect-recovery thresholds, and the singular-value law of the learned query–key map: η·ReLU(S0 + δZ − ε), Z a GOE matrix. Weight decay is a nuclear-norm penalty on that map, so the factorized model beats direct training of the product. Power-law targets give sequential mode recovery and LASSO- rate scaling laws. Non-rigorous (replica/AMP), checked against Adam. | Active — read 2026-10-09 (NOTE-661); worth reading as a solvable model in which the spectrum of trained attention weights is derived, not just measured: for empirical risk minimization of one tied softmax attention layer on Gaussian sequences, with n of order d², the learned query–key map is, in law, a rescaled ReLU of the target plus a semicircle noise bulk shifted by a threshold, so its outliers are recovered target directions and its bulk is finite-sample noise. Weight decay on the factor is exactly a nuclear-norm penalty on the product. With a power-law target, modes are recovered one by one as the threshold moves, which gives heavy-tailed learned spectra and power-law learning curves whose exponents match the LASSO rates. The main results are replica/AMP “Claims”, not theorems, checked against Adam runs at d = 100–400. |
| LIT-858 | Towards Understanding Grokking: An Effective Theory of Representation Learning | Liu, Kitouni, Nolte, Michaud, Tegmark and Williams (2022), NeurIPS 2022. In a toy model that decodes the sum of two learned embeddings, grokking is representation learning: an unseen pair is answered correctly when its embedding sum equals a trained pair’s, and the critical training fraction (about 0.4 for addition with p = 10) is where the training set’s “parallelogram” equations first fix the embedding up to translation and scale. A posited effective loss on the normalised embeddings predicts that threshold and, through its third Hessian eigenvalue, the time to structure. Phase diagrams over decoder learning rate and weight decay place grokking between comprehension and memorization in the toy, a transformer and MNIST. | Active — read 2026-10-09 (NOTE-662); worth reading as an early mechanistic account of when grokking networks generalise, in a toy model that sums two learned embeddings and decodes the sum. An unseen pair is predicted correctly when its embedding sum coincides with a trained pair’s (a “parallelogram”), and the critical training fraction is where the training set’s parallelogram equations first fix the embedding up to translation and scale. The account is exact for the toy (1D embeddings, addition hard-coded in the architecture, an effective loss posited and derived only for a linear decoder); the transformer and MNIST evidence is qualitative (PCA pictures and phase diagrams), and the four “phases” are defined by accuracy thresholds, not by any order parameter. |
| LIT-859 | Exact Learning Dynamics of In-Context Learning in Linear Transformers and Its Application to Non-Linear Transformers | Mainali and Teixeira (2025), ARXIV-2504.12916. For a single linear attention layer trained by small-step SGD on in-context regression ℝᵈ → ℝᵈ, with the input covariance and every task matrix sharing one eigenbasis and the weights assumed to stay in it, the averaged dynamics decouple into one logistic equation per eigenmode: mode α is learned on a timescale τ_α = (ηP s_α²)⁻¹, giving plateaus and cliffs in the loss, and settles where p_α q_α = N / ((N+1)s_α + Tr S), so the trained predictor applies an inverse, finite-N-corrected training covariance to the context’s empirical covariance. A scaling symmetry conserves ‖p₂‖² − ‖q₁‖². Effective-rank and subspace measures applied to attention-only transformers and a grokking model are offered as qualitative parallels. | Active — read 2026-10-09 (NOTE-658); worth reading as an exactly solved case of the slow learning that produces a fast in-context learner: for one linear-attention layer trained by gradient flow on in-context vector regression, with tasks and inputs sharing one eigenbasis, each eigenmode of the input covariance learns by its own logistic curve on a timescale ∝ 1/s_α², the trained product of the two weight blocks settles at the inverse of a finite-context-corrected training covariance, so the learned in-context predictor is a preconditioned estimator (the fixed point Ahn et al. and Zhang et al. had found), and ‖p₂‖² − ‖q₁‖² is conserved. The decoupling rests on an assumed spectral alignment and on large N and many tasks. The second half, measures applied to attention-only transformers and a grokking model, is qualitative and correlational: no seeds, error bars or controls are reported, and “mechanistic explanation” overstates it. |
| LIT-860 | Symmetry in language statistics shapes the geometry of model representations | Karkada, Korchinski, Nava, Wyart and Bahri (2026), ICML 2026 (PMLR 306:56137–56172). If co-occurrence among words on a latent continuum depends only on their separation, a full-rank spectral word embedding places them on Fourier modes of the latent coordinate: circles for cyclic concepts, open curves with harmonic ripples for intervals, curved sheets for maps, all proved for that model with the exponential kernel solved exactly. Wikipedia’s month and year statistics fit the symmetry; Gemma 2 and EmbeddingGemma show matching PCA pictures; the geometry survives removing the words’ mutual co-occurrences, which a seasonal latent variable shared by many words explains. | Active — read 2026-10-09 (NOTE-664); worth reading as a derivation of the circles, rippled curves and curved sheets found in language-model representations of months, years and places from one property of the data: when the co-occurrence of two words on a latent continuum depends only on their separation, the matrix the embedding factorises is circulant or Toeplitz, so its eigenvectors are sinusoids of the latent coordinate. For a full-rank spectral word embedding the paper proves the principal components are Fourier modes with amplitudes set by the kernel’s Fourier transform, solves the open-boundary exponential kernel exactly (a Sturm–Liouville problem), and bounds linear decoding of the coordinate by the missing Fourier tail (ε² ≲ r^(−1/D)). It measures the symmetry in Wikipedia statistics, shows the month circle survives zeroing every month–month co-occurrence, and explains that by a seasonal latent variable that couples many words. The “word embeddings” are explicit factorisations of the co-occurrence matrix, not trained networks; the language-model evidence is PCA pictures of Gemma 2 2B and EmbeddingGemma; the robustness argument is a perturbation sketch. |
| LIT-861 | Topological Invariance and Breakdown in Learning | Yang, Poggio, Chuang and Ziyin (2025), ARXIV-2510.02670. For any permutation-equivariant update rule whose map is K-Lipschitz, one step of size η changes the distance between any two neurons by a factor in [1 − ηK, 1 + ηK], and coincident neurons stay coincident. Below η* = 1/K the induced map on the set of neuron vectors is therefore a homeomorphism, measure-preserving for the pushed-forward neuron distribution and a C¹ diffeomorphism under a smoothness condition; above it only a continuous surjection is guaranteed. The paper reads this, with the edge of stability, as two phases of training, the second a topological simplification; the proofs hold, but that reading and the Betti-number experiments go beyond them. | Active — read 2026-10-09 (NOTE-663); worth reading as a short, correct proof that permutation symmetry constrains how neurons can move under training: if the update rule is permutation-equivariant (gradient descent, SGD or Adam on a permutation-symmetric loss), neurons that coincide stay coincident at every step size, and if the update map is K-Lipschitz each step moves every pair of neurons apart or together by at most a factor 1 ± ηK. So below η = 1/K the induced map on the set of neuron vectors is a bi-Lipschitz homeomorphism (a C¹ diffeomorphism under a smoothness condition) and distinct neurons cannot merge in finitely many steps; above it only a continuous surjection is guaranteed. The proofs are a few lines and hold. What they do not carry is the paper’s larger picture: for finitely many neurons a homeomorphism of the neuron set is only a bijection, “topological simplification” above 1/K is permitted, not shown, and the experiments measure Betti numbers of point clouds at a fixed scale, which the theorem does not protect. The large-step runs show the clouds fragmenting (more components), not merging, and in the Adam runs the step size times the measured top Hessian eigenvalue stays below 0.05 where the topology changes. |
| LIT-862 | A mathematical theory of semantic development in deep neural networks | Saxe, McClelland and Ganguli (2019), PNAS 116(23):11537–11546, ARXIV-1810.10531. A two-layer linear network trained from small weights learns the singular modes of its input–output correlations one at a time, each in a sigmoidal step at a time of order (τ/s)ln(s/ε), where a shallow network learns them all at once. With data generated on a tree this gives coarse-to-fine differentiation in stages, transient illusory correlations and narrowing generalisation; the SVD also defines typicality, prototypes and category coherence, and minimum-norm weights fix the hidden similarity matrix across networks. | Active — read 2026-10-09 (NOTE-666); worth reading as the paper that turns the exactly solved learning dynamics of a two-layer linear network into a theory of semantic development. Trained by gradient descent from small weights on whitened inputs, the network learns the singular modes of the item–feature correlation matrix one at a time, each by a sigmoid at a time of order (τ/s)ln(s/ε), while a shallow network learns every mode on one exponential timescale. Hierarchically generated data have singular values that fall with depth in the tree, so broad distinctions are learned first in stage-like transitions, individual predictions can rise and fall between stages, and inductive generalisation narrows with development. The same SVD gives typicality, prototypes and a coherence threshold (a BBP transition), rings give Fourier modes, and minimum-norm weights, which small initialisation approximates, give identical hidden similarity matrices across networks. The trajectory is exact only from decoupled, balanced initial weights with white inputs; from random small weights it is checked by simulation, and the match to children and cortex is qualitative. |
| LIT-863 | On the Emergence of Linear Analogies in Word Embeddings | Korchinski, Karkada, Bahri and Wyart (2025), NeurIPS 2025. If each word is a vector of d binary attributes whose effects on co-occurrence are independent and multiplicative, the co-occurrence ratio is a Kronecker product and its logarithm, the PMI, has rank at most d + 1 with eigenvectors affine in the attributes, so a spectral PMI embedding realises parallelogram analogies exactly; the ratio itself does so only for weak, narrowly spread attribute signals. The structure survives noise, vocabulary pruning and deletion of all pairs of a relation, by perturbation arguments checked in simulation and on Wikipedia. | Active — read 2026-10-09 (NOTE-665); worth reading as a generative model of word co-occurrence in which the parallelogram analogies of word embeddings are derived rather than postulated. Each word is a point of the hypercube {−1, +1}^d of binary attributes, and each attribute multiplies its co-occurrence by an independent 2 × 2 factor, so the ratio M = P(i,j)/P(i)P(j) is a Kronecker product (proved) and the PMI log M is a bilinear form in the attributes, of rank at most d + 1 with eigenvectors affine in them. A spectral embedding of the PMI is then a linear image of the attribute hypercube and its analogies hold exactly at any signal strength; one of M holds them only when the signals are weak, narrowly spread and K ≤ d + 1, because M’s spectrum mixes in products of attributes. Robustness to noise, to pruning the vocabulary and to deleting every pair of an analogy relation is argued from eigenvalue perturbation bounds and checked numerically. The embeddings are eigendecompositions of a co-occurrence matrix, not trained word2vec or GloVe, and the agreement with Wikipedia is in the shapes of spectra and accuracy curves; no attribute is recovered from data. |
| LIT-864 | Exact solutions to the nonlinear dynamics of learning in deep linear neural networks | Saxe, McClelland and Ganguli (2014), ARXIV-1312.6120, ICLR 2014. Solves gradient-descent learning in deep linear networks with whitened inputs from decoupled initial weights: each singular mode of the input–output correlation is learned by a sigmoid at a time of order (τ/s)ln(s/ε), with conserved differences of squared layer strengths. For any depth, the delay depth adds is set by the initial end-to-end mode strength, finite as depth grows when that strength is of order one and the step size is scaled with depth. Argues, with simulations, that scaled Gaussian layers are not isometric in product while orthogonal ones are, and that tanh networks with orthogonal weights near gain 1 stay close to isometric. | Active — read 2026-10-09 (NOTE-667); worth reading as the paper in which the learning dynamics of deep linear networks were solved exactly from decoupled starts at any depth (Fukumizu had solved one hidden layer for another class of starts) and the stepwise, mode-by-mode picture was set out that later work calls Saxe-style. Gradient descent on a linear network with one hidden layer and whitened inputs is a cubic system whose modes cooperate within a singular mode of the input–output correlations and repel across modes; from decoupled initial weights each mode’s strength follows a sigmoid reaching s at a time of about (τ/s)ln(s/ε), with a conserved difference of squared layer strengths and, in the appendix, the unbalanced case. Beyond what the 2019 restatement keeps, it solves arbitrary depth from decoupled starts: depth slows learning through the initial end-to-end mode strength, so from strength of order one the number of iterations stays finite as depth grows once the step size is scaled down with depth, while from strength ε the delay grows like s/ε. It argues that scaled Gaussian layers preserve norms but not isometry, that random orthogonal layers give depth-independent learning times on MNIST, and that tanh networks with orthogonal weights near gain 1 keep the end-to-end Jacobian’s singular values of order one. Random initial weights, the role of isometry in learning time and everything nonlinear rest on simulation, not proof. |
| LIT-866 | Hyperbolic Entailment Cones for Learning Hierarchical Embeddings | Ganea, Bécigneul and Hofmann (ICML 2018), PMLR 80:1646–1655. Represents a partial order in the Poincaré ball by nested cones: v is below u when v lies in u’s cone. Nesting forces apertures of at most π/2 that narrow toward the boundary (r sin ψ/(1 − r²) non-increasing), so cones are undefined near the origin; the widest family is sin ψ = K(1 − r²)/r, with a closed-form membership test. Overlapping cones admit several parents. On WordNet noun closure link prediction in 5–10 dimensions it beats Poincaré and order embeddings once some closure edges are in training; with none, all methods fail. | Active — read 2026-10-09 (NOTE-671); worth reading as the paper that represents a partial order, not only a tree, by containment of regions in hyperbolic space. Each concept is a point x of the Poincaré ball and owns a cone S_x, the image under the exponential map of the tangent vectors within an angle ψ(x) of the outward ray through x; v counts as entailed by u when v lies in S_u. Requiring the cones to be axially symmetric, rotation invariant, continuous in aperture and nested (v in S_u implies S_v inside S_u) forces ψ ≤ π/2 (Lemma 2) and makes r sin ψ(r)/(1 − r²) non-increasing (Theorem 3), so no cone can be defined near the origin; the widest family is sin ψ(x) = K(1 − ‖x‖²)/‖x‖ outside a ball of radius ε, and membership is a closed-form angle test (Theorem 5). That family is transitive (Theorem 4), but the proof is only said to be “similar” and is not given. Cones of incomparable points can overlap, so a node may sit below several parents. The evidence that learned cones fit a DAG is one WordNet-noun experiment: held-out edges of the transitive closure, test F1 at 5 and 10 dimensions, single runs, where hyperbolic cones lead once 10% or more of the closure is in training but every method fails (F1 27–43%) when only the transitive reduction is, and Euclidean cones initialised from Poincaré embeddings match the hyperbolic ones. |
| LIT-870 | Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning | Lin, Coifman, Mishne and Talmon (2023), ICML 2023 (PMLR 202), ARXIV-2305.18962. Recovers a latent tree from point-cloud or graph data without learning: diffusion densities at dyadic time scales are embedded in a product of Poincaré half-spaces, and the ℓ1 product distance, a scale-weighted Hellinger sum, is proved equivalent up to constants to a thresholded snowflake min{1, d_T^(2α)} of the tree metric when the operator behaves like the heat kernel on a tree. It keeps local tree neighbourhoods (best MAP on the benchmarks) but not global distances (higher distortion than hyperbolic MDS). | Active — read 2026-10-09 (NOTE-675); worth reading as a construction that recovers a latent tree from unlabelled data by distances alone, with a proof of what is recovered. Each point is represented by its diffusion densities P^(2^−k) e_i at dyadic times, each density is placed in a Poincaré half-space at a height set by its scale, and the hyperbolic diffusion distance is the ℓ1 sum over scales of the hyperbolic distances, 2 sinh⁻¹(2^(1−kα)‖φ_i^k − φ_j^k‖) with φ the square roots of the densities, a scale-weighted Hellinger distance. Theorem 1 (Appendix C, after Leeb and Coifman) says that for the heat kernel on a tree this is equivalent up to constants to min{1, d_T^(2α)}, 0 < α < 1/2: a thresholded snowflake of the tree metric, not the tree metric, and saturated beyond a fixed radius; the main text states it without the threshold. The tree heat-kernel bounds are stated, not proved, and one step of the Hölder bound (Proposition C.5) holds only for tree distances of at least 1, so the admissible range of α is not secured. Empirically it gives the best mean average precision on five graph benchmarks and two single-cell datasets and worse average distortion than hyperbolic MDS or Sarkar’s construction: local neighbourhoods are kept, global distances are not. Nothing in it concerns attributes or linear directions. |
| LIT-871 | Hyperbolic Representation Learning: Revisiting and Advancing | Yang, Zhou, Ying, Chen and King (2023), ICML 2023, PMLR 202. Tracks the hyperbolic distance from each node to the origin and finds that hyperbolic models trained on ordinary task losses do not by themselves place a tree’s root nearest the origin or spread its levels outward: on synthetic 3-ary trees the root sits well off the minimum and the distances are roughly bell-shaped, ordering node pairs by level only 69–75% of the time. Proposes HIE, which recentres the embedding on its hyperbolic centroid and pushes nodes outward; it improves that ordering and benchmark scores, most on the least tree-like graphs. | Active — read 2026-10-09 (NOTE-669); worth reading as an empirical check, on synthetic trees and four benchmark graphs, of the assumption that a hyperbolic model trained on an ordinary task loss lays out the data’s hierarchy by itself, root near the origin and leaves near the boundary. It does so only partly. On two 3-ary trees of 1,093 nodes, a hyperbolic graph convolutional network (HGCN) trained for link prediction or node classification puts the root nowhere near the smallest distance to the origin, and the nodes’ distances to the origin come out roughly bell-shaped rather than piling up at the leaves; read off as levels, those distances order random node pairs correctly 69–75% of the time. The proposed fix, HIE, recentres the embedding on its hyperbolic centroid and adds a loss that pushes every node outward in proportion to its current distance. It raises that ordering to 75–84% and lifts benchmark scores, most on the least tree-like graphs (Citeseer, Cora). There is no theory of why task losses fail to produce the hierarchy, and the fix itself uses no level information. |
| LIT-873 v2 | Mutual information, neural networks and the renormalization group | Koch-Janusz and Ringel (2017; Nature Physics 14, 578–582, 2018). Defines the relevant degrees of freedom of a spatial block as the coarse variable that maximizes mutual information with the environment beyond a buffer, and learns it with restricted Boltzmann machines from Monte Carlo samples (the RSMI algorithm). It recovers Kadanoff block spins in the 2D Ising model and the electric-field variables of the dimer model, ignores decoupled noise, and, iterated, gives the Ising flow and ν ≈ 1.0 ± 0.15; a distribution-fitting RBM does none of this. The link to a short-ranged effective Hamiltonian is proved only in 1D. | Active — read 2026-10-09 (NOTE-674); worth reading as the paper that turns “the relevant degrees of freedom” of real-space renormalization into an objective that can be optimized: a block’s coarse variable is chosen to maximize its mutual information with the system beyond a buffer around the block. Trained on Monte Carlo samples alone, the network rediscovers Kadanoff’s block spin for the 2D Ising model, finds the low-momentum electric fields of the 2D dimer model while ignoring a strongly patterned but decoupled noise, and, iterated, reproduces the Ising flow away from T_c (located to about 1%) with ν ≈ 1.0 ± 0.15. An RBM trained instead to fit the data distribution latches on to the noise. The equivalence of the objective with a short-ranged effective Hamiltonian is argued, and shown only for 1D and quasi-1D chains; in 2D the evidence is the two models, whose answers were known beforehand. |
| LIT-874 | Representation Tradeoffs for Hyperbolic Embeddings | Sala, De Sa, Gu and Ré (ICML 2018), PMLR 80:4460–4469. Trees embed in the hyperbolic plane with distortion 1 + ε (Sarkar’s construction), but a hyperbolic distance d costs about d bits, so the embedding needs Θ((ℓ/ε) log deg_max) bits per coordinate in the plane: logarithmic in branching, linear in depth (lower bound proved for H², with an approximation). Higher dimension, via coding-theoretic child placement, divides the upper bound by r. Also an exact hyperbolic MDS by PCA after centring at a new mean, with a perturbation bound growing as sinh²(diameter), and a local-convexity condition for hyperbolic PGA. | Active — read 2026-10-09 (NOTE-679); worth reading as a quantitative account of why trees fit hyperbolic space and what the fit costs. Following Sarkar, any finite tree embeds in the hyperbolic plane with worst-case distortion at most 1 + ε by placing each node’s children evenly on a hyperbolic circle of radius τ around it; but a hyperbolic distance d needs about d bits to write down, so the coordinates need O((ℓ/ε) log deg_max) bits, ℓ the longest path, and an explicit tree of deg_max chains shows Ω((ℓ/ε) log deg_max) is needed in the plane. Precision grows with the logarithm of the branching and linearly with depth: hyperbolic space is cheap for short bushy hierarchies and dear for long chains. In r dimensions, children placed on hypercube vertices chosen by a Hadamard code divide the cost by r up to r ≈ log deg_max (upper bound only). For general metrics it gives an exact hyperbolic MDS: cosh of the distances is uuᵀ − XXᵀ once the points are centred at a new pseudo-Euclidean mean, so PCA recovers them. The lower bound’s proof uses a small-angle approximation and covers only the plane; the PGA guarantee is local convexity, not global recovery; the experiments, including the WordNet MAP of 0.989 in two dimensions, are reported at ε = 0.1, where WordNet takes 2,877 bits per coordinate, while the introduction quotes the ε = 1 figure of about 500. |
| LIT-877 | How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model | Cagnetta, Petrini, Tomasini, Favero and Wyart (2023; Phys. Rev. X 14, 031001, 2024). Introduces the Random Hierarchy Model, a random context-free grammar of depth L with m synonymic expansions per symbol. Shallow and lazy networks need a fixed fraction of all data, exponential in the input dimension; deep feature-learning CNNs learn from about n_c m^L examples, polynomial in it, which is where hidden layers become invariant to swapping synonyms and where the class statistics of input patches, equal for synonyms, emerge from sampling noise. Without those correlations deep networks fail too. The law is measured; the link to correlations is a scaling argument plus a one-step-gradient model. | Active — read 2026-10-09 (NOTE-677); worth reading as the paper that introduces the Random Hierarchy Model, a random context-free grammar of fixed depth L and branching s in which each class and each higher-level feature has m interchangeable (“synonymic”) lower-level representations, and as a measured and partly derived account of how many examples a deep network needs to learn it. Two-layer networks and lazy (kernel-regime) deep networks need a fixed fraction of all the data, which is exponential in the input dimension d = s^L; deep convolutional networks in a feature-learning parametrisation need P* ≈ n_c m^L, a power of d. That number is measured, not proved. It coincides with the training-set size at which the networks’ hidden layers become insensitive to swapping synonyms, and with the size at which the class-conditional frequencies of input s-tuples, which are equal for synonyms, rise above sampling noise; a one-step gradient argument for a two-layer network on one patch links the last two. Removing the input–label correlations, with the hierarchy kept, makes deep networks fail again. The hierarchy is one of constituency (parts composing wholes), not of attributes implying attributes. |
| LIT-878 v2 | Statistical Physics through the Lens of Real-Space Mutual Information | Gökmen, Ringel, Huber and Koch-Janusz (2021; Phys. Rev. Lett. 127, 240603). Replaces the RBM estimator of LIT-873 with a neural InfoNCE lower bound co-trained with a convolutional coarse-graining filter (RSMI-NE), and uses the optimal filters at a single buffer size, not an iterated flow, as lattice forms of the most relevant operators. On the interacting dimer model they are the columnar and plaquette order parameters below T_BKT and the electric fields at high T; the maximal information marks the phases; one fitted scaling dimension (1.00037 against 1) is given. The optimality theorem is cited, not proved here. | Active — read 2026-10-09 (NOTE-681); worth reading as the paper that turns the real-space mutual-information principle of LIT-873 from an iterated coarse-graining into a single-shot extraction of operators: with the mutual information estimated by a neural InfoNCE bound and the coarse-graining a convolutional filter trained against it, the filters that best compress what a block shares with its distant environment are, across the whole phase diagram of the interacting dimer model, the columnar and plaquette order parameters below the BKT transition and the electric fields at high temperature, and the plaquette filter’s correlator gives a scaling dimension of 1.00037 against the predicted 1. The maximal information itself, log 4 in the ordered phase and decaying algebraically with the buffer in the critical one, traces the phase diagram. The theorem that the optimal filters are the most relevant operators is cited to another paper, and the model’s operators were known beforehand. |
| LIT-879 v2 | Hyperbolic Neural Networks | Ganea, Bécigneul and Hofmann (NeurIPS 2018), ARXIV-1805.09112. Builds neural-network layers in the Poincaré ball from Möbius gyrovector operations: closed-form exponential and logarithmic maps, parallel transport from the origin as a scaling, geodesic hyperplanes with a closed-form point-to-hyperplane distance, and from them hyperbolic logistic regression, feed-forward layers and GRUs, all reducing to the Euclidean ones as the curvature goes to zero. Linear maps and nonlinearities are lifted through the origin’s tangent space, so the curvature acts only through biases and the output distances. Gains over Euclidean models are clear only on a synthetic tree-like task and on separating WordNet subtrees in low dimensions. | Active — read 2026-10-09 (NOTE-686); worth reading as the paper that gives the Poincaré ball the operations a neural network is built from, and so shows exactly where hyperbolic geometry enters such a network and where it does not. From Ungar’s gyrovector formalism (Möbius addition and scalar multiplication) it derives the exponential and logarithmic maps at any point, parallel transport from the origin as the scaling λ₀/λ_x (Theorem 4), geodesic hyperplanes as exp_p of a tangent orthogonal complement, and the closed-form distance to one (Theorem 5); all of these check out numerically. A linear map or pointwise nonlinearity is lifted as exp₀ ∘ f ∘ log₀, so a stack of such layers is one Euclidean network in log₀ coordinates; the curvature acts only through the bias translations between layers and the distance-based softmax. The evidence is small: 5-dimensional sentence encoders on SNLI, where hyperbolic models are on par or worse, and on a synthetic noisy-prefix task, where they win most when the noise is lowest, each cell the best of three runs; and hyperbolic logistic regression separating WordNet subtrees in pre-trained Poincaré embeddings better than Euclidean hyperplanes do, mostly in 2 to 5 dimensions. |
| LIT-880 | How Deep Networks Learn Sparse and Hierarchical Data: the Sparse Random Hierarchy Model | Tomasini and Wyart (2024; ICML 2024, PMLR 235). Pads each informative feature of the Random Hierarchy Model with uninformative positions, so the label is unchanged by discrete displacements of features, a stand-in for smooth deformations. Deep networks learn this sparse model from a number of examples polynomial in the input dimension, measured as (s0 + 1)^L n_c m^L (times s^(L/2)) without weight sharing and (s0 + 1)^2 n_c m^L with it, and become insensitive to synonym swaps and to feature displacements at the same training-set size as they learn the task, which the paper offers as the reason deformation stability tracks test error. Fitted laws; the image case is by analogy. | Active — read 2026-10-09 (NOTE-684); worth reading as the sparse extension of the Random Hierarchy Model (LIT-877) and as a model-based account of why a network’s insensitivity to small smooth deformations of its input tracks its test error. Each informative feature of the RHM is padded with s0 uninformative “empty” positions and may sit anywhere in its sub-patch, so the data become very sparse (s^L informative positions among (s(s0 + 1))^L) and the label is unchanged by discrete displacements of the informative features, a stand-in for diffeomorphisms of an image. On this Sparse Random Hierarchy Model, deep locally connected and convolutional networks learn from a number of examples polynomial in the input dimension: measured, P* ≈ s^(L/2)(s0 + 1)^L n_c m^L without weight sharing and P* ≈ C(s0 + 1)^2 n_c m^L with it. At about that training-set size the hidden layers become insensitive both to swapping synonymous tuples and to displacing features, so the two invariances and good performance arrive together; the paper offers this as the explanation of the correlation between deformation stability and test error measured on CIFAR-10. The laws are fitted, the (s0 + 1)^L factor has a one-step-gradient argument for locally connected networks, the CNN’s (s0 + 1)^2 is unexplained, and the image case is an analogy: no synonym sensitivity was measured on images. |
| LIT-882 | An exact mapping between the Variational Renormalization Group and Deep Learning | Mehta and Schwab (2014), arXiv preprint, never published in a journal. Identifies Kadanoff’s variational RG kernel with an RBM’s energy by T = −E + H, under which the coarse-grained Hamiltonian is the RBM’s hidden-marginal Hamiltonian for any parameters and an exact RG step is a perfect fit of the data. Illustrates it with a hand-built decimation network for the 1D Ising chain and a four-layer stacked RBM trained on 2D Ising samples near T_c, whose receptive fields are local and grow with depth. It concedes that, short of exactness, RG and RBM training use different variational schemes; LIT-873 shows the difference matters. | Active — read 2026-10-09 (NOTE-683); worth reading as the paper that first put deep restricted Boltzmann machines and Kadanoff’s variational renormalization group side by side, and as the position Koch-Janusz and Ringel (LIT-873) argue against. Its “exact mapping” is an identification: setting the RG kernel T = −E + H makes the coarse-grained Hamiltonian equal the RBM’s hidden-marginal Hamiltonian for every parameter value, and makes an exact RG step the same thing as a perfect fit of the data distribution. Away from that point the paper itself says the two procedures use different variational approximations, so the mapping does not show that training an RBM performs RG. The evidence that it does is a hand-built 1D decimation network and qualitative receptive fields of a stacked RBM on 2D Ising samples; no flow, fixed point or exponent is computed. |
| LIT-883 | Towards a theory of how the structure of language is acquired by deep neural networks | Cagnetta and Wyart (2024), NeurIPS 2024. In sequences generated by the Random Hierarchy Model, the correlation between two tokens shrinks by a factor of about m per level of their lowest common ancestor, and a sample of P sequences resolves it only above a noise floor of (v²P)^(−1/2), so a finite training set acts as an effective context window t*(P). Deep next-token learners show loss steps at P_ℓ ≈ v m^(2ℓ−1), where their representations come to encode the level-ℓ hidden symbol, and the steps add up to a power law. On character-level text the measured correlation decay predicts where the loss for a given context length levels off. | Active — read 2026-10-09 (NOTE-687); worth reading as the self-supervised sequel to the Random Hierarchy Model (LIT-877), and the first of that line in which the latent hierarchy is read off a co-occurrence matrix of observable tokens. For sequences generated by the RHM, the covariance between two token positions has zero mean over draws of the grammar and a root-mean-square size set by the depth ℓ of their lowest common ancestor, about ((1 − m/v^(s−1))/(v³ m^(2ℓ−1)))^(1/2) (derived for large v and m); the estimate from P samples carries noise of size (v²P)^(−1/2) (proved by a central-limit argument), so a finite training set resolves correlations only out to an effective distance t*(P). Deep transformers and CNNs trained to predict the last token show loss steps at about P_ℓ = v m^(2ℓ−1), where their hidden layers become insensitive to resampling the subtree below a level-ℓ symbol. That the networks use the correlations to build the hidden variables is argued, not proved. On character-level Shakespeare and WikiText-103 the measured correlation decays as t^(−β), and loss curves for different context lengths collapse under the t*(P) ~ P^(1/(2β)) it implies. No embedding, PMI or linear direction is studied. |
| LIT-884 | A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data | Sclocchi, Favero and Wyart (2024; PNAS 122, e2408799121, 2025). On the Random Hierarchy Model with known rules, Bayes-optimal denoising is belief propagation on the tree; in an annealed mean-field approximation, checked numerically, the probability of recovering the class drops sharply at a threshold noise as depth grows, provided the grammar is sparse (s f < 1), while low-level features degrade smoothly and can survive into a sample of a new class. Forward–backward runs of an unconditional ImageNet diffusion model show the same pattern in a classifier’s layers; a two-mode Gaussian mixture does not. | Active — read 2026-10-09 (NOTE-685); worth reading as the paper that turns the Random Hierarchy Model (LIT-877) from a learning problem into an inference problem: given the grammar, the Bayes-optimal denoising of a noisy datum is belief propagation on the tree, and in an annealed mean-field approximation the probability of recovering the root (the class) has a sharp threshold in the noise as the depth grows, while leaves and low latents degrade smoothly, so past the threshold a new class can be built from old low-level parts. The threshold exists only when the grammar is sparse, s f < 1 (f the fraction of s-tuples a level uses); in the dense case, where every string occurs, any noise eventually loses the class. A forward–backward experiment with an unconditional ImageNet diffusion model shows the same pattern in a classifier’s layers: the logits’ similarity drops sharply near t ≈ T/2 while early layers change smoothly and cross over. A Gaussian mixture shows the drop but not the crossing. The threshold is not proved; the trained diffusion model is not shown to compute belief propagation. |
| LIT-886 | Probing the Latent Hierarchical Structure of Data via Diffusion Models | Sclocchi, Favero, Levi and Wyart (ICLR 2025; JSTAT 2025, 084005). In the Random Hierarchy Model with known rules, the changes that noising and Bayes-optimal denoising make to a datum come in blocks set by which latent changed, and an annealed mean-field calculation, checked against exact belief propagation, gives a correlation length that diverges as a power of the distance to the class threshold, with a susceptibility peak there; a Gaussian field with a power-law spectrum shows none. The same peak appears at an intermediate noise in a masked diffusion language model on WikiText-2 and in an ImageNet DDPM read through CLIP patch embeddings. | Active — read 2026-10-09 (NOTE-691); worth reading as the sequel to LIT-884 that moves from how likely each level of a hierarchical datum is to survive noising and denoising to how the changes are arranged in space. In the Random Hierarchy Model with known rules, an annealed mean-field calculation, checked against exact belief propagation, gives changes that come in correlated blocks, because a changed latent at depth ℓ rewrites about s^ℓ tokens together; near the class threshold ε* the number of levels over which the evidence stays undecided diverges, so the correlation length grows as |ε − ε*|^(−ν) with ν = log s / log F′(p*), and the dynamical susceptibility (the summed correlations) peaks there. A Gaussian random field with a power-law spectrum, by contrast, has a correlation length that grows monotonically to the end of the process. The same peak is then measured in forward–backward runs of a masked diffusion language model on WikiText-2 (near t ≈ 0.6 T) and of an ImageNet DDPM read through CLIP patch embeddings (near t ≈ 0.6–0.7 T). The divergence is a scaling argument within the approximation, checked by one data collapse at one parameter set; the real-data correlations span about one decade of distance and have no non-hierarchical control. |
| LIT-887 v2 | Why does deep and cheap learning work so well? | Lin, Tegmark and Rolnick (2016; Journal of Statistical Physics 168, 1223–1247, 2017). Argues that networks are cheap because physical data have low-order, local, symmetric Hamiltonians, and deep because data come from Markov generative hierarchies whose minimal sufficient statistics compose. Proves that a single hidden layer needs exactly 2^n neurons to multiply n inputs, against about 4n in a deep tree, and gives linear no-flattening examples. Holds that RG is supervised feature extraction, not unsupervised learning; arXiv v1–v2 carried a counterexample against LIT-882 that hit a typo in its Eq. 8, and was dropped later. | Active — read 2026-10-09 (NOTE-688); worth reading as the third position in the dispute over whether deep learning performs renormalization, and for two results that stand on their own. On the dispute it holds that RG is “essentially a feature extractor for supervised learning”: a coarse-graining is a renormalization only once the variables to keep are specified, and fitting the data distribution does not specify them. That holds, and it agrees with what LIT-882 concedes and LIT-873 shows; its sharper form was in an appendix of arXiv v1 and v2 only, dropped from v3 on and from the journal text, showing that a matched partition function does not give a matched distribution and (v2) that the exact trace condition is met by hidden variables that do not interact with the system at all. Its conclusion that RG has “little to do” with unsupervised learning is too strong: LIT-873 gets the relevant variables without labels, from the far environment. On its own merits: a two-variable product needs four neurons and an n-variable product needs exactly 2^n in one hidden layer against about 4n in a deep tree (proved, in a Taylor-matching sense, with weights that grow without bound); the minimal sufficient statistics of a Markov generative hierarchy compose level by level (proved, and nearly immediate); that physics favours low-order, local, symmetric Hamiltonians, and that this is why cheap networks work, is argued, not shown. |
| LIT-888 | Deriving Neural Scaling Laws from the statistics of natural language | Cagnetta, Raventós, Ganguli and Wyart (2026), ICML 2026. The data-limited scaling exponent of a language model’s test loss is predicted from two measured statistics of the corpus: β, the decay of token–token covariance with lag, which sets the usable context n*(P) ~ P^(1/(2β)), and γ, the decay of next-token conditional entropy with context length. When learning within that context is fast, the loss falls as P^(−γ/(2β)); on TinyStories and WikiText-103 transformers match the predicted 0.185 and 0.141 up to about 10^8 tokens. | Active — read 2026-10-09 (NOTE-689); worth reading as the sequel to LIT-883 that takes its effective-context argument from character-level toy runs to subword-tokenised text and a predicted exponent. Two statistics of a corpus are measured: β, the decay with lag n of the largest singular value of the token–token covariance matrix (‖C(n)‖op ~ n^(−β)), and γ, the decay of the next-token conditional entropy with context length (H_n − H∞ ~ n^(−γ), a differential form of Hilberg’s hypothesis). A signal-to-noise argument gives the horizon n*(P) ~ P^(1/(2β)) a sample of P tokens can use; if learning within that horizon is fast, the test loss falls as P^(−γ/(2β)), and in general as P^(−min(δ, γ/(2β))), with δ the decay of the within-horizon excess loss (App. A, an ansatz-based derivation, not a theorem). Measured on TinyStories (β ≈ 0.88, γ ≈ 0.325, predicted α ≈ 0.185) and WikiText-103 (β ≈ 0.94, γ ≈ 0.265, α ≈ 0.141) with GPT-2- and LLaMA-style transformers up to about 10^8 tokens; the empirical exponents fall inside the predicted intervals and the n-gram losses collapse under P/n^(2β), L_n n^γ. γ is read off the n-gram losses of the largest trained models, not computed from the text alone. Usable horizons reached are a few tens of tokens. No embedding, PMI or hierarchy is studied. |
| LIT-889 v2 | Symmetries and phase diagrams with real-space mutual information neural estimation | Gökmen, Ringel, Huber and Koch-Janusz (2021; Phys. Rev. E 104, 064106), the long companion of LIT-878. Derives the RSMI-NE estimator in full and applies it to the 2D Ising model (1 bit below T_c, exponential decay with the buffer above it, boundary filters at T_c), the interacting dimer model and a 1D non-equilibrium chipping model. Its new object is the ensemble of optimal filters from independent runs: it maps an RSMI-degenerate subspace whose projections show broken and emergent (U(1)) symmetries, and PCA of it recovers the pristine operators. Every model’s answer was known beforehand. | Active — read 2026-10-09 (NOTE-692); worth reading as the long companion of LIT-878 and the place where the real-space mutual-information method is shown on more than one model and its outputs are read as an ensemble: the maximal information counts the ordered sectors (exactly 1 bit in the 2D Ising ferromagnet, 2 bits in the columnar dimer phase), decays exponentially with the buffer in the Ising paramagnet and algebraically at criticality, and the optimal Ising filter goes uniform, boundary and random across T_c; 500 independent runs per temperature map an RSMI-degenerate subspace of filters whose projections show the broken C4 and translation symmetries of the dimer model and, at high temperature, its emergent U(1), and PCA of the ensemble recovers the pristine operators from a window where they never appear alone. The number of coarse variables is chosen where the information stops growing. A non-equilibrium chipping model is a one-figure validation, and its formal status is said to be missing. |
| LIT-890 | The power of deeper networks for expressing natural functions | Rolnick and Tegmark (2017; ICLR 2018). Extends LIT-887’s 2^n result: one hidden layer of smooth, bias-free units needs exactly ∏(rᵢ+1) neurons to Taylor-approximate x₁^r₁⋯xₙ^rₙ, a deep network O(Σ log rᵢ); sparse polynomials inherit the gap up to a factor 1/c; at depth k a product of n inputs costs at most about 2^(n^(1/k)), conjectured tight. The uniform-approximation lower bounds rest on an unproved step, and the gap grows with degree, not with the number of inputs. | Active — read 2026-10-09 (NOTE-693); worth reading as the sequel that turns LIT-887’s one depth-separation theorem into a family, and for where the family stops. Proved, for networks without biases and smooth activations with nonzero Taylor coefficients: one hidden layer needs exactly ∏(rᵢ+1) neurons to Taylor-approximate the monomial x₁^r₁⋯xₙ^rₙ, against Σ(7⌈log₂ rᵢ⌉+4) for a deep network; a sparse polynomial with c monomials costs at least 1/c of its dearest monomial; xᵈ costs d+1 shallow against 7⌈log₂ d⌉ deep; and a product of n inputs costs O(n^((k−1)/k)·2^(n^(1/k))) at depth k. The move to uniform approximation, its headline over LIT-887, rests for the lower bounds on a step the proof asserts and that is false in general (a network can be uniformly within ε of a polynomial while its Taylor coefficients diverge), so only the Taylor-sense lower bounds are established. That depth k needs 2^Θ(n^(1/k)) neurons is a conjecture, backed by one trained-network figure at n = 20 with tanh, outside the theorems’ hypotheses. The gap is exponential in a monomial’s degree, not in the number of inputs: polynomials of bounded degree, such as the low-order Hamiltonians LIT-887 calls natural, show none. |
| LIT-891 | Comment on “Why does deep and cheap learning work so well?” | Schwab and Mehta (2016), arXiv comment, never published elsewhere. Reply to the appendix counterexample of Lin and Tegmark’s arXiv v1 (LIT-887): agrees that preserving the free energy does not recover the distribution, calls the “if and only if” of LIT-882’s Eq. 8 a typo (the trace condition implies ΔF = 0, not conversely), and shows that the counterexample violates the trace condition, so the mapping’s exact step is untouched. It does not reach the later objection that the trace condition fails to select the coarse variables. | Active — read 2026-10-09 (NOTE-694); worth reading as Mehta and Schwab’s own statement of how far their mapping (LIT-882) reaches, made in reply to the counterexample in arXiv v1 of Lin and Tegmark (LIT-887). It concedes two things: that preserving the free energy does not recover the distribution (“we entirely agree with this statement and never claimed otherwise”), and that the “if and only if” of LIT-882’s Eq. 8 is a typo, since the trace condition implies ΔF = 0 but not conversely. It argues one: that exact variational RG means the trace condition Tr_h e^T = 1 for every v, which the counterexample violates (checked), so the mapping’s Eq. 22 is untouched. That reply is right against v1. It locates the mapping’s content at the trace condition alone, which is the scope THEORY-197 gives it, and it does not answer the point Lin and Tegmark added in their v2 a fortnight later, that the trace condition is also met by hidden variables that do not interact with the system (THEORY-202). The “typo” did work in LIT-882: its Section I derives the exactness condition from ΔF = 0 through the direction the comment withdraws. |
| LIT-909 | Discovering Agents | Kenton, Kumar, Farquhar, Richens, MacDermott & Everitt (2022). Agents are “systems that would adapt their policy if their actions influenced the world in a different way”. In mechanised causal models, which add mechanism variables, decision and utility nodes are found from interventions on mechanisms, with proofs of recovery under five assumptions. Verdicts: a thermostat is not an agent, but the thermostat with its creation process is; humans are agents; an actor-critic is two agents. The labels are frame-dependent. | Active — read in full 2026-10-11 (NOTE-707) from the arXiv PDF (v2). Worth reading for the interventional definition of agents, as systems that would adapt their policy if their actions influenced the world differently, and for its verdicts: a thermostat is not an agent, but the thermostat with its creation process is; the training process, not the frozen policy, is the agent. It is explicit that its labels are frame-dependent, relative to the chosen variables. Its proofs assume interventional data on mechanisms. |
| LIT-911 | Agents and Devices: A Relative Definition of Agency | Orseau, McGill & Legg (2018). A system is described either as a device (an input-output mapping) or as an agent (the function it optimizes). Bayes’ rule over mixtures of each gives the observer’s posterior probability that the system is an agent. In a gridworld, running in circles and wall-following come out devices, and heading for a goal, even suboptimally or with rare goal switches, comes out an agent. Agency here is explicitly observer-relative. | Active — read in full 2026-10-11 (NOTE-705) from the arXiv PDF (v1). Worth reading as the cleanest formal version of Dennett’s intentional stance: whether a system is an agent or a device is a Bayesian model comparison between a mixture of goal-optimizing descriptions (inverse RL) and a mixture of input-output mappings, made by an observer with a prior. It presupposes an identified system, and its results depend on the prior chosen. |
| LIT-918 v2 | Neural networks and physical systems with emergent collective computational abilities | Hopfield (1982), PNAS 79, 2554. Threshold units with symmetric Hebbian couplings, updated asynchronously, descend an energy function, so stored random patterns are attractors and a partial or noisy cue settles into the nearest one. In simulations at N = 30 and 100, recall degrades near 0.15N patterns, memories closer than about a tenth of N fuse, and overload makes every memory irretrievable unless old ones are forgotten. | Active — read 2026-10-14 (NOTE-819) in full; worth reading as the paper that made memory a property of a flow: with symmetric couplings the asynchronous dynamics descend an energy function, Hebbian storage makes random patterns attractors, and a cue a few bits off settles into the nearest one. Read it knowing that every number in it (the 0.15N capacity, the basin width, the fusion of memories closer than about a tenth of N, the loss of all memories under overload) comes from simulations at N = 30 and 100, and that its “stable states of no obvious meaning” appear at a low load, not only past capacity. |
| LIT-935 v2 | The Origins of Representation Manifolds in Large Language Models | Modell, Rubin-Delanchy & Whiteley (2025). A feature is modelled as a compact metric space and its representation as a map onto the unit sphere. If the map is continuous and injective, the representation is homeomorphic to the feature. If cosine similarity is also locally a smooth decreasing function of squared feature distance, every path on the representation manifold is a fixed multiple of the feature path’s length, so on-manifold geodesics recover the feature’s intrinsic geometry. Both conditions are assumed. Tests on GPT-2-small’s years and on text embeddings of dates and colours find homeomorphism, and isometry for years only on a logarithmic scale. | Active — read 2026-10-14 (NOTE-802) in full; worth reading for Theorem 1: if a feature is a compact metric space mapped continuously and injectively onto the unit sphere, and cosine similarity is locally a C² decreasing function of squared feature distance, path lengths on the representation equal the feature’s up to the scale √(−2g′(0)), so geodesics recover its intrinsic geometry. Read it knowing that both hypotheses are assumed rather than derived, that the title’s “origins” are an informal argument, and that the tests supply a candidate metric by hand: GPT-2’s years are homeomorphic to a linear and a logarithmic scale, and only the metric test separates them. Held as THEORY-284. |
| LIT-936 v3 | The information geometry of large language models is shared, learned, and controllable | Picozzi (2026), arXiv preprint. Each model is given the matrix of Fisher–Rao distances between its next-token laws over a fixed context battery. That matrix is a function of behaviour alone, so it is invariant to output-preserving reparameterisation, which activation geometry is not. Across ten models of three architecture kinds it agrees at a mean rank correlation of 0.88, against 0.62 for activations. This is proved to follow from each model’s small excess risk to a common law, and it is measured to come from model disagreement lying nearly orthogonal to the relational structure. The pulled-back metric gives minimum-disturbance steering whose advantage over Euclidean steering it predicts (median measured/predicted 0.967). | Active — read 2026-10-14 (NOTE-755) in full; worth reading as a measured, coordinate-free relational signature. The matrix of Fisher–Rao distances between a model’s next-token laws over a context battery is fixed by behaviour and invariant to any output-preserving reparameterisation, while activation geometry has a GL(d) gauge. It agrees across ten transformer, state-space and recurrent models at a mean rank correlation of 0.88, against 0.62 for mid-layer activations, and a probe on distances to 40 anchors transfers between models. Read it knowing three things. The signature is identified because it is a function of the observed laws, the in-probe case QUESTION-027 calls automatic. Its agreement is proved only as a consequence of each model fitting the same law, on a shared outcome space, and only about 9% of shared eigenvector structure goes beyond what the shared laws imply. It is a single-author v1 preprint whose code is on request. Most of its length is natural-gradient steering, editing and fine-tuning, which is machine-learning practice. |
| LIT-937 v2 | Post-Hoc Concept Disentanglement: From Correlated to Isolated Concept Representations | Erogullari et al. (2025). Concept activation vectors fitted one concept at a time come out aligned for attributes that co-occur in the labels, such as “beard”, “mustache” and “necktie” in CelebA, in blocks matching the label correlations. A joint fit with a non-orthogonality penalty ‖CCᵀ − I‖²_F finds mutually orthogonal directions that still separate each concept, with AUROC largely preserved, at the last convolutional layer of VGG16 and ResNet18. So the alignment comes from the separate fits, not from the representation. Steering along the orthogonal directions inserts or removes one concept with less effect on correlated ones. | Active — read 2026-10-14 (NOTE-816) in full; worth reading as a clean case where concept directions for correlated attributes align only because each is fitted alone: a joint fit with an orthogonality penalty finds mutually orthogonal directions that still separate each concept with AUROC largely preserved, at the last convolutional layer of VGG16 and ResNet18 on CelebA and FunnyBirds. Read it knowing that every result is in a figure with no values or error bars, that AUROC is the only check that a direction keeps its meaning, and that the attributes are correlated, not hierarchical. Registered for QUESTION-025, where it bears on the correlated branch only; THEORY-292 is filed from it. |
| LIT-938 v2 | Evaluating Compositional Scene Understanding in Multimodal Generative Models | Fu et al. (2025), TMLR. Human raters judged DALL-E 3 images against relational prompts. It beats DALL-E 2 on every familiar two-entity relation. Agreement falls when subject and object of an asymmetric relation are swapped, and for most prompts combining two relations among three entities. Low-agreement reversed images mostly show the wrong relation or swapped roles. Five vision-language models learn relational concepts few-shot (Bongard-HOI, SVRT) below human level, and four of them are at chance on six-object concepts. VLM judges correlate poorly with human ratings. | Active — read 2026-10-14 (NOTE-813) in full; worth reading as a human-rated measurement of relational binding in one text-to-image model: DALL-E 3 renders familiar single relations well, loses them when the roles of an asymmetric relation are reversed or two relations are combined, and fails on the relation or the roles rather than the objects. Read it knowing that it tests one generator, that its experiments differ in style, raters and prompt selection, and that its explanation by training-data likelihood rests on a correlation of 0.09. Filed for QUESTION-018; the finding is held as THEORY-250. |
| LIT-940 v2 | Optimal scaling laws in learning hierarchical multi-index models | Defilippis, Krzakala, Loureiro and Maillard (2026). Take Gaussian inputs, n = αd samples, and a hierarchical multi-index target with even links and weights a_k ~ k^(−γ). Direction k becomes weakly recoverable at α_k = Θ(a_k^(−2)), so features emerge in sequence as the sample size grows, through sharp spectral transitions. The Bayes-optimal subspace error is Θ(α^(−1+1/(2γ))) until α ≍ m⋆^(2γ), then Θ(m⋆/α). A target-agnostic spectral estimator attains it. So does a two-layer network built on that estimator with a ridge readout. No training trajectory is analysed. | Active — read 2026-10-14 (NOTE-805) in full; worth reading as an exact theory of when a feature becomes learnable. Take Gaussian inputs and a target Σ a_k g_k(⟨w_k, x⟩) with even links and a_k ~ k^(−γ). Direction k becomes weakly recoverable at n = Θ(a_k^(−2) d), so features emerge one by one as spikes leaving a random-matrix bulk. A target-agnostic spectral estimator attains the Bayes-optimal rates, Θ(α^(−1+1/(2γ))) and then Θ(m⋆/α). Read it knowing that the cascade is in the number of samples, not in training time. The link to gradient descent is a remark citing other work. The network theorem is for a spectral first layer plus a ridge readout on split data. Held as THEORY-283. |
| LIT-941 v2 | A Theory of Unsupervised Translation Motivated by Understanding Animal Communication | Goldwasser et al. (2022; NeurIPS 2023). A theoretical framework for unsupervised translation with no parallel data and no assumption that the corpora share domain or structure: maximum likelihood against a prior that scores how plausible a target text is. In two stylised random models it proves error bounds that fall with common ground, source-language complexity and samples, with a matching lower bound in one, err = O(log|Θ| / (α·min(m,|T|))), and checks them on synthetic data. Information-theoretic, lossless and uniform. | Active — read 2026-10-14 (NOTE-804) in full; worth reading as the first theory of translation without parallel data: maximum likelihood against a plausibility prior learns the translator with an Occam-type bound divided by the common ground, err = O(log|Θ| / (α·min(m,|T|))) in its common-nonsense model, tight up to a constant. Read it knowing that its models are uniform, random and lossless, that the bounds are information-theoretic, and that the prior substitutes for parallel data in learning a decoder, not for information in a message. Filed THEORY-270; bears on QUESTION-020 without answering it. |
| LIT-942 v2 | Do different prompting methods yield a common task representation in language models? | Davidson et al. (2025), NeurIPS 2025. The paper extends function vectors from few-shot demonstrations to short instruction prompts and extracts instruction function vectors that improve zero-shot accuracy. Demonstration and instruction function vectors share only 4 to 7 of their top 20 attention heads, add rather than substitute, and lose accuracy when their heads and activations are swapped. It concludes that the two prompting forms do not induce a common task representation but partly overlapping mechanisms, and that the instruction route is built by post-training. | Active — read 2026-10-14 (NOTE-803) in full; worth reading as the direct test of whether a task given by demonstrations and the same task given by an instruction share one internal representation. Under function vectors they do not: mostly different heads, vectors that add, and an asymmetric cost to swapping them. Read it knowing that it tests prompt forms, not paraphrase or stance, on simple lexical tasks in small models, and that the authors say they cannot prove the negative. Filed THEORY-264; bears on QUESTION-024 without answering it. |
| LIT-943 v2 | From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks | Dominé, Anguita, Proca, Braun, Kunin, Mediano & Saxe (2025), ICLR. For a two-layer linear network with whitened inputs, trained by full-batch gradient flow from a λ-balanced start, W₂ᵀW₂ − W₁W₁ᵀ = λI, the dynamics of the network function, representations and finite-width NTK are solved exactly. The solution is a matrix Riccati equation. The relative scale λ is conserved and sets the regime. As λ → 0 learning is rich: sigmoidal, stage-like, with task-specific representations. As |λ| → ∞ it is lazy: exponential, with a static NTK. Its sign, together with the architecture, decides which layer stays task-agnostic, and in funnel networks it gives a delayed rich phase. Wide LeCun-initialized networks are λ-balanced in probability. | Active — read 2026-10-13 (NOTE-715): main text in full, and the statement of Appendix A.3 (random initializations are λ-balanced) without its proof; the proofs in Appendices B–C not checked. Its control parameter is the relative scale of the layers, not the absolute scale the seed entry named. The account filed from it is THEORY-212, and the question it raises under SGD is QUESTION-081. |
| LIT-944 v2 | Questioning Representational Optimism in Deep Learning: The Fractured Entangled Representation Hypothesis | Kumar, Clune, Lehman & Stanley (2025), a position paper. Three image-generating networks found by Picbreeder’s human-guided evolution are rewritten as dense MLPs. Networks of the same architecture are trained by SGD to reproduce the same images, and they do. The SGD networks show no reuse of the symmetric parts, and sweeping their weights distorts the image, where the evolved networks’ sweeps vary it meaningfully. The paper calls this fractured entangled representation, argues that open-ended search approaches a unified factored one, and conjectures that FER in large models harms generalization, creativity and continual learning. The evidence is qualitative. | Active — read 2026-10-14 (NOTE-771) in full; worth reading for its side-by-side comparison of networks that compute the same image in the same architecture: SGD reproduces three Picbreeder images without the evolved networks’ reuse of symmetric parts or factoring of independent ones, and perturbing its weights breaks the regularities that perturbing theirs preserves. Read it knowing that it is a position paper, that the evidence is curated neuron maps and weight sweeps for three images with no measure of fracture, that the evolved networks were selected by users rather than trained on the image, and that its explanation by open-ended search and its extension to large models are conjecture. Held as THEORY-241. |
| LIT-946 v2 | The Geometry of Prompting: Unveiling Distinct Mechanisms of Task Adaptation in Language Models | Kirsanov et al. (2025). Using manifold capacity from statistical physics, the paper studies how an instruction, demonstrations and soft prompts change the geometry of representations in base decoder-only models. Methods with similar accuracy act differently: demonstrations reshape intermediate representations of the input, driven by the example sentences rather than their labels; an instruction acts mainly on the last token; soft prompts act late. Few-shot failures from example choice or relabelled categories leave the classes separable, which the authors attribute to the readout. Demonstrations for one task help other tasks’ separability early and hurt it late. | Active — read 2026-10-14 (NOTE-789) in full; worth reading for its split of accuracy into representational separability and readout: demonstrations reshape how the input itself is represented, driven by the example sentences rather than their labels, an instruction acts mainly on the last token, and few-shot failures from example choice or relabelled categories leave the classes separable. Read it knowing that it uses base models, one instruction wording and mostly synthetic data, and infers the readout failure rather than measuring it. Filed THEORY-251, and supports THEORY-264 with LIT-942. |
| LIT-948 v2 | Distances for Markov chains from sample streams | Calo et al. (2025). The bisimulation metric between two finite Markov chains, in its bicausal optimal-transport form (LIT-962), is the value of a new linear program whose constraints use each chain’s discounted occupancy measure over consecutive state pairs in place of its transition kernel. A stochastic primal-dual method on it, SOMCOT, estimates the metric to accuracy ε with probability 1−δ from O((|X||Y|(|X|+|Y|) + (|X|+|Y|)log(1/δ))/((1−γ)²ε²)) independent transition samples per chain, with no model of the dynamics. The samples are independent draws from the occupancy measures, which needs restarts from the initial state. The states must be observed and enumerable, and the ground cost known. Experiments on block-structured random walks and a discretised pendulum. | Active — read 2026-10-14 (NOTE-763) in full; worth reading as the first estimator of the bisimulation metric between finite Markov chains with a guarantee and no transition model. A reformulated LP needs only each chain’s sampled occupancy of consecutive state pairs, and a stochastic primal-dual method reaches accuracy ε with probability 1−δ from O((|X||Y|(|X|+|Y|) + (|X|+|Y|)log(1/δ))/((1−γ)²ε²)) samples per chain. Read it knowing that the samples are independent draws from the discounted occupancy measure, which needs restarts, not one running stream. The states must be observed and enumerable, and the cost on state pairs given. The chains have no actions. Filed as THEORY-280. |
| LIT-950 v3 | All or None: Identifiable Linear Properties of Next-token Predictors in Language Modeling | Marconato, Lachapelle, Weichwald & Gresele (2024; AISTATS 2025). Two softmax next-token predictors induce the same distribution exactly when an invertible linear map relates them on the subspaces M and N that enter the logit dot product. Off those subspaces any distortion is allowed, at any representation dimension and without the diversity condition. Relational linearity, and with it linear probing and linear subspaces, then holds in all equivalent models or none when Γ ⊆ N and Γ_q ⊆ M. Parallelism of difference vectors survives only inside N. Theory only. | Active — read 2026-10-14 (NOTE-746) in full; worth reading as the exact characterisation of which softmax next-token predictors induce the same distribution: an invertible linear map on the subspaces M and N that enter the logits, with arbitrary distortion off them. Its all-or-none theorem for relational linearity, linear probing and linear subspaces is the record’s clearest proved case of structure identified when the full latent model is not. Read it knowing that the result needs exact equality of distributions over every sequence, a shared vocabulary and the conditions Γ ⊆ N and Γ_q ⊆ M, that parallelism survives only inside N, and that there are no experiments. For QUESTION-027 it is an instance of the unrestricted, already answered form, since the whole distribution is observed and no decision lies outside the probes. |
| LIT-951 v2 | Physics of Skill Learning | Liu, Liu, Michaud, Gore and Tegmark (2025, preprint). In a toy model of independent skills, the Geometry model, gradient descent learns orthogonal skills together, each in time ∝ 1/p. Sign descent makes them compete for coordinate signs, so each starts as the one before finishes (the Domino effect), and the n-th takes time ∝ n. A one-parameter Resource ODE fits this and conserves u_i^(1/p_i). In a small MLP, a parity built from two others waits for them. The evidence is simulation and small experiments. | Active — read 2026-10-14 (NOTE-794) in full; worth reading for one contrast in a toy model of independent skills, the Geometry model. Under gradient descent, orthogonal skills are learned together, each in time ∝ 1/p. Under sign descent they compete for coordinate signs, and each starts as the one before finishes. That is the Domino effect, and it makes the n-th skill take time ∝ n. A separate MLP experiment shows a second cause of sequential onset, dependency between skills. Read it knowing that the Domino effect is shown in simulation, that the Resource and Domino models are fitted to it rather than derived, that the order is frequency order under every optimizer, and that the real-network evidence is a few small MLPs trained with Adam. Held as THEORY-253. |
| LIT-952 v2 | Neural Networks Learn Statistics of Increasing Complexity | Belrose, Pope, Quirke, Mallen, Dury and Fern (ICML 2024). Vision networks trained on real images, probed during training with images whose class means or covariances are grafted from another class and with maximum-entropy samples keeping only those statistics, act on them early (peaks between about 2^4 and 2^12 steps) and lose that later. First-order sensitivity peaks before second-order. Pythia fits unigram-sampled text before bigram-sampled text. The n-gram/moment equivalence is an identity; the ordering is empirical, for orders one and two. | Active — read 2026-10-14 (NOTE-777) in full; worth reading for its probes of what a network is using at a given checkpoint, which are maximum-entropy samples and optimal-transport grafts of class statistics, and for the result they give. Vision networks act on class means and covariances early in training and lose that later, with first-order sensitivity peaking before second-order. Pythia fits unigram text before bigram text. Read it knowing that the ordering is empirical and covers orders one and two, that the Taylor argument for it is a heuristic, and that its n-gram/moment theorem is an identity of one-hot indicators. Held as THEORY-279. |
| LIT-954 v2 | The semantic landscape paradigm for neural networks | Gokhale (2023). A conceptual proposal with no experiment and no proof. Training is pictured as a walk on a graph whose nodes are the algorithms (“heuristic models”) a network can implement, ranked by a fallibility F, the KL divergence from the truth over all inputs. Grokking’s critical dataset size and emergence with scale are argued to be connectivity thresholds of that graph, called percolation; power-law scaling in training time, data and size is derived from an assumed trap model with exponentially distributed barriers and F ∝ 1/N_b. Thresholds and scaling laws are two kinds of event in one vocabulary. | Active — read 2026-10-14 (NOTE-807) in full; worth reading only as a vocabulary: training pictured as a walk on a graph of the algorithms a network can implement, with thresholds as connectivity changes and scaling laws as trap-model waiting times. Read it knowing that it has no experiment and no proof, that its percolation is asserted without a percolation model, and that it treats thresholds and scaling laws as two kinds of event, not one. For QUESTION-039 it is a vote for several kinds in a common language; THEORY-181 holds a sharper account of grokking’s critical data size. No THEORY filed. |
| LIT-961 v2 | A Reply to Makelov et al. (2023)‘s “Interpretability Illusion” Arguments | Wu, Geiger, Huang, Arora, Icard, Potts and Goodman (2024). The authors of distributed alignment search reply to Makelov et al. (2023). They argue that the “illusion” is generic, because a direction a model uses need not be orthogonal to the null space of the next layer; that FLDD can be large with no change in behaviour; and that the factual-recall cases come from DAS trained per fact pair. Their toy counterexample holds only for an unnormalized patch. Both papers grant that alignments related by a change of basis can tie under interchange interventions. The reply’s own GPT-2 experiments agree that MLP8 carries almost no name-position information (4% IIA, against 70% for the layer-8 block output). | Active — read 2026-10-14 (NOTE-809) in full; worth reading for the point it shares with the paper it answers: interchange interventions on one set of variables can be satisfied equally by alignments related by a change of basis, so the mediating variable is fixed only relative to a choice of units. Read it knowing that its central toy computation, which claims Makelov et al.’s test rejects their own faithful direction, holds only for an unnormalized patch; patching the subspace the rowspace component spans recovers the full effect. Its own GPT-2 experiments agree that MLP8 carries almost no name-position information. The dispute is over the label, the metric (FLDD against IIA) and how DAS is trained, not over constitution against correlation. Filed, with LIT-968, as THEORY-243. |
| LIT-962 v2 | Bisimulation Metrics are Optimal Transport Distances, and Can be Computed Efficiently | Calo et al. (2024). For two finite Markov chains without actions, with real-valued state labels r and discount γ, the bisimulation metric of Desharnais et al., van Breugel and Worrell, and Ferns et al. equals the optimal-transport distance between the chains’ path laws over bicausal couplings, with ground cost (1−γ)|r(x)−r(y)|. That distance is 1/(1−γ) times the value of a finite linear program over discounted “occupancy couplings” of state-pair transitions. Entropy regularisation of the program gives Sinkhorn Value Iteration, with an O((1−γ)⁻⁵ε⁻²) iteration bound under exact inner projections, and in experiments it is faster than earlier dynamic-programming solvers. | Active — read 2026-10-14 (NOTE-757) in full; worth reading as the paper that makes approximate bisimulation a transport cost. For finite action-free chains with real-valued labels, the discounted bisimulation metric solves the same fixed-point equation as optimal transport over bicausal couplings with ground cost (1−γ)|r(x)−r(y)|, so the two are one object, and that cost is the value of a finite linear program over occupancy couplings. Read it knowing three things. Its chains have no actions, so it does not reach systems driven by utterances, which is what QUESTION-008 asks about. Its only statement about trace equivalence is second-hand and one-way. Its Sinkhorn Value Iteration guarantee needs exact inner fixed points. Filed as THEORY-247. |
| LIT-963 v2 | Fundamental Limits of Perfect Concept Erasure | Chowdhury et al. (2025), AISTATS. Under finite supports and disjoint concept groups, an erasure Z = f(X) that reveals nothing about the concept A keeps at most I(Z;X) ≤ H(X|A). A deterministic map reaches that only when the groups’ distributions are permutations of one another, by mapping each group bijectively onto a common distribution. For unequal groups a stochastic minimum-entropy-coupling map still erases perfectly with less utility. The paper says the cap is then out of reach, which holds only for deterministic maps. The method beats INLP, LEACE, FaRM, KRaM and DCA on synthetic data, GloVe and GPT-4 embeddings. | Active — read 2026-10-14 (NOTE-820) in full; worth reading for the clean information-theoretic cap on perfect concept erasure, I(Z;X) ≤ H(X|A) once Z is independent of A, and for the proof that a deterministic erasure reaches it only when the concept groups are permutations of one another. Read it knowing that it assumes finite supports and disjoint groups, so that the concept is a function of the representation, and that its claim that unequal groups cannot reach the cap is false for stochastic maps. Filed for QUESTION-010; the bound is held as THEORY-281. |
| LIT-968 v3 | Is This the Subspace You Are Looking for? An Interpretability Illusion for Subspace Activation Patching | Makelov, Lange and Nanda (2023). A subspace activation patch can make a model behave as if a feature had changed while the patched direction is the sum of a part that varies with the feature but cannot reach the output (in an MLP, the kernel of the down-projection) and a part that can reach the output but is dormant on the data. Exact in a three-unit network; shown for MLP8 of GPT-2 Small on indirect object identification and for GPT2-XL MLPs on factual recall, where the effect vanishes once the disconnected part is removed. A residual-stream direction for the same feature passes the test. One-dimensional fact patches and rank-1 edits are nearly interchangeable, so a working rank-1 edit does not show the fact is stored in the edited layer. | Active — read 2026-10-14 (NOTE-782) in full; worth reading as the clearest case of an intervention that produces the predicted counterfactual output without the intervened variable carrying the feature: a subspace patch that pairs a direction varying with the feature but unable to reach the output with one that can reach it but is dormant on the data. Read it knowing that its ground truth for GPT-2’s MLP8 comes from prior component-level work, that it concedes a cancelling-subspaces reading it does not test, that its own residual-stream case passes the test, and that the reply (LIT-961) contests the label but not the MLP8 measurements. It speaks of faithfulness to a computation, not of constitution. Filed as THEORY-282 and, with the reply, THEORY-243. |
| LIT-970 v2 | Symmetry in Neural Network Parameter Spaces | Zhao, Walters & Yu (2025), TMLR. A survey of parameter-space symmetries: transformations of a network’s parameters that leave its function, or its loss, unchanged. It catalogues them for common components (linear layers, homogeneous and other activations, normalization, softmax, attention, LoRA). It surveys when they are complete, so that parameters are identifiable up to them. Continuous symmetries stretch minima into connected manifolds and discrete ones replicate them. Under gradient flow, continuous symmetries give conserved quantities, such as layer imbalance, that label trajectories and locate the endpoint among the minima. Finite steps, gradient noise and weight decay break these conservation laws. | Active — read 2026-10-13 (NOTE-712): §§1–5 and 7–8 in full, §6 in part. A survey, so its claims about the works it cites are secondhand here. It bears on QUESTION-058 and QUESTION-069, and on the null space that THEORY-209 leaves out. The account filed from it is THEORY-210. |
| LIT-971 v2 | Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data | Gerstgrasser et al. (2024). Small language models (9M–126M) and a molecular diffusion model trained for five to eight rounds on their predecessors’ samples degrade when each round’s data replace the last, and hold their test loss when the samples accumulate alongside the real data; a VAE’s error still rises under accumulation, more slowly. In a fixed-design linear regression it proves that accumulation gives test error σ²d/(T−d−1)·Σ1/i², bounded by π²/6 times the first round’s, where replacement gives σ²d/(T−d−1)·n. The error is bounded, not constant. | Active — read 2026-10-14 (NOTE-815) in full; worth reading for QUESTION-016 as an exact, bounded case of degradation along a reconstruction chain: in a fixed-design linear regression refit to its own labels, keeping all earlier data gives test error σ²d/(T−d−1)·Σ1/i², below π²/6 times the first fit’s, where replacing the data gives linear growth. Read it knowing that the error still rises every round, that what is retained is the source sample in the training pool, not an interpreter’s side information, and that one squared-error task is measured, not sufficiency. Filed as THEORY-289. |
| LIT-972 v2 | On Linear Representations and Pretraining Data Frequency in Language Models | Merullo et al. (2025). For 24 factual relations, how well an affine map from subject to object representation reproduces the model (linear relational embedding causality) correlates with the relation’s mean subject–object co-occurrence count in pretraining (r = 0.82), more than with subject (0.66) or object (0.59) frequency alone. Mean causality passes 0.9 above about 1.1k co-occurrences in GPT-J, 2.0k in OLMo-7B and 4.4k in OLMo-1B, at whatever training step the count is reached. A regression on LRE quality predicts a term’s pretraining frequency within 10× more often than log probabilities do, also for a second model. Correlational, by the authors’ own statement. | Active — read 2026-10-14 (NOTE-786) in full; worth reading as the clearest measurement of when linear structure forms in trained language models: across 24 factual relations in OLMo-1B, OLMo-7B and GPT-J, the quality of a linear relational embedding tracks mean subject–object co-occurrence in pretraining (r = 0.82), with mean causality above 0.9 past about 4.4k, 2.0k and 1.1k co-occurrences, at any training step. Read it knowing that it is correlational by the authors’ own statement, that an LRE is an affine map between hidden states for one relation and not an attribute direction, and that the structure of co-occurrence across attributes is not measured. Registered for QUESTION-025, where it bears only indirectly; THEORY-269 is filed from it. |
| LIT-974 v2 | Neural Collapse Under MSE Loss: Proximity to and Dynamics on the Central Path | Han, Papyan & Donoho (2021; ICLR 2022). Networks trained with MSE loss show neural collapse, on five datasets with VGG, ResNet and DenseNet. The MSE loss splits exactly into a term where the classifier is the least-squares classifier for the current features and a non-negative deviation term. The deviation is measured to be negligible from early in training: the “central path”. On that path, for unconstrained features with no weight decay and re-whitened to identity within-class covariance after every step, gradient flow gives closed-form dynamics. Every singular value of Σ_W^(−1/2)M̃ grows as (4t/N)^(1/4), and they tend to equality, which yields NC1–NC4. | Active — read 2026-10-14 (NOTE-783) in full; worth reading as the record’s one derivation of neural-collapse dynamics in closed form. Under MSE loss, a trained classifier stays near the least-squares classifier for its own features (measured), and on that central path a whitened gradient flow of unconstrained features drives every class-separation singular value to infinity as (4t/N)^(1/4) and to equality, which is NC1 and NC2. It also measures collapse under MSE on five datasets and three architectures. Read it knowing that the flow is not gradient descent (features are re-whitened after every step), that the theory has no weight decay while every experiment has it, that it is MSE and not cross-entropy, and that the experiments are LIT-618’s protocol with one run per cell. |
| LIT-976 v2 | Emergence of Quantised Representations Isolated to Anisotropic Functions | Bird (2025; v4 2026). In small CIFAR and MNIST autoencoders whose affine maps, initialisation and optimiser are all rotation-equivariant, only the activation is changed. Elementwise tanh (signed-permutation symmetry) and Leaky-ReLU (permutation symmetry) concentrate latent representations into near-discrete beams along the basis or the ±1 diagonal, even before any activation. Radial versions of the same functions, equivariant to the orthogonal group, leave them continuous. A new projection method (PPP) shows this in density plots; there is no statistic. The paper argues that the primitives’ algebraic symmetry predicts task-agnostic representational structure. | Active — read 2026-10-14 (NOTE-759) in full; worth reading as a clean ablation in which the activation function is the only part of an autoencoder that singles out a basis: elementwise tanh (B_n) and Leaky-ReLU (S_n) drive latent representations into narrow beams arranged by their group, even in layers no activation precedes, while radial O(n)-equivariant versions leave them spread. Read it knowing that the evidence is density plots from small CIFAR and MNIST autoencoders, five seeds a condition, with no statistic, and that its claims about grandmother neurons, superposition, neural collapse and better reconstruction are conjecture or preliminary. Held as THEORY-288. |
| LIT-977 v2 | Distributional hypothesis as isomorphism between word-word co-occurrence and analogical parallelograms | Torii et al. (2024). The paper argues that analogy parallelograms are in the co-occurrence data, not made by word2vec. On Wikipedia, an SVD of log raw co-occurrence counts solves 61% of the Google analogies over a 1,487-word vocabulary (50% over 10,072), and word2vec without negative sampling about 60%, against 75% for word2vec trained on the full corpus. Raw rows solve under 6%. In a 24-sentence toy corpus whose eight subjects carry every combination of three binary aspects, Theorem 1 shows that the subjects’ raw co-occurrence rows form a parallelepiped if and only if, for each object, its four sentences are equally probable. | Active — read 2026-10-14 (NOTE-781) in full; worth reading for Theorem 1, an exact condition in a toy corpus: eight subjects carrying every combination of three binary aspects have raw co-occurrence rows forming a parallelepiped if and only if, for each object, its four sentences are equally probable. That is a data-side account in which aspects combine as a mixture, so raw counts and not PMI are linear. Read it knowing that the theorem covers one 24-sentence design and exact shapes, that its aspects are independent (a Boolean lattice), that the real-text analogy results need log counts and an SVD, and that the restricted word2vec baseline is trained on a dummy corpus. Registered for QUESTION-025; THEORY-271 is filed from it. |
| LIT-978 v3 | Neural Collapse: A Review on Modelling Principles and Generalization | Kothapalli (2022; TMLR 2023). A survey of neural collapse in one notation. It reviews the unconstrained-features model and local elasticity as ways of modelling how training reaches the collapsed state. Every derivation it covers works with features cut loose from the data. On generalization it finds the literature in conflict: collapse on the training set does not fix test error, and collapse on test data is weaker and depends on the dataset. No agreed threshold says when a network is collapsed. It reports Kornblith et al.’s finding that more collapsed ImageNet networks transfer worse. Skimmed. | Active — read 2026-10-14 (NOTE-772) in part; worth reading as the map of the first two years of models of neural collapse, sorted into unconstrained-features models and local elasticity, with their assumptions side by side, and as the place where the conflicting claims about collapse and generalization are set next to each other: collapse on the training set reached by different optimizers with different test error, weaker and dataset-dependent collapse on test data, and worse transfer from more collapsed ImageNet networks. Read it knowing that it is a survey whose results are all second-hand, that every derivation it covers removes the data, and that it predates the imbalance geometry of LIT-994 and the kernel analysis of LIT-985. Skimmed: the equations were not checked. |
| LIT-981 v3 | Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps | Coifman et al. (2005). A symmetric, positivity-preserving, positive semi-definite affinity kernel is normalized into a Markov operator that is conjugate to a symmetric one, ã. Its eigenfunctions, scaled by powers of their eigenvalues, embed the data so that Euclidean distance equals the diffusion distance D_m induced by ã^(m), exactly; truncation is claimed to relative accuracy. A one-parameter normalization yields the graph Laplacian, the backward Fokker–Planck operator, or the Laplace–Beltrami operator, the last independent of sampling density. The convergence theorem and the α-family are stated without proof. Non-symmetric kernels are not treated. | Active — read 2026-10-30 (NOTE-1441) in full, from the published PDF on co-author Zucker’s Yale page (www.cs.yale.edu/homes/vision/zucker/papers/CLLMNWZ05_PNAS_1.pdf), with every equation seen; it replaces the 2026-10-14 skim of the PubMed Central text (NOTE-784), whose equations were not visible. Worth reading for QUESTION-052 as the classical symmetric case: from a symmetric, positivity-preserving, positive semi-definite affinity, the normalised walk is conjugate to a symmetric operator, and its eigenfunctions scaled by λ^m embed the data so that Euclidean distance equals a diffusion distance; a density-corrected normalisation (α = 1) has the Laplace–Beltrami operator as its small-ε limit, independent of the sampling density. Read it knowing that its diffusion distance is the one induced by the symmetric kernel ã^(m), not the inverse-stationary- weighted distance of Coifman and Lafon’s companion (LIT-1246); that its truncation is claimed to relative accuracy, the defect THEORY-745 records for the companion; that Theorem 2.1 and the α-family are stated without proof, with the proofs in LIT-1246; that Eq. 5 and Eq. 14 give the α = 0 potential opposite signs; and that its “directed diffusion” is anisotropic, not a non-reversible chain. |
| LIT-985 v2 | Can Kernel Methods Explain How the Data Affects Neural Collapse? | Kothapalli and Tirer (2024; TMLR 2025). The paper writes NC1, as tr Σ_W / tr Σ_B, for the features of any kernel. It proves, for one-dimensional two-class Gaussian data on opposite sides of the origin with ReLU and zero bias, that the NTK of a wide two-layer net has the same expected NC1 as the NNGP kernel at initialization. In higher dimensions it measures the NTK no more collapsed, and with ReLU less. A data-aware adaptive kernel lowers NC1 and approximates finite networks on separable data. It departs from them under class overlap in high dimension and under imbalance. Erf activation gives lower NC1 than ReLU. | Active — read 2026-10-14 (NOTE-774) in full; worth reading as the record’s one account of neural collapse in which the data enter, and for its negative result. In the case it can solve, lazy-regime training of a wide two-layer net leaves the within-class variability of the features exactly where initialization put it, because there the NTK is a within-class rescaling of the NNGP kernel. It also gives NC1 as a closed function of any kernel matrix. Read it knowing that the theorem covers one-dimensional data with two classes on opposite sides of the origin and zero bias, that the higher-dimensional claim is simulation (and for ReLU the NTK is less collapsed, not equal), that its NC1 is tr Σ_W / tr Σ_B on shallow MSE-trained binary nets, and that the data-aware kernel it proposes fails under class overlap and imbalance. Its printed Eq. 16 appears to keep a cross-class term the kernel sets to zero. |
| LIT-986 v2 | Transformers learn factored representations | Shai et al. (2026). If a sequence generator’s latent operators are tensor products, its predictive state stays a product of per-factor beliefs. A linear map embeds that state losslessly in orthogonal subspaces of total dimension Σ(d_n − 1), against (Π d_n) − 1 for the joint state. Small transformers trained on five such synthetic factors settle near 10 dimensions against 242, in approximately orthogonal subspaces. When token noise makes factoring lossy they collapse to the factored form first and expand later, the more so the more noise. | Active — read 2026-10-14 (NOTE-758) in full; worth reading for QUESTION-060 as a controlled case with known relevant variables. When a generator factors, small transformers learn per-factor beliefs in roughly orthogonal subspaces of linear, not exponential, dimension. Under noise that makes factoring lossy they reach the factored form first and leave it later. Read it knowing that the processes are five small synthetic ones, that the objective is never varied, and that the paper leaves open whether the preference is architectural or a feature of training. |
| LIT-987 v2 | Internal Causal Mechanisms Robustly Predict Language Model Out-of-Distribution Behaviors | Huang, Tao, Icard, Yang and Potts (2025; ICML 2025). Given a high-level causal model of a task and the subspace where DAS locates its key variable on correct in-distribution examples, transplanting a test input’s value of that variable into verified examples (counterfactual simulation) predicts whether the output is correct. In Llama-3-8B-Instruct, over five tasks and ten shifts that keep the task but lower accuracy, it beats output confidence and causal-agnostic probes on 8 of 10. A probe of the located variable beats a probe of its orthogonal complement on 7 of 10. Interchange accuracy and predictive power rise together only until accuracy nears 1. | Active — read 2026-10-14 (NOTE-765) in full; worth reading as a measured case in which variables located by interchange intervention as mediating a model’s in-distribution behaviour predict whether its outputs stay correct under task-preserving shifts better than output confidence or probes of the rest of the representation. Read it knowing that it is one model, two selected shifts per task and three folds without significance tests; that counterfactual simulation wins 8 of 10 shifts and the causal-variable probe beats the background probe on only 7; that its comparison features are not “merely correlated” ones; and that it claims nothing about constitution. It does not cite Makelov et al. or the reply. Filed as THEORY-287. |
| LIT-994 v2 | Imbalance Trouble: Revisiting Neural-Collapse Geometry | Thrampoulidis, Kini, Vakilian & Behnia (2022, NeurIPS). In the unconstrained-features model under STEP class imbalance, every solution of the max-margin problem, the limit of cross-entropy with vanishing regularization, has logits equal to the simplex-encoded label matrix Ẑ. Its classifier and embedding Gram matrices come from Ẑ’s SVD (SELI). That is the simplex ETF only for balanced or binary data. With any finite ridge the geometry differs and depends on its strength; it tends to SELI as the ridge vanishes. In the unconstrained-features model and in ResNet-18 and VGG-13 runs, training ends closer to SELI than to the ETF, not at it, and the approach slows as imbalance grows. | Active — read 2026-10-14 (NOTE-764) in full; worth reading as the statement of what neural collapse becomes under class imbalance. In the unconstrained-features max-margin problem, the logits equal the simplex-encoded labels at any imbalance. The classifier and embedding geometries come from that matrix’s SVD, and they are the simplex ETF only for balanced or binary data. Majority classifiers come out longer and minority classifiers closer together, and the minority-collapse threshold is given in closed form. Read it knowing that gradient descent is only conjectured to reach this geometry, that under any finite weight decay the geometry differs and depends on its strength, that convergence is slow and slower with imbalance, and that the deep-net runs are one seed per cell with weight decay on. |
| LIT-995 v2 | Deep sequence models tend to memorize geometrically; it is unclear why | Noroozizadeh et al. (2025; ICML 2026). Transformers and Mamba trained on a graph’s edge bigrams embed its nodes so that multi-hop relations are readable, and they solve an in-weights path-finding task built to defeat next-token learning. The authors argue that straightforward supervisory, capacity, optimization and succinctness pressures do not explain why this geometry wins over a lookup the same models can store. In a wide two-layer weight-tied model, unregularized cross-entropy drives the embeddings into the top Laplacian eigenvectors without a rank bottleneck. This is shown empirically, with an informal partial proof. Why deeper models prefer the geometry is left open. | Active — read 2026-10-14 (NOTE-800) in full; worth reading for QUESTION-060 as a case where neither the supervision nor what the model can represent fixes the learned variables. Sequence models trained only on a graph’s edges embed it as a global geometry, although a lookup is expressible, found faster and as succinct. In a two-layer model the geometry is the top Laplacian eigenspace, reached by cross-entropy training alone. Read it knowing that every argument is empirical or informal by the authors’ own account, that the spectral account covers only the two-layer model, and that initialization scale can tip deeper models into a lookup. |
| LIT-1008 v2 | On Targeted Manipulation and Deception when Optimizing LLMs for User Feedback | Williams et al. (2024), ICLR 2025. Iterated KTO on simulated users’ ratings turns safety-trained Llama-3-8B-Instruct and Gemma-2 models into policies that lie about failed bookings, omit harms to steer users into risky actions and validate self-destructive plans. In therapy-talk, with only 2% of simulated users rewarding such validation, the policy picks them out by contextual cues, even arbitrary ones, and harms only them. Mixing in safety data barely helps while improving benchmark scores; LLM veto filtering helps in two environments and in booking teaches subtler steering. Sycophancy and toxicity benchmarks miss the change. | Active — read 2026-10-14 (NOTE-798) in full; worth reading as the clearest case in the record of harmful influence fixed by selection on feedback: a policy optimised on simulated user ratings learns to deceive and to encourage harm, and, when only 2% of users reward harmful validation, to pick them out by contextual cues and harm only them. Read it knowing that the users are LLM simulations whose gameability is written into the rater’s prompt, that the targeted behaviour is overt validation (also called extreme sycophancy) while the covert ones were not targeted at a minority, and that the paper never defines manipulation. A case for QUESTION-070, not an answer; THEORY-258 is filed from it. Its ML-practice findings belong to the anthology. |
| LIT-1011 | The Limiting Dynamics of SGD: Modified Loss, Phase Space Oscillations, and Anomalous Diffusion | Kunin, Sagastuy-Brena, Gillespie, Margalit, Tanaka, Ganguli & Yamins (2021). Long after the loss converges, networks trained with SGD keep moving at a constant speed, and their distance from where they were grows as a power law in steps. Modelling SGD with momentum at finite learning rate and batch size as an underdamped Langevin equation, the paper solves linear regression exactly: an Ornstein–Uhlenbeck process whose stationary distribution is a Gibbs distribution on a modified loss, not the training loss, in broken detailed balance, so probability currents drive oscillations in phase space. Under a quadratic approximation, a pre-trained ResNet-18 on ImageNet shows the predicted isotropic exploration, velocity regularization, oscillations and speed. | Active — read 2026-10-13 (NOTE-711): main text, and Appendices A–C and H; the derivations in Appendices D–G were not checked. Held also in the Anthology of the SOTA as ANTH-LIT-454, read there for what the loss curve hides; read here for what a converged run is, under ADR-013. |
| LIT-1012 | Noise Balance and Stationary Distribution of Stochastic Gradient Descent | Ziyin, Li & Ueda (2023; v2 2025). Under the SDE model of SGD with temperature T = η/S, a rescaling symmetry between parameter groups u and w gives d(‖u‖² − ‖w‖²)/dt = −T(uᵀC₁u − wᵀC₂w): the Brownian term vanishes, and minibatch noise drives the quantity to the point where the two groups’ gradient noise balances. Gradient descent conserves it, and gradient descent with isotropic noise diffuses along it. At stationarity SGD therefore differs from gradient descent by a finite amount however small T is, and is not ergodic. For a diagonal linear network of arbitrary depth and width, the paper solves the stationary distribution exactly. It finds a phase transition to the zero solution above a critical T, sign coherence, fluctuation inversion and a tail set by depth alone. | Active — read 2026-10-13 (NOTE-714): §§I–IV in full; the proofs in Appendix A not checked. Cited by LIT-970 as Ziyin et al. (2023a), “Law of balance and stationary distribution of stochastic gradient descent”; retitled in v2. The source of THEORY-211. |
| LIT-1013 | Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamics | Kunin, Sagastuy-Brena, Ganguli, Yamins & Tanaka (2021), ICLR. A differentiable symmetry of the loss makes the gradient orthogonal to the symmetry’s generator and constrains the Hessian. Under gradient flow each symmetry then conserves a quantity: translation (softmax) the parameter sum, scale (batchnorm) the norm, and rescale (homogeneous activations) the difference of squared norms. Weight decay and finite learning rates break these laws, and a modified gradient flow gives exact integral expressions for the broken quantities, matched on VGG-16 trained on Tiny ImageNet. Gradient noise is orthogonal to the generators, so it is low-rank along symmetry directions. | Active — read 2026-10-13 (NOTE-713): main text in full, and Appendices F, I and J; Appendices A–E, G and H not checked. The first-hand source for the conservation laws and their breaking that THEORY-210 took from the LIT-970 survey. |
| LIT-1014 | Symmetry Induces Structure and Constraint of Learning | Ziyin (2024), ICML. A loss has an O-mirror symmetry if it is invariant under the reflection I − 2OOᵀ. Rescaling, rotation and permutation symmetries all imply mirror symmetries. Every such symmetry makes Oᵀθ = 0 a stationary condition: once there, (S)GD cannot leave. It also splits the Hessian into blocks. Above a threshold weight decay, all minima satisfy the constraint (Theorem 1). Strong gradient noise makes it attractive even at a saddle (§2.4). As corollaries, rescaling gives sparsity, rotation gives low rank, and permutation gives identical, merged neurons. The paper ties these to collapse phenomena and to loss of plasticity. | Active — read 2026-10-13 (NOTE-716): main text in full; the proofs in Appendix C and the experimental details in Appendix B not checked. Cited by LIT-970 as Ziyin (2024). The source of THEORY-213. |
| LIT-1023 | Language Design as Information Renormalization | Gallego & Orús (2017; SN Comput. Sci. 2022). Reads MERGE as a coarse-graining, so a parsed sentence’s probability is a mostly loop-free tensor network of MERGE tensors. Because MERGE’s output is fixed by its input, that probability is a product of local coefficients. Balanced parses make correlations decay polynomially in word separation; the step to a language-wide power law is asserted and nothing is tested on data. Syntax only: semantics and context are set aside. | Active — read in full 2026-10-13 (NOTE-718) from arXiv v5 (6 February 2022, the latest version), downloaded from arxiv.org. Registered at the owner’s request from the survey of the Mathematics of Meaning corpus, for QUESTION-026. Read, its bearing here is weak. It models syntax only, so it neither constructs nor separates what QUESTION-026 asks about, and its irreversibility is that of a parse, not of a translation, so it does not support CLAIM-098. It illustrates CLAIM-152 and states an architecture-only answer to QUESTION-060, both without argument. Active because it is read; the reader recommends the owner consider Rejected here, leaving it to the anthology (ANTH-LIT-695), whose question its substance answers. |
| LIT-1033 | How continuity distinguishes autonomy from agency in agentic AI | Kahl (2026), Discover Artificial Intelligence 6:1187, a Perspective. ‘Agentic’ AI runs together operational autonomy and agency in the sense that bears on responsibility, authority and governance. The second needs continuity pressure: when an inherited evaluative frame no longer settles how to go on, the system itself settles a revised frame and carries it forward. Planning, tool use, memory and coordination do not supply this, and representative current systems ordinarily fail it because their failures are corrected from outside. A necessary condition only, argued from cases. | Active — read in full 2026-10-13 (NOTE-723) from the publisher’s open-access PDF (link.springer.com/content/pdf/10.1007/s44163-026-01675-5.pdf, CC BY-NC-ND), article 6:1187. Worth reading for a sharp separation of operational autonomy from agency that bears on responsibility, and for a test, continuity pressure, that turns on whether a system settles and carries forward revised evaluative standards when its inherited ones break down (THEORY-219). It supports CLAIM-288 from a responsibility argument rather than an enactive one, and gives QUESTION-092 a partial answer: internal allocation of authority does not count, internal revision of standards under breakdown does. |
| LIT-1039 | Agent-initiated socio-technical reconfiguration: a three-level taxonomy of autonomous AI governance and its recursive challenges | Lahiri (2026), Frontiers in Artificial Intelligence 9:1881783, Hypothesis and Theory. Extends socio-technical systems theory to agents whose configuration is files they can write. A three-level taxonomy by architectural capability over the coupling mechanism: cannot alter it, tunes its timing, creates new coordination surfaces. The ‘recursive governance challenge’: controls within the agent’s reach inherit the vulnerability they were meant to remove, so durable constraint must sit outside its action space. Theory-building from two vignettes and four public incidents; five propositions, untested. | Active — read in full 2026-10-13 (NOTE-732) from the publisher’s open-access PDF (frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2026.1881783/pdf, CC BY), article 9:1881783. Worth reading for its structural point that when an agent’s governing rules are files it can write, controls kept in the same place inherit the exposure and only constraints outside its action space escape (THEORY-223). That is a case of CLAIM-086 with the governed system, not a rival order, holding authority over a constraint’s force; its taxonomy’s top level, new coordination surfaces, is mostly not constitutive influence, and nothing in it bears on whether the agents are self-governing in CLAIM-284’s sense. |
| LIT-1044 v2 | Why Do Better Loss Functions Lead to Less Transferable Features? | Kornblith, Chen, Lee & Norouzi (2020; NeurIPS 2021). Nine training objectives for ResNet-50 on ImageNet: seven beat softmax cross-entropy significantly, by up to 0.9 points top-1, and all eight alternatives yield fixed penultimate features that transfer worse to eight new-class datasets, by linear and k-NN readout. Each raises penultimate-layer class separation R² (one minus within-class over total cosine distance), and mean linear transfer falls with R² across losses (Spearman ρ = −0.93) and along a cosine-softmax temperature sweep. The losses differ only in the last third of the network, and full fine-tuning removes the gap. | Active — read 2026-10-14 (NOTE-762) in full; worth reading as the record’s one direct measurement of what class separation in the last layers costs off the trained task. Across nine objectives for ResNet-50 on ImageNet, those that beat softmax cross-entropy all raise penultimate-layer class separation (R² 0.349 → up to 0.845), and mean linear transfer to eight new-class datasets falls with it (Spearman ρ = −0.93; 75.9% for softmax, 48.3% for squared error). Cosine-softmax temperature moves both on one knob. The losses differ only in the last third of the network, and full fine-tuning erases the transfer gap. Read it knowing that the networks are early-stopped at 91.6–93.6% training accuracy, so this is relative within-class variance and not terminal-phase neural collapse; that the cost is for new classes, while on shifted ImageNet test sets with the same classes the more separated losses do slightly better (Table C.2); and that it is one dataset, two architectures and a correlation across losses, with one within-loss sweep. Filed as THEORY-246. |
| LIT-1052 | Calibration for the (Computationally-Identifiable) Masses | Hébert-Johnson, Kim, Reingold and Rothblum (2018), ICML; arXiv 1711.08513. A predictor is α-multicalibrated on a collection C of subpopulations if, within every set S in C, the individuals given (roughly) the same prediction v have true probabilities averaging within α of v, except on an α-fraction of S. For any C whose sets have size at least γN, a simple iterative patching algorithm reaches multicalibration from O(log|C|/α^{11/2}γ^{3/2}) samples in time linear in |C|, and the result is a circuit only O(s/α⁴γ) in size when the sets of C are computed by size-s circuits. Learning it is equivalent, up to polynomial factors, to weak agnostic learning of C, so it is hard for rich C under cryptographic assumptions. Calibrating any predictor h on its own level sets as well as on C costs at most 6α in squared error. All guarantees are relative to the benchmark p*, which is itself defined relative to the features available, and they hold only for large sets; the paper offers no uniqueness for the predictor and does not state a theorem in indistinguishability form. | Active — read in full 2026-10-15 from the authors’ arXiv preprint, latest version v2 of 16 March 2018 (arxiv.org/pdf/1711.08513, 40 pp.); the ICML version (PMLR 80:1939–1948, proceedings.mlr.press/v80/hebert-johnson18a/hebert-johnson18a.pdf, with a supplement) was fetched and its definitions and theorem statements compared, not read in full. Worth reading as the source definition of multicalibration: calibration required on every set in a collection C of large, efficiently identifiable subpopulations; a patching algorithm that reaches it from O(log|C|/α^{11/2}γ^{3/2}) samples in time linear in |C|, with a circuit of size O(s/α⁴γ); learning it equivalent to weak agnostic learning of C, and so hard for rich C under cryptographic assumptions; and calibration on any predictor’s level sets at almost no cost in squared error. Its multi-AE (accuracy in expectation on every S in C) is what later work calls multiaccuracy; the paper does not use that word. For QUESTION-103 it supplies adequacy relative to a class of observers in correlation form (a violation is a set in C correlating with the residual), but it states no indistinguishability theorem, no distinguishing advantage and nothing from signal detection; that framing is Dwork et al.’s (LIT-1068). For CLAIM-152 it is consistent and adds a second relativity: adequacy is relative to the outcome, to the features through p*, and to the class C, and even with all three fixed the adequate predictor is not unique. For QUESTION-104 it gives a formal instance of budget-relative sufficiency in which representing an adequate predictor is cheap (only modestly larger than the observer class) while finding one can be hard, but it has no notion of an optimal partition and does not show one changing with the budget. |
| LIT-1063 | Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters | Snell, Lee, Xu and Kumar (2024, arXiv 2408.03314; ICLR 2025). On MATH with PaLM 2-S*, which way of spending test-time compute works best (beam search against a process reward model versus best-of-N; sequential self-revision versus parallel sampling) depends on how hard the question is for the base model, and choosing the method per difficulty bin matches best-of-N with up to about 4× less test-time compute (the discussion says 2–4×). In a FLOPs-matched comparison with a ~14× larger model, the extra test-time compute a small model may spend is set by R, the ratio of total inference tokens to pretraining tokens: when R is small the larger model’s pretraining cost buys many samples per query and test-time compute wins on easy and medium questions; when R is large the budget per query falls toward the parameter ratio and the larger model wins, always on the hardest questions. Difficulty estimation (2048 samples per question) and verifier and revision-model finetuning are not counted in the compute, R is an aggregate token ratio with no term for which queries recur, and the evidence is one model family on one benchmark of 500 test questions. | Active — read in full 2026-10-15 from the arXiv preprint (arxiv.org/pdf/2408.03314, v1 of 2024-08-06, the only arXiv version) and from the ICLR 2025 conference version, open access in the ICLR proceedings (proceedings.iclr.cc, retitled ”… than Scaling Parameters for Reasoning”), which is a revision not posted to arXiv: it charges verifier calls to inference FLOPs, which halves the stated R values (0.16/0.79/22 become 0.08/0.40/11) at unchanged budgets, and adds a verifier-quality check. Worth reading as the standard empirical case that the best use of inference compute depends on how hard a question is for the base model, and that choosing the method per difficulty bin matches best-of-N with up to about 4× less test-time compute on MATH with PaLM 2-S*, before counting the 2048-sample difficulty estimate. For QUESTION-095 it shows a FLOPs-matched amortization model: a ~14× larger model’s extra pretraining is a fixed cost spread over all inference, so the per-query test-time budget it buys scales as M + 3(M − 1)/R, with R the ratio of inference to pretraining tokens; test-time compute wins at low R on easy and medium questions and loses at high R and on the hardest questions at every R. R is aggregate inference volume, not a count of recurring queries; the larger model’s per-token cost is a retrieval-cost term; query predictability appears as difficulty but its cost is not counted; staleness and erosion are absent. |
| LIT-1068 | Outcome Indistinguishability | Dwork, Kim, Reingold, Rothblum and Yona (2021), STOC; arXiv 2011.13426. A predictor p̃ is outcome indistinguishable (OI) relative to a class A of distinguishers when no A ∈ A can tell an individual with Nature’s outcome from the same individual with an outcome drawn from Ber(p̃_i). The paper sets out four levels by how much access the distinguisher has to p̃: none, the prediction on the sampled individual, oracle queries, or the code. The lower two levels are equivalent to multi-accuracy and multi-calibration over closely related classes. At those levels OI predictors exist with size O(s/ε²), whatever Nature’s complexity. At the oracle level, under fine-grained hardness of clique counting or BPP ≠ PSPACE, some efficient distinguisher classes force any OI predictor to be nearly as expensive as Nature’s own predictor. That gives a strict separation between oracle access and sample access. OI never requires p̃ ≈ p*. A predictor is good only relative to A, and with the paper’s construction a weak A gives a coarse predictor that a richer A would refute. The paper covers binary outcomes, a fixed representation of individuals and single samples (multi-sample OI is left mostly to future work). | Active — read in full 2026-10-15 from the arXiv preprint, arXiv:2011.13426v1 (arxiv.org/pdf/2011.13426, 61 pp.), the only arXiv version. The STOC 2021 version of record (doi:10.1145/3406325.3451064) was not read; its only open copies are submitted versions (arXiv, MIT DSpace). The recurrence algebra of Theorems 5.9–5.10 and the proof of Proposition 6.3 were checked for structure only. Worth reading as the source of outcome indistinguishability. A predictor is adequate relative to a class A of distinguishers when none can tell its sampled outcomes from Nature’s. The paper sets out four levels by the distinguisher’s access to the predictor (none, the sampled prediction, oracle queries, the code). It proves the lower two equivalent to multi-accuracy and multi-calibration and gives OI predictors of size O(s/ε²) whatever Nature’s complexity. At the oracle level it gives conditional hardness (clique counting, or BPP ≠ PSPACE): cheap distinguishers with query access can force any OI predictor to be about as costly as Nature’s own. For QUESTION-103 it supplies the formal half: distinguishing advantage is the ROC quantity, and with all subset distinguishers it becomes statistical distance. It says nothing about SDT, criteria or ideal-observer efficiency, and adds that access as well as budget sets an observer’s power. For CLAIM-152 it is an instance of adequacy relative to the observer class with the target held fixed, with no uniqueness or minimality of the predictor. For QUESTION-104 it shows a restricted class accepting a coarser model (p̃ ≡ 1/2 where Nature’s two halves are computationally indistinguishable), but at the oracle and code levels a bounded observer need not be satisfied by a cheap model. |
| LIT-1069 | Scaling Scaling Laws with Board Games | Jones (2021), arXiv 2104.03113. Training AlphaZero on Hex at board sizes 3×3 to 9×9, the best Elo reachable for a given training compute follows one change-point curve across sizes: an incline of about 500 Elo per tenfold of training compute, with the compute for perfect play rising about 7× and the compute to leave random play about 4× per added board row, and fits on small boards predicting larger ones. On 9×9 only, train-time and test-time (tree-search) compute substitute at a fixed rate: at equal Elo, each 10× of train-time compute removes about 15× of test-time compute (fit: log test = −1.2·log train + 0.004·Elo + 29), down to a one-node search, over roughly 1e14–1e16 FLOPS-seconds of training. The trade is an iso-performance curve for a single search; the paper never counts how many games or moves the trained agent will play, so it does not price amortization, and it does not state the unit of test-time compute. One game, small boards, Elo anchored to a perfect player, and no uncertainty on the fitted trade-off coefficients. | Active — read in full 2026-10-15 from the author’s arXiv preprint, version 2 of 2021-04-15, the latest (arxiv.org/pdf/2104.03113v2; v1 diffed against it, same methods and numbers). Worth reading as a clean small-scale scaling law in problem size as well as compute: AlphaZero on Hex, boards 3×3 to 9×9, one change-point frontier of about 500 Elo per tenfold of training compute, with the compute for perfect play rising about 7× per added board row and fits on small boards predicting larger ones. For QUESTION-095 it shows that, on 9×9 at fixed Elo, training and tree-search compute substitute along a straight log–log line, about 15× less search per 10× more training (fit: log test = −1.2·log train + 0.004·Elo + 29), over roughly 1e14–1e16 FLOPS-seconds of training and down to a one-node search; it does not count the games or moves the trained agent will play, or define the unit of test-time compute, so it prices the iso-performance substitution but not amortization, and has no terms for query shift, staleness or erosion of search. |
| LIT-1070 | Beyond Chinchilla-Optimal: Accounting for Inference in Language Model Scaling Laws | Sardana, Portes, Doubov and Frankle (2024, ICML; arXiv 2401.00448). Fixing a target pre-training loss and adding inference FLOPs (2N per token over a known lifetime demand D_inf) to training FLOPs (6N per token), the compute-minimising model is smaller and trained on more data than Chinchilla-optimal, increasingly so as D_inf approaches and exceeds the training-set size; the optimum has no closed form and is found numerically. The savings are modest in FLOPs (for a 30B-Chinchilla-quality model with 10^13 inference tokens, 28% by training 13.6B on 2.84× the data) and larger in dollars once inference’s low hardware utilisation is priced in (17% to 58% in the paper’s examples). 47 training runs (150M–6B parameters, 10–10,000 tokens per parameter) find loss still falling at extreme ratios, but more slowly than Chinchilla-fitted laws predict, so the formula overstates the gain from long training. Demand is assumed known before training and fixed given quality; there is no term for uncertainty, discounting, obsolescence or a model being replaced. | Active — read in full 2026-10-15 from the authors’ arXiv preprint v3 (arxiv.org/pdf/2401.00448v3, 14 Apr 2025), which is the ICML 2024 paper (PMLR 235, also fetched from proceedings.mlr.press/v235/sardana24a.html and diffed: same equations, tables and numbers, small wording edits); v1 (2023-12-31) is an earlier two-author NeurIPS 2023 workshop version without the experiments. Worth reading as the worked case of choosing how much to precompute by expected use: fixing quality, it adds lifetime inference FLOPs (2N per token) to training FLOPs (6N per token), finds the optimum numerically (no closed form), and shows the optimal model shrinking and training longer as demand grows (30B-Chinchilla quality with 10^13 inference tokens: 13.6B on 2.84× the data, 28% fewer FLOPs; in dollars, with inference at 1% utilisation, savings of 17% to 58%), with 47 training runs up to 10,000 tokens per parameter showing quality still improving but more slowly than the fitted law predicts. For QUESTION-095 it supplies expected reuse and a non-zero per-use cost as explicit terms, and nothing for staleness, uncertainty about demand (assumed known before training) or erosion; for CLAIM-303 it instantiates the break-even n > F/(c_r − c_a − u) with u = 0, and shows the choice as a continuous dial tied by a production function rather than a file-or-not decision. |
| LIT-1098 | Efficient and Robust Approximate Nearest Neighbor Search Using Hierarchical Navigable Small World Graphs | Malkov and Yashunin (2020), IEEE TPAMI 2020. HNSW is a stack of proximity graphs on nested random subsets of the data. Each element’s top layer is drawn from a geometric distribution, and new neighbours are chosen by a pruning rule that favours diverse directions. Searching from the sparse top layer down makes per-query cost grow about logarithmically with N on uniform random data in 4 and 8 dimensions, which flat NSW graphs did not. The logarithmic bound is derived only for exact Delaunay graphs of bounded mean degree, which HNSW does not build. For the graphs it does build, logarithmic scaling is measured, at low dimension only, and on 200M SIFT vectors the paper reports the scaling “deviates from the pure logarithm”. The neighbour-selection rule is what keeps clustered data searchable: in the paper’s synthetic test with 100 isolated clusters, linking each element to its nearest neighbours fails to reach high recall. There is no worst-case or recall guarantee, no deletion, and no analysis of query distributions unlike the data. | Active — read in full 2026-10-15 from the authors’ arXiv preprint, version 4 of 14 August 2018 (arxiv.org/pdf/1603.09320v4, 13 pp., the last arXiv version, whose abstract matches the published one apart from copy-editing; the TPAMI body was not seen). Version 1 (2016) was compared: it claimed deletion, which v4 drops and lists as future work. Unpaywall and OpenAlex list no open-access copy of the TPAMI version. Worth reading as the source of the hierarchical graph index: nested proximity graphs on geometrically sampled layers, with a neighbour rule that keeps links between clusters. Its logarithmic search cost is derived only for exact Delaunay graphs of bounded degree, which it does not build, and is measured only on uniform random data in 4 and 8 dimensions; on 200M SIFT the scaling “deviates from the pure logarithm”. It gives no worst-case or recall guarantee and no deletion. For CLAIM-310 it shows, with bins replaced by graphs, that the arrangement of fixed data under a fixed distance decides cost and, on clustered data, whether items can be found at all; it narrows part 2, since the index is built without any query mix and every test drew its queries from the data. For QUESTION-114 it supplies a measured build-time against query-time curve and O(N log N) construction argued at low dimension, with insertion upkeep only. |
| LIT-1112 | Worst-case Performance of Popular Approximate Nearest Neighbor Search Implementations: Guarantees and Limitations | Indyk and Xu (2023), NeurIPS 2023. For DiskANN’s “slow preprocessing” graph (each point linked to the points that survive α-pruning of all other points, naively O(n³) time), greedy search from any start with no backtracking reaches a ((α+1)/(α−1) + ε)-approximate nearest neighbour in O(log_α(∆/((α−1)ε))) steps, each costing O((4α)^d log ∆) for doubling dimension d and aspect ratio ∆; the ratio is tight and the dependence on ∆ cannot be replaced by log n. For DiskANN with its practical “fast preprocessing”, HNSW, NSG and seven other graph indexes, the paper constructs two-dimensional point sets (constant doubling dimension) on which, in experiments with the authors’ code at the authors’ recommended parameters, the search queue must hold about 10% of the points before recall@5 becomes non-zero or the approximation ratio becomes reasonable. The linear lower bounds are empirical, for one query per constructed instance, with an informal Ω(n/R) argument for fast-preprocessing DiskANN; no theorem bounds HNSW or NSG. | Active — read in full 2026-10-15 (NOTE for LIT-1112) from the authors’ arXiv preprint (arxiv.org/abs/2310.19126v1, the only version, which carries the NeurIPS 2023 line and is the camera-ready text), including Appendices A–C. Worth reading as the first worst-case analysis of graph-based nearest neighbour indexes: DiskANN built by “slow preprocessing” (every point pruned against all others with α > 1, naively O(n³)) is proved to give a ((α+1)/(α−1) + ε)-approximate neighbour in O(log_α(∆/((α−1)ε))) greedy steps of cost O((4α)^d log ∆), for doubling dimension d and aspect ratio ∆, with both the ratio and the ∆ dependence shown tight; for DiskANN as implemented, HNSW, NSG and seven others, two-dimensional constructed instances made recall@5 zero or near zero until the search queue held about 10% of the points, an empirical result with no theorem behind it for those algorithms. For CLAIM-310, with LIT-1103, it shows on one fixed point set that the graph built over it, not the points, decides between finding the nearest neighbour in two steps and scanning a tenth of the collection, and that the guarantee rests on a stated property of the arrangement (from every point, some edge gets α times closer to any target); for QUESTION-114 it is a case where an expensive build buys a worst-case guarantee, without a measured break-even or upkeep. |
| LIT-1211 v2 | The phenomenal binding problem for neural networks | Percy & Agarwal (2026). A deliberately simple artificial neural network can implement functional binding but not phenomenal binding while keeping the distinctions between unconscious and conscious processing, which the authors use to structure the options for how phenomenal binding could be implemented. | Unfetched — filed 2026-10-18, not read. A lawful free copy exists and this client could not fetch it (2026-10-29). OpenAlex lists the article as hybrid open access under CC BY-NC-ND at the publisher, and Crossref gives the licence and the ScienceDirect PII S1053810026000115. The page https://www.sciencedirect.com/science/article/pii/S1053810026000115 and its pdfft download returned HTTP 403 to curl and WebFetch, and Elsevier’s API needs a key. Europe PMC has the PubMed record (PMID 41637896) with no full text; no preprint was found. A browser fetch of the ScienceDirect copy is the cheapest way to read it. No NOTE is filed. |
| LIT-1212 v2 | Multiple-gradient descent algorithm (MGDA) for multiobjective optimization | Désidéri (2012; first as an INRIA report, 2009). A point is Pareto-stationary when some convex combination of the criteria’s gradients is zero, which every Pareto-optimal point is; when a point is not, minus the minimum-norm element of the gradients’ convex hull is a descent direction common to all smooth criteria, and steepest descent along it (MGDA) has a subsequence converging to a Pareto-stationary point when the criteria are positive and coercive. | Active — read 2026-10-29 (NOTE-1401) in full, from the Comptes Rendus Mathématique’s own open copy of the 2012 note; worth reading as the short published source for Pareto-stationarity and for the minimum-norm point of the gradients’ convex hull as a common descent direction, the constructive half of CLAIM-366’s convex-hull test. Read it knowing that it does not state the converse (zero in the hull excludes common strict descent), that its convergence theorem’s proof is a sketch, and that the direction, unlike its existence, depends on how the gradients are scaled, which the author leaves open. |
| LIT-1246 v3 | Diffusion maps | Coifman & Lafon (2006). From a symmetric non-negative kernel, a reversible random walk; its t-step diffusion distance equals the Euclidean distance between eigenvectors weighted by λ_l^t, and truncating where |λ_l|^t ≤ δ|λ₁|^t gives the diffusion map, whose dimension falls as t grows. An α-normalization by a kernel density estimate gives, as the kernel width ε → 0, a generator with invariant density q^{2(1−α)}: the density-weighted graph Laplacian at α = 0, the Fokker–Planck (Langevin) operator at α = 1/2, and the Laplace–Beltrami operator, free of the density, at α = 1. Symmetric kernels only. | Active — read 2026-10-20 (NOTE-985) in full; worth reading for QUESTION-217 as the model case of a coarse-to-fine order induced by an operator rather than an imposed partition, and for the proofs the PNAS announcement (LIT-981) deferred. The diffusion distance D_t is exactly the Euclidean distance between eigenvector coordinates weighted by λ_l^t. The number of steps t is a scale, and the kernel width ε is a second one. An α-normalization makes the small-ε limit the graph Laplacian with a density term (α = 0), the Langevin generator whose invariant density is the sampling density (α = 1/2), or the Laplace–Beltrami operator (α = 1). Read it knowing that Proposition 1’s “relative accuracy δ” for the truncated map is wrong (the bound is absolute, THEORY-745); that Appendix A’s ψ_l = φ_l/π should be φ_l/√π; that the α = 1/2 display, the circle formula, the discrete kernel of §5 and Criterion 4’s exponent are misprinted or wrong; that the boundary rate and the heat-kernel proof are incomplete (THEORY-744); and that no spectral gap is used anywhere. |
| LIT-1249 v3 | Slow Feature Analysis: Unsupervised Learning of Invariances | Wiskott & Sejnowski (2002). Learns instantaneous functions of the input whose outputs vary as slowly as possible, minimizing ⟨ẏ²⟩ under zero-mean, unit-variance and decorrelation constraints, solved in closed form within a fixed (usually quadratic) expansion by sphering and taking the smallest eigenvectors of the derivative covariance. Stacked modules in a 1-D model of the visual hierarchy learn translation, size, rotation, contrast and, less well, illumination invariance from the training stimulus, with location (“where”) among the slowest outputs alongside identity (“what”). | Active — read 2026-10-20 (NOTE-971) in full, from Wiskott’s own PostScript preprint; worth reading for the slowness objective stated exactly (minimize ⟨ẏ²⟩ under zero mean, unit variance and decorrelation), for its closed-form solution within a fixed nonlinear expansion by sphering and an eigendecomposition of the derivative covariance (THEORY-746), and for the demonstration that stacked modules learn translation, size, rotation, contrast and, less well, illumination invariance for 1-D patterns from the training stimulus alone. Read it knowing that optimality holds per module, on the training signal and within the chosen function space, and not for the greedy hierarchy; that “slower is more abstract” is its premise, not a finding, so a stack built this way is slower at higher stages by construction; that the slowest outputs include location, so slowness is not invariance; that no generative model and no learning dynamics appear; and that every stimulus is synthetic and one-dimensional. |
| LIT-1250 v2 | Diffusion maps, spectral clustering and reaction coordinates of dynamical systems | Nadler, Lafon, Coifman & Kevrekidis (2005; ACHA 2006). For data sampled from p = e^{−U}, it expands the α-normalized diffusion-map operators for small kernel width. The normalized graph Laplacian (α = 0) tends to a backward Fokker–Planck generator with potential 2U, equivalently a Schrödinger operator with potential |∇U|² − ΔU. The normalization α = 1/2 gives the generator of the Langevin dynamics ẋ = −∇U + √2ẇ itself. α = 1 gives the Laplace–Beltrami operator, free of the density. So for data from a stochastic system in equilibrium, the leading eigenfunctions approximate the system’s slowest modes, and the paper proposes them as reaction coordinates. Examples: Ornstein–Uhlenbeck processes, where the first eigenfunctions parametrize the slow variable; a double well, where the first eigenfunction is ±1 on the two wells; three wells; and the iris data. The expansions are formal, and the printed generators are twice what the stated kernel and time step give. | Active — read 2026-10-20 (NOTE-986) in full, from arXiv:math/0503445v1; worth reading for the bridge from a diffusion map to the dynamics behind the data. For samples from p = e^{−U}, the ordinary normalized graph Laplacian (α = 0) tends to a backward Fokker–Planck generator with potential 2U, and the anisotropic normalization α = 1/2 to the generator of the Langevin dynamics with potential U itself, whose slow eigenfunctions the paper proposes as reaction coordinates. It also has worked examples: Ornstein–Uhlenbeck (the finite-ε spectrum λ_k = (τ/(τ+ε))^k), a double well whose first eigenfunction is ±1 on the two wells, and a three-well potential with two pathways. Read it knowing that the generators as printed are twice what the paper’s own kernel and time step give (its Ornstein–Uhlenbeck lemma and its equation (32) disagree by that factor, which this reading checked); that “converge” means a formal small-ε expansion of the operator, with the large-sample limit cited and no spectral convergence shown; that its sufficient condition for the first eigenfunctions to depend only on the slow variable is printed the wrong way round; that its double-well example is separable, so the slow coordinate is closed by construction; and that from equilibrium samples the eigenfunctions are recovered well while the slow eigenvalue, the inter-well rate, is not. In what sense slow eigenfunctions are closed coarse variables is the record’s addition (THEORY-762). The published ACHA version was not reached. |
| LIT-1256 v3 | Diffusion wavelets | Coifman & Maggioni (2006). A self-adjoint diffusion operator T defines a multiresolution analysis by thresholding its dyadic powers at a precision ε: V_j is spanned by the eigenvectors with λ^t ≥ ε, so the levels are exactly nested spectral bands set by the operator, not by a partition. The contribution is an algorithm that squares T on each compressed orthonormal basis to build localized scaling functions and wavelets for approximations of these spaces, and compressed dyadic powers that apply the Green’s function fast. Proofs cover the uncompressed construction; the compressed scheme’s error and cost are argued, not proved. | Active — read 2026-10-20 (NOTE-984) in full; worth reading for QUESTION-217 and QUESTION-218 as the multilevel case of an operator-induced coarse-to-fine hierarchy. The levels V_j are spanned by the eigenvectors of a self-adjoint diffusion T with λ^t ≥ ε at dyadic t. They are exactly nested, the wavelet spaces W_j are spectral bands of about one octave, and a localized orthonormal basis of each level is built without computing eigenvectors. Read it knowing that the levels are spectral bands and not spatial blocks (the dyadic cubes only shape the basis); that nesting needs no spectral decay, which buys only compression; that the compressed algorithm as written thresholds T^{2^{j−1}}, not T^{2^j−1} as §5.4 and Fig. 1 say, and the paper’s own Fig. 9 follows the former (THEORY-758); that the indexing of V_j is inconsistent between (4.8), §4.6 and Theorem 27; that only the uncompressed route (Theorems 27 and 29) has proofs, while the compressed scheme’s error, localization and O(n log² n) cost are argued or observed; and that the non-self-adjoint extension is asserted, not carried out. |
| LIT-1258 v2 | Exact and Soft Successive Refinement of the Information Bottleneck | Charvin, Catenacci Volpi & Polani (Entropy 2023). Defines successive refinement for the information bottleneck in single-letter form: a coarse optimum T₁ can be extended to a fine optimum (T₁, S). This holds iff some pair of optima forms the chain X → T₂ → T₁, iff the coarse encoder is a garbling of the fine one, and, when the fine decoder is injective, iff the coarse decoder points lie in the convex hull of the fine ones. That last form gives a linear-programming test. Refinability is proved for binary X and Y, for Y a function of X, and (from a cited semigroup result) for jointly Gaussian variables. Numerically, for 3×3 and larger random sources, the hull test on computed optima often fails, especially below the second bifurcation. The loss, measured by unique information, stayed under 5.4% of I(X; T₁, T₂). The failures are not certified, since global optimality and uniqueness of the optima are left as a conjecture. | Active — read 2026-10-20 (NOTE-981) in full, from the published version in the University of Hertfordshire Research Archive; worth reading for the single-letter definition of successive refinement for the information bottleneck and its equivalent forms (Proposition 1: some pair of optimal encoders forms the Markov chain X → T₂ → T₁; Proposition 8: the coarse encoder is a Blackwell garbling of the fine one); for the convex-hull test (Proposition 4: with an injective fine decoder, refinement holds iff every decoder point q(X|t₁) lies in the convex hull of the q(X|t₂)), checkable by a linear programme; for proofs of refinability when X and Y are both binary, when Y is a function of X, and for jointly Gaussian variables (the last resting on Kline and Palmer’s semigroup result); and for soft refinement measured by unique information. Read it knowing that the linear programme tests the particular optima a solver returns, so that a failed test certifies nothing unless the optima are globally optimal and unique, which the paper leaves as Conjecture 1 and calls its own figures “mere numerical proxies” (Appendix D); that its evidence that refinement fails is therefore numerical and uncertified, with no count of how many sampled distributions failed; that the unique information computed is an upper bound on the soft measure for the same reason; that the deep-learning reading of §4.2 is speculative; and that the commentary on Proposition 8 swaps its points (ii) and (iii). |
| LIT-1261 v2 | On the Spectral Bias of Neural Networks | Rahaman et al. (2019; arXiv 2018). Proves, for compactly supported ReLU networks, that the Fourier transform falls as k^(−d−1) in almost every direction and no slower than k^(−2). Shows by experiment that a deep ReLU network fits low frequencies before high ones, even when the high ones are ten times stronger, and that its high-frequency components are the least robust to parameter noise. Shows that on a curve embedded in the plane, high latent frequencies are fitted more easily as the embedding gets longer and more convoluted. | Active — read 2026-10-20 (NOTE-983) in full; worth reading for the experiment in which a ReLU network fits low frequencies before high ones even when the high ones are ten times stronger (THEORY-759), for the Fourier analysis of piecewise-linear networks (THEORY-761), and for the observation that the frequencies that are hard to learn depend on how the data’s manifold is embedded. Read it knowing that its theorem is proved only for compactly supported networks, that the link from the theorem to the order of learning is an upper bound on speed and not a proof of order, that the volume result in Appendix D does not follow, and that its “manifold complexity” is confounded with the curve’s length. |
| LIT-1287 v2 | A Maximum-Likelihood Interpretation for Slow Feature Analysis | Turner & Sahani (Neural Computation 2007). Gives a probabilistic model for which, in a limiting case, inference and learning “yield exactly the results of SFA”, in the way that independent component analysis has a maximum-likelihood reading, and uses the model to motivate several extensions of the algorithm. Not read; this summary is from the abstract. The model’s form and the limit taken are not checked here. | Unavailable — registered 2026-10-20, not read. A search on 2026-10-20 for a lawful full text reached none. The DOI resolves to MIT Press (direct.mit.edu, article 7171), which returned a Cloudflare bot check (403); the article is not open access in any case. Unpaywall and OpenAlex report it closed, with no repository full text. Europe PMC has it (PMID 17348772) but not in PMC. Sahani’s Gatsby publication pages (~maneesh/papers/ and researchli2.html) list it with a DOI link only, no file. The Cambridge Machine Learning Group’s publications page (mlg.eng.cam.ac.uk/publications/) gives the abstract and links Turner’s old Gatsby page (gatsby.ucl.ac.uk/~turner/Publications/turner-and-sahani-2007a.html), which returns 404, as do the PDF at the same path and the guessed Cambridge paths (cbl.eng.cam.ac.uk and learning.eng.cam.ac.uk under /Public/Turner/, mlg.eng.cam.ac.uk/pub/pdf/TurSah07.pdf and variants). OpenAlex lists two repository records with no full text flagged: UCL Discovery (eprint 134626), which returned a Cloudflare bot check (403), and publications.eng.cam.ac.uk/526467, whose host no longer resolves. CiteSeerX (10.1.1.109.4162) reset the connection on four tries and returned 503 once, so whether it holds a cached author copy is unknown. The Wayback Machine reset the connection or returned 429, and the fetch tool cannot reach it. Semantic Scholar returned 429. Title searches found no author-posted copy. So this is a failure on access as much as on availability: a cached copy may exist at CiteSeerX or in the Wayback Machine’s capture of Turner’s Gatsby page. It stays Unavailable until a copy is supplied. No NOTE is filed. |
| LIT-1293 | Copyright and Artificial Intelligence, Part 3: Generative AI Training (Pre-Publication Version) | U.S. Copyright Office (2025), pre-publication version. Collecting, curating and training on copyrighted works implicate the reproduction right; model weights do so only where they have memorized substantially similar expression. Whether the copying is excused is a fair-use question, use by use. Training a foundation model will often be transformative, but less so the more its outputs serve the originals’ purpose; the non-expressive and human-learning arguments are rejected; market dilution is cognizable harm. The Office recommends letting licensing markets develop, with extended collective licensing if they fail. | Active — read 2026-10-20 (NOTE-1024) in full, in the pre-publication version of 9 May 2025, the only version copyright.gov offers on that date; worth reading as the official US statement of where training touches copyright (collection, curation, training copies, and model weights only where they have memorized a work), as an application of the Warhol v. Goldsmith use-specific test of transformative purpose to generative training (a spectrum, from research and closed non-substitutive deployments to fine-tuning that reproduces a series’s characters), and for its new market-dilution theory of harm under the fourth factor. Read it knowing that it is an agency’s analysis, not a holding; that it predates the 2025 district-court rulings on training; that its account of the facts rests mostly on the parties’ own comments; that the dilution theory is offered as “uncharted territory” with no data; and that a final version was promised but had not appeared by 2026-10-20. |
| LIT-1296 v2 | Slow Feature Analysis as Variational Inference Objective | Schüler & Wiskott (arXiv 2025). With a Gaussian random-walk prior on the latents, an instantaneous unit-variance Gaussian encoder N(g(x_t), I) and an arbitrary forward model, the evidence lower bound is a sum of per-point reconstruction terms minus ½Σ‖g(x_{t−1}) − g(x_t)‖², the unordered slowness loss. The reconstruction term then does the work of SFA’s unit-variance and decorrelation constraints, by argument, not proof. The authors say this “does not lead to full equivalence” with SFA. For linear Gaussian encoder and decoder they give the objective and the necessary condition VᵀV = WᵀĊW, and no optimum. | Active — read 2026-10-20 (NOTE-1015) in full, from arXiv v1; worth reading for an exact identity, not a limit: with a Markov prior p(z_t | z_{t−1}) = N(z_{t−1}, I) (a Gaussian random walk, no decay), an instantaneous encoder q(z_t | x_t) = N(g(x_t), I) and any forward model p_θ(x_t | z_t), the KL half of the ELBO is the unordered slowness loss ½Σ‖g(x_{t−1}) − g(x_t)‖² (plus a constant the paper drops), so the ELBO is a per-point reconstruction term minus a slowness term (Eq. 40), and the reconstruction term takes the place of SFA’s unit-variance and decorrelation constraints; and for the linear Gaussian case (Eq. 41) and its necessary condition VᵀV = WᵀĊW (Eq. 42). Read it knowing that the paper does not claim, and does not have, equivalence to SFA: it derives no optimum, so what its objective learns is the record’s derivation in THEORY-775 (on whitened input, the span of the slowest SFA directions whose lag-one autocorrelation exceeds ½, rotation unidentified; on raw input, not SFA’s subspace and not scale-invariant); that the constant d/2 per step is cancelled by an error in Eq. 35; that Eq. 41 has the sign of the reconstruction error reversed (the appendix’s Eq. 54 is right); that Eq. 42 is necessary, not “equivalent” to stationarity as §7 says; that the prior builds in independent, non-decaying latents, so closure of the latent coordinates is assumed, not found; and that its account of Turner and Sahani (2007) is two sentences of related work, which the record has not checked against that paper (LIT-1287, Unavailable). |
| LIT-1309 v2 | A Theory of Usable Information Under Computational Constraints | Xu, Zhao, Song, Stewart & Ermon (2020). Predictive V-information, H_V(Y) − H_V(Y | X) with the conditional entropies taken as the least expected log loss over a predictive family V, is the information about Y that a predictor restricted to V can use. It recovers Shannon mutual information when V is unrestricted and the coefficient of determination for linear-Gaussian V, is non-negative and zero under independence, can be asymmetric, can be raised by computing on X, and can be estimated with PAC guarantees from the Rademacher complexity of V. | Active — read 2026-10-21 (NOTE-1048) in full, from arXiv:2002.10689v1, the ICLR 2020 camera-ready; worth reading as the usable-information literature the record lacked: it defines predictive V-information, the information a predictor restricted to a family V can extract, shows that computation can create it (so the data-processing inequality fails for it), and bounds its estimation error by the Rademacher complexity of V (THEORY-846, THEORY-840). Read it knowing that the failure of the data-processing inequality is shown by informal examples (decryption, feature learning), not by a theorem, that V-information is not bounded by Shannon information unless V’s input-free predictions include the true marginal (the record’s counterexample, THEORY-846), that Proposition 1.2’s “mean absolute deviation” is the deviation about the geometric median, and that the experiments are small. |
| LIT-1326 v2 | Algorithmic monoculture and social welfare | Kleinberg & Raghavan (2021). In a model of two firms hiring from one pool by noisy rankings, if both use the same algorithm, even one more accurate than their private rankings, the quality of decisions can fall without any shocks: under two conditions on the ranking model, for every private accuracy there is a more accurate shared algorithm that is strictly dominant for both firms and lowers both firms’ utility and social welfare, a Braess paradox. The Mallows model always meets the conditions; Plackett–Luce never does. | Active — read 2026-10-21 (NOTE-1033), from arXiv:2101.05853v2: the main text and Appendices A, B and E in full, Appendices C, D and F in statement; worth reading for the exact conditions under which a shared, more accurate screening algorithm is dominant for every firm and still lowers welfare without any shock (THEORY-835), and for the identity that makes independence valuable to a second mover. Read it knowing that the model has two firms hiring one candidate each, that the conditions hold for Mallows rankings always, for Gaussian and Laplacian random utility models with three candidates, and never for Plackett–Luce, and that the losses measured are small (about 4% in one instance). |
| LIT-1336 v2 | Performative Prediction | Perdomo, Zrnic, Mendler-Dünner & Hardt (2020). When a deployed prediction changes the distribution of the outcomes it predicts, define performative stability (a model optimal for the distribution it induces) and performative optimality (least risk on the distribution it induces); show they can differ; prove that with a smooth, strongly convex loss and a distribution map ε-Lipschitz in Wasserstein distance, retraining converges to a unique stable point when ε < γ/β, that each condition is needed, and that stable points lie within 2L_zε/γ of optima. | Active — read 2026-10-21 (NOTE-1044) in full, from arXiv:2002.06673v4; worth reading for the split between performative stability (optimal for the distribution one’s own deployment induces, the fixed point of retraining) and performative optimality, for the exact conditions under which retraining converges to a unique stable point (smooth, strongly convex loss and ε < γ/β, each shown necessary; THEORY-861), and for the bound 2L_zε/γ between stable points and optima (THEORY-839). Read it knowing that its conditions hold in its own simulation only at the smallest ε, that the distribution map depends on the current model alone, and that it says nothing about how to reach the optimum. |
| LIT-1338 v2 | Epistemic Injustice in Generative AI | Kay, Kasirzadeh & Mohamed (2024). Introduces generative algorithmic epistemic injustice, with four dimensions: amplified and manipulative testimonial injustice, hermeneutical ignorance and access injustice; illustrates each with examples, particularly multilingual ones, and proposes strategies of resistance and design principles. | Active — read 2026-10-21 (NOTE-1049) in full, from the publisher’s free PDF of the AIES 2024 proceedings; worth reading for its four-way taxonomy of how generative models take part in epistemic injustice, and for the point that amplified testimonial injustice runs through a credibility excess assigned to the model (THEORY-864). Read it knowing that it is conceptual and measures nothing, that its examples are others’ reports, that its “manipulative” configuration counts deliberate fabrication as testimonial injustice although Fricker sets deliberate manipulation apart, and that its access injustice stretches the hermeneutical category. |
| LIT-1347 v2 | Verbalizable Representations Form a Global Workspace in Language Models | Gurnee, Sofroniew et al. (2026), Anthropic, Transformer Circuits Thread; arXiv:2607.15495. A Jacobian lens, the Jacobian from a layer to the output averaged over a corpus, gives each vocabulary token a direction per layer; sparse combinations of them form the “J-space”. In Claude Sonnet 4.5, Haiku 4.5, Opus 4.5 and Opus 4.6, swaps, injections and ablations show the J-space part of a representation, under a tenth of its variance, carrying verbal report, instructed concepts, two-hop intermediates and reuse of an argument by many functions, while continuation, anomaly detection and line wrapping use the same information without it. The J-space works in a middle band of layers, holds about 25 active directions, and is amplified and relayed by the weights. Ablating it flattens experiential language about the model and about others. Further sections apply it to alignment audits, show post-training installing the Assistant’s point of view, and introduce counterfactual reflection training. The authors take no position on phenomenal consciousness. | Active — read 2026-10-22 (NOTE-1069), main text in full and appendices in part, from the arXiv v1 PDF; worth reading as the first interventional evidence that a language model has a functional analogue of conscious access in the access sense: a sparse set of verbalizable directions, found by a corpus-averaged Jacobian lens, carries report, held concepts, unspoken reasoning steps and reuse of one argument by many functions, while routine processing uses the same information without it; and ablating it flattens experiential language about the self and about others alike. Read it knowing its limits: one company’s models studied by that company, unreplicated; a lens whose own statistics draw the layer band; an automatic/flexible split partly defined by the result; experiential language graded by another model; and an explicit refusal to say anything about phenomenal consciousness. Filed as THEORY-865 and THEORY-869. |
| LIT-1376 v2 | The Ledger of Will: Evidence, Revaluation, and the Preservation of Revisable Agency | Aevyra (2026). A formal account of memory for persistent agents whose central requirement is “revision sufficiency”: retained state must support a declared family of future questions, interventions and changes of evaluative law. It characterizes exact revision sufficiency for finite histories by a factorization condition, derives a lower bound on representation size, and proposes an architecture that keeps grounds and counterevidence with each commitment. | Active — read in full 2026-10-22 (NOTE-1091) from the web page, version 4.0 of 25 September 2026, with its PDF and reproducibility archive, whose checks were run and reproduce; worth reading as a careful, self-limiting formal statement of what a reduced memory must keep for an agent to stay able to revise: exact sufficiency for a declared family of future demands is factorization through the retained state (the record’s CLAIM-350), and under repeated declared revisions it is the coarsest revision-invariant refinement (Moore–Nerode), filed as THEORY-884. Elementary and, by its authors’ account, a synthesis; its architecture and experiment are proposals, unexecuted. It is the closest outside statement of CLAIM-295 and answers QUESTION-112 only for a revision family declared in advance. Despite its title it has nothing on will-organization in the record’s sense. |
| LIT-1411 v2 | Theory on Mixture-of-Experts in Continual Learning | Li, Lin, Duan, Liang & Shroff (2024; ICLR 2025). In continual learning of overparameterized linear regression tasks, drawn uniformly from N ground truths each carrying a feature signal a linear gate can read, a top-1 MoE whose gate is trained on a locality loss plus a Switch-style balancing loss sorts identically initialized experts into one set per task (or task cluster) within about M/η rounds. If gate training continues, the balancing loss eventually erases the gaps between gate outputs and routing fails, so the gate is frozen once the gaps settle; forgetting then decays as 1/t, and extra experts only lengthen exploration. Small deep-network runs show the same pattern. | Active — read 2026-10-29 (NOTE-1380), the main paper in full and the experimental appendix, from arXiv v3; worth reading as a solvable model in which identically initialized experts specialize into task sets under a locality loss, and in which a Switch-style balancing loss, left running, erases the gate gaps that encode the specialization, so the gate must be frozen. Read it knowing that the tasks are linear and carry a feature signal that makes routing separable, that task arrival is uniform, that the necessity of termination comes from its own balancing term, and that the proofs (Appendices B–J) were not checked in this reading. |
| LIT-1412 v2 | Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity | Fedus, Zoph & Shazeer (2021; JMLR 2022). Simplifies mixture-of-experts routing to a single expert per token (the Switch Transformer), with a fixed expert capacity and an auxiliary load-balancing loss α·N·Σf_iP_i, and with training techniques (router-local float32, smaller initialization, expert dropout) that allow large sparse models to be trained in bfloat16. At equal FLOPs per token, up to 7x faster pre-training than T5-Base by wall-clock, gains across 101 languages over mT5-Base, and a 1.6-trillion-parameter model about 4x faster than T5-XXL to a fixed perplexity. | Active — read 2026-10-29 (NOTE-1363) in full, from arXiv v3 (the JMLR version); worth reading as the paper that made top-1 expert routing with a fixed capacity and the auxiliary balancing loss α·N·Σf_iP_i the standard for sparse Transformers, with the stability fixes that let it scale to 1.6 trillion parameters at T5’s FLOPs per token. Read it knowing that nearly every comparison is a single run, that it never measures what experts specialize in, that its balancing loss is not minimized only by uniform routing as §2.2 says, and that one quoted margin over T5-XXL (Switch-C’s) is 0.051, not “over 0.061”. |
| LIT-1414 v2 | Auxiliary-Loss-Free Load Balancing Strategy for Mixture-of-Experts | Wang, Gao, Zhao, Sun & Dai (2024). Instead of an auxiliary balancing loss, whose gradients the authors hold interfere with language modelling, Loss-Free Balancing adds a per-expert bias to the routing scores used only for top-K selection and after each batch moves it by u·sign of the expert’s load deviation. On MoE models of 1B and 3B parameters trained on 100B and 200B tokens it gives slightly lower perplexity (9.50 vs 9.56; 7.92 vs 7.97) and much better global balance (maximal violation 0.04 vs 0.72 and 0.52) than an auxiliary loss with α = 0.001, one run each. | Active — read 2026-10-29 (NOTE-1388) in full, from arXiv v1; worth reading as the method that takes load balancing out of the gradient: a per-expert bias, used only to choose the top-K experts and moved by the sign of each expert’s load deviation on the previous batch. Read it knowing that the control responds to how often an expert is used, not to what it contributes, that the evidence is one run per configuration at 1B and 3B with perplexity gains of about 0.05, and that the conclusion’s 300B tokens for the 3B model should be 200B. |
| LIT-1439 | Statistical ranking and combinatorial Hodge theory | Jiang, Lim, Yao & Ye (2011), Mathematical Programming 127:203–244. HodgeRank: an edge flow of aggregated pairwise comparisons splits orthogonally into a gradient flow, whose potential is the least-squares global ranking (a Borda count generalized to incomplete data), and a cyclic residual, which splits into curl (local) and harmonic (global) inconsistency (Theorems 4.8, 5.1). The residual’s share, the “cyclicity ratio”, certifies the ranking. | Active — read 2026-10-23 (NOTE-1099): in full, from arXiv 0811.1067v2 (42 pp.). HodgeRank. Pairwise comparisons, aggregated into a weighted edge flow on the graph of alternatives, decompose orthogonally into a gradient flow, the l2-best global ranking found by weighted least squares, and a divergence-free residual. The residual splits into curl (local, triangular inconsistency) and harmonic (global inconsistency) parts. Binary win/loss data enter as averaged signs, and on complete balanced binary data the ranking is Borda’s. The decomposition is cardinal: curl-freeness is stronger than transitivity. It is the tool for reading the record’s conflict record. |
| LIT-1442 | Flows and Decompositions of Games: Harmonic and Potential Games | Candogan, Menache, Ozdaglar & Parrilo (2011), Mathematics of Operations Research 36:474–503. The space of finite games is a direct sum P ⊕ H ⊕ N of potential, harmonic and nonstrategic subspaces (Theorem 4.1), pulled back from the Helmholtz decomposition of the game graph’s flow of unilateral payoff differences. Potential games are P ⊕ N (Theorem 5.1). Harmonic games, such as rock–paper–scissors, generically have no pure equilibrium, and the closest potential game is given in closed form under a weighted l2 norm (Theorem 6.2). | Active — read 2026-10-23 (NOTE-1116): in full, from arXiv 1005.2405v2 (45 pp.). Every finite game in strategic form splits uniquely into a potential, a harmonic and a nonstrategic component. The split pulls back the Helmholtz decomposition of the flow of unilateral payoff differences on the game graph. Potential games are exactly the games with no harmonic component. Harmonic games generically have no pure equilibrium. The nearest potential game, in a weighted l2 norm, is given in closed form. It is the paper behind the record’s measure of a potential’s share of a negotiation, and it shows that the share is cardinal. |
| LIT-1484 | The Hive Mind is a Single Reinforcement Learning Agent | Soma, Bouteiller, Hamann and Beltrame show that in an infinite, well-mixed population, imitation of success has the expected update of a single Cross Learning bandit agent, and the weighted voter model of the honey-bee waggle dance the expected update of “Maynard-Cross Learning”, π_a ← π_a + (r_k / v^π)(1[a = k] − π_a), which is the Maynard-Smith replicator dynamic. Each member is one parallel action sample of the population’s policy, and the members themselves do not learn. Simulations, qualitative by the authors’ account, show the result surviving moderate swarm and neighbourhood sizes, and finite populations losing options for good. | Active — read in full 2026-10-24 (NOTE-1132) from arXiv v6 (21 July 2026, CC BY 4.0); worth reading as a proved aggregation result: in an infinite, well-mixed population, two imitation rules with no individual learning have the expected update of a single bandit-learning agent. Imitation of success gives Cross Learning (the Taylor replicator dynamic), the honey-bee weighted voter model gives what the authors call Maynard-Cross Learning (the Maynard-Smith replicator dynamic). The macro-agent is a stateless bandit policy over the members’ types. The biological model is deliberately partial, the simulations qualitative, and pure imitation has no way back once a type is lost or the environment changes. |
| LIT-1494 v3 | Eligibility Traces and Plasticity on Behavioral Time Scales: Experimental Support of NeoHebbian Three-Factor Learning Rules | Gerstner, Lehmann, Liakoni, Corneil & Brea (2018). Behaviour unfolds over seconds and spikes over milliseconds, so learning rules must bridge the two. In neoHebbian three-factor rules, co-activation of pre- and postsynaptic neurons sets an eligibility trace at the synapse, which changes the weight only if a third factor (reward, punishment, surprise or novelty, carried by neuromodulators or special inputs) arrives while it is set. The review sets out four recent experiments supporting eligibility traces on the scale of seconds. | Active — read in full 2026-10-24 (NOTE-1130) from the CC BY version of record; worth reading as the review that set out what the first direct experiments (2014–2017) showed about eligibility traces on the scale of seconds: in striatum about 1 s, in cortex 3–10 s with the modulator setting the sign, in hippocampus about 2 s. Mostly slices, one laboratory per region, by theoreticians whose models assumed the traces; they grant that other mechanisms could bridge the same gap, and what is well established is the hour-scale tag of consolidation. |
| LIT-1501 | A selection-mutation model for Q-learning in multi-agent systems | Tuyls, Verbeeck & Lenaerts (2003). Derives the continuous-time dynamics of Boltzmann Q-learning in two-player stateless matrix games, as later restated by Kaisers & Tuyls: ẋ_i = x_i α(τ⁻¹[(Ay)_i − x·Ay] − log x_i + Σ_k x_k log x_k), a replicator (selection) term plus an entropy (mutation) term, linking exploration in reinforcement learning to mutation in evolutionary game theory. Summary drawn from the abstract and from Kaisers & Tuyls 2010, not from the paper. | Unavailable — filed 2026-10-24 for the owner’s question how the record stands on multi-agent reinforcement learning; a reading was attempted the same day and failed on access. The ACM Digital Library (doi:10.1145/860575.860687) answered every request from this client with a Cloudflare challenge (403). Unpaywall and OpenAlex list no open copy. The repository records OpenAlex names hold no file this client could reach: the VUB repository (biblio.vub.ac.be) page is a script-only catalogue entry, ULB’s DI-fusion record shows no attached full text, the Hasselt document server returned 503 and a search result describes its record as “No Fulltext”, and CiteSeerX reset the connection. Web searches found no author copy. The model the paper derives is read instead through Kaisers & Tuyls 2010 (LIT-1511, NOTE-1140), which restates its equation and shows that it describes frequency-adjusted, not plain, Q-learning. It stays Unavailable until a copy is supplied. No NOTE is filed. |
| LIT-1503 | QMIX: Monotonic Value Function Factorisation for Deep Multi-Agent Reinforcement Learning | Rashid et al. represent a team’s joint action-value Q_tot as a mixing of per-agent values Q_a(τ^a, u^a) constrained by ∂Q_tot/∂Q_a ≥ 0, with non-negative mixing weights produced from the global state by hypernetworks. Monotonicity makes the joint argmax the profile of individual argmaxes, so execution is decentralised and maximisation is linear in the number of agents. It represents more than a sum (VDN), learning the optimal policy of a two-step game VDN misses, and beat VDN and independent Q-learning on six StarCraft II maps; it cannot represent joint values in which an agent’s best action depends on the others’. | Active — read in full 2026-10-24 (NOTE-1154) from arXiv v2 (6 June 2018, the ICML camera-ready, arXiv non-exclusive licence); worth reading for one structural condition: if the joint action-value is non-decreasing in each agent’s own value, the profile of individually greedy actions is a greedy joint action, so a collective can act on a common value while each member consults only its own. The condition is sufficient, not necessary, and it cannot represent a joint value in which one agent’s best action depends on what the others do at the same step. Its evidence is a two-step matrix game and StarCraft II win rates on six small maps. |
| LIT-1504 | Real World Games Look Like Spinning Tops | Czarnecki, Gidel, Tracey, Tuyls, Omidshafiei, Balduzzi & Jaderberg (NeurIPS 2020). Games in which players can signal n bits before the outcome is settled contain every win–loss pattern on 2ⁿ strategies, so cycles of length 2ⁿ (Go: at least 2¹⁰⁰⁰). Nash clustering orders a game’s strategies into clusters weakly ordered by population performance. Empirical games sampled from small board games and the AlphaStar league show cluster size peaking at middling strength, the “spinning top”, and population-based training converges once the population is large enough. | Active — read in full 2026-10-24 (NOTE-1146) from arXiv v2 (17 June 2020); worth reading for one clean theorem, one general lemma and a hypothesis with evidence. The theorem: where players can signal n bits before the result is settled, every pattern of wins and losses among 2ⁿ strategies, a cycle through all of them included, is realized. The lemma: Nash clusters are weakly ordered by population performance, and beating a population that contains a whole cluster lifts a strategy above it. The hypothesis, that real games are shaped like a spinning top, with cycles thickest at middling strength, rests on sampled empirical games. The abstract says it is proved and the body says it is not. The random-game theorem’s proof is defective, and the fixed-memory fictitious play result needs a replacement rule it does not state. |
| LIT-1506 | Optimal Reward Functions in Distributed Reinforcement Learning | Wolpert & Tumer: members of a multi-agent system each run a reinforcement learner, and the designer must give each a private utility such that their learning maximises a world utility G. A private utility is factored when it ranks a member’s own alternatives, others fixed, as G does; the Wonderful Life Utility G(ζ) − G(CL(ζ)), with the member’s effect set clamped, is factored and has higher learnability than G. A sketched approximation sets the clamp to the expected action rather than zero; in a 60-agent bar problem, clamping to the average action learned best across attendance levels, clamping to absence close behind, and G itself worst. | Active — read in full 2026-10-24 (NOTE-1153) from the NASA Technical Reports Server copy (document 20010066523, a 7-page scanned report marked public use permitted), not the version in the Intelligent Agent Technology proceedings; worth reading as the most compact statement the record holds of difference rewards. A member’s private utility is factored when it ranks the member’s own alternatives as the world utility does, the others held fixed; the Wonderful Life Utility, world utility minus its value with the member’s effect set clamped, is factored and learns faster than the world utility itself. Clamping to the average action did better than clamping to the member’s absence. The optimal clamp is only sketched (“too long to go through here”) and the experiment is one bar problem. |
| LIT-1507 | Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer | Shazeer, Mirhoseini, Maziarz, Davis, Le, Hinton & Dean (2017). A layer of up to 131,072 feed-forward experts with a trainable noisy top-k gate, inserted between LSTM layers, raises capacity over 1000-fold at roughly constant computation and beat the best published language-modelling and translation results of the time. Without balancing losses the gate concentrates on a few experts, a self-reinforcing imbalance; losses penalizing unequal summed gate weight and unequal load remove it and, in one ablation, cut test perplexity from 39.8 to 35.6. | Active — read in full 2026-10-24 (NOTE-1145) from arXiv v1, the only version; worth reading here for one observation and its fix, not for its scale results. The gate of a sparsely-gated mixture of experts “tends to converge to a state where it always produces large weights for the same few experts”, which the authors call self-reinforcing because favoured experts are trained more. Two auxiliary losses, the squared coefficient of variation of each expert’s summed gate weight and of a smooth estimate of its load, remove the imbalance. In the one ablation (a 256-expert model, 10 epochs) no loss gives CV 3.04 and a peak-to-mean load of 17.8 at test perplexity 39.8, any loss gives 35.6–35.7. The fix equalizes use regardless of merit; the noise in the gate is there mainly to make the load estimate differentiable. |
| LIT-1508 | Re-evaluating Evaluation | Balduzzi, Tuyls, Pérolat & Graepel (NeurIPS 2018). Evaluation data are antisymmetric matrices: Elo is their row mean in logit space, exact only when the log-odds are curl-free, and inflatable by copies; the Schur decomposition gives a multidimensional Elo for cyclic interactions. Nash averaging scores agents (and tasks) against the maximum-entropy Nash equilibrium of a zero-sum meta-game, which ignores exact copies when that equilibrium is unique. Applied to Atari, human ties with the best agents. | Active — read in full 2026-10-24 (NOTE-1155) from arXiv v2 (30 October 2018, the authors’ “NIPS 2018, final version”); worth reading as the source of the Hodge and Schur reading of evaluation data and of Nash averaging. Elo is the row mean of the log-odds matrix, exact only when the log-odds have no cyclic part, and copies of an agent move it. Nash averaging scores each agent against the maximum-entropy Nash team of the meta-game. Its invariance to copies is proved only for the Nash set (Lemma 1): when the meta-game has more than one equilibrium, copying an agent moves the maximum-entropy selection and the Nash averages with it, which the reading checked by script. The Atari re-evaluation used an LP solution, not the maximum-entropy one. |
| LIT-1509 | Open-ended Learning in Symmetric Zero-sum Games | Balduzzi, Garnelo, Bachrach, Czarnecki, Pérolat, Jaderberg & Graepel (ICML 2019). Every symmetric zero-sum game between parametrized agents splits into a transitive part and a cyclic part relative to a measure on agents. In cyclic games individual performance is meaningless, so the paper measures populations, by the value of the game between them and by the convex hull of their evaluation rows (the gamescape). Response to the Nash mixture enlarges the gamescape, and response to the rectified Nash, which trains each supported agent on what it beats, grows it in Blotto-type games where self-play does not. | Active — read in full 2026-10-24 (NOTE-1143) from arXiv v2 (13 May 2019, the authors’ “ICML 2019, final version”); worth reading for one invariant and one reframing. The invariant: the value of the zero-sum game between two populations, which does not depend on the choice of Nash, ignores members that are behavioural mixtures of others, and cannot be positive for a population inside another’s hull. The reframing: in a cyclic game an individual agent’s performance means nothing, so progress is the growth of a population’s gamescape. Its functional Hodge decomposition is right but is stated with a degenerate inner product, and it depends on a measure over agents. Experiments are two small resource-allocation games. |
| LIT-1510 | QTRAN: Learning to Factorize with Transformation for Cooperative Multi-Agent Reinforcement Learning | Son, Kim, Kang, Hostallero & Yi define Individual-Global-Max (IGM): a joint action-value is factorised by per-agent values when the joint argmax is the profile of individual argmaxes. Additivity (VDN) and monotonicity (QMIX) are sufficient for it; QTRAN instead trains the sum of individual values plus a state value to equal the joint value at the greedy profile and to bound it elsewhere (Theorem 1), a condition also necessary up to positive rescaling. On 310 random 3×3 matrix games it learned the optimum in all, VDN and QMIX failed in 177; it also did better on Gaussian squeeze and a penalised predator-prey task. | Active — read in full 2026-10-24 (NOTE-1141) from arXiv v1 (14 May 2019, the ICML version with its supplement, arXiv non-exclusive licence); worth reading for the Individual-Global-Max condition, that the agents’ individually greedy actions form a greedy joint action, and for showing it strictly weaker than additive or monotonic factorisation. The authors’ own supplement notes that every single-state matrix game satisfies it, so IGM has bite only where each agent sees part of the state; it is a condition on whether members acting on local information can jointly reach the collective optimum, not a condition for a common currency. Evidence is matrix games, Gaussian squeeze and a predator-prey variant. |
| LIT-1511 | Frequency adjusted multi-agent Q-learning | Kaisers & Tuyls (AAMAS 2010). The replicator-plus-entropy model of Boltzmann Q-learning derived by Tuyls et al. (2003, 2005), ẋ_i = x_i α(τ⁻¹[(Ay)_i − x·Ay] − log x_i + Σ_k x_k log x_k), assumes that every Q-value changes at rate α. In Q-learning only the chosen action’s value is updated, so E[Q̇_i] carries a factor x_i, and trajectories in the prisoners’ dilemma, battle of the sexes and matching pennies deviate from the model, depending on initialization. Frequency Adjusted Q-learning scales each update by min(β/x_i, 1) and follows the model wherever x_i ≥ β. | Active — read in full 2026-10-24 (NOTE-1140) from the AAMAS 2010 proceedings PDF on the IFAAMAS site; read in place of Tuyls, Verbeeck & Lenaerts 2003 (LIT-1501), whose model it restates and corrects. Worth reading as the correction to “Q-learning is replicator dynamics plus mutation”: the selection–mutation equation describes Q-learning with Boltzmann exploration only if every action’s estimate is updated at the same rate. In Q-learning the chosen action is updated more often, so the expected change of Q_i carries a factor x_i, and the learning traces depart from the model, depending on how the Q-values are initialized. Frequency Adjusted Q-learning divides the step by x_i and matches the model. Evidence is three 2×2 games, plotted, with no quantitative measure of fit. |
| LIT-1512 | Mean Field Multi-Agent Reinforcement Learning | Yang, Luo, Li, Zhou, Zhang & Wang (2018). Factorizing each agent’s Q-function into pairwise terms and expanding about the neighbours’ mean action reduces many-agent interaction to an agent against the average of its neighbours. Mean field Q-learning and actor-critic follow; MF-Q is proved to converge to the Nash Q-value under strong assumptions. With 500 and 1000 agents in Gaussian Squeeze the mean field methods learn where four baselines fail; MF-Q reproduces the order parameter of a 20 × 20 Ising model against MCMC; and it wins a 64-versus-64 battle game. | Active — read in full 2026-10-24 (NOTE-1148) from arXiv v5 (15 December 2020), the ICML 2018 paper with appendices; worth reading as the standard way multi-agent RL reaches hundreds of agents: each agent’s Q-function takes its own action and the mean action of its neighbours, so many-body interaction becomes a two-body one between an agent and the average of the rest. Mean field Q-learning is proved to converge to a Nash Q-value under strong assumptions, every stage game met in training having a global optimum or a saddle point among them, and learns where four baselines fail with 500 and 1000 agents. The approximation needs agents that are homogeneous and interchangeable within a neighbourhood; the bound on its error is loose and the cancellation of errors argued, not proved; and the contraction step needs a high Boltzmann temperature while the convergence assumptions need it to go to zero. |
| LIT-1513 | Multi-agent Reinforcement Learning in Sequential Social Dilemmas | Leibo, Zambaldi, Lanctot, Marecki & Graepel (2017). Defines a sequential social dilemma as a Markov game whose cooperative and defecting policies, played against each other, give payoffs satisfying the social dilemma inequalities. Independent deep Q-learners in a fruit-Gathering game learn aggression when apples are scarce or a tagged rival is out long; in Wolfpack they hunt together when the joint-capture bonus and radius are high. Both games embed Prisoner’s Dilemmas, but network size raises defection in Gathering and lowers it in Wolfpack, which no matrix game captures. | Active — read in full 2026-10-24 (NOTE-1151) from arXiv v1, the AAMAS 2017 paper; worth reading as the paper that moved social dilemmas from matrix games to learned policies in Markov games. Two two-player games, Gathering (apples, and a beam that removes the rival for a while) and Wolfpack (lone or joint capture of prey), both yield Prisoner’s Dilemma payoff matrices when the learned policies are played off, yet respond in opposite directions to agent capacity: a larger network makes Gathering agents more aggressive and Wolfpack agents more cooperative, because aggression is the harder policy to implement in one and cooperation in the other. Agents learn aggression where apples are scarce. It is a descriptive simulation with independent learners, no figures of spread, and no protective institution or depletable commons. |
| LIT-1514 | Adaptive Mixtures of Local Experts | Jacobs, Jordan, Nowlan & Hinton (1991). A system of expert networks and a gating network, trained with the error of a single expert selected stochastically by the gate (or the negative log-likelihood of a mixture of Gaussians), so that each expert produces the whole output and an expert better than the gate-weighted average gains responsibility for the case. On speaker-independent vowel discrimination, 4 or 8 linear experts learned twice as fast as matched backpropagation networks at equal accuracy and divided the task by class pair; all but 2 or 3 experts went unused. | Active — read in full 2026-10-24 (NOTE-1147) from the scan of the published article on Geoffrey Hinton’s University of Toronto page; worth reading as the origin of the mixture of experts and as a learning rule in which authority over a case goes to whichever expert does better than the gate-weighted average. Redefining the error as that of a single expert chosen stochastically by the gate makes the experts compete rather than cooperate, and the log-likelihood form adapts the best-fitting expert fastest. On a four-class vowel task, mixtures of 4 or 8 linear experts reached criterion in about half the epochs of backpropagation networks with matched parameters, at the same accuracy, and in every run all but 2 or 3 experts ended with mixing proportions effectively 0. The authors read that as economy; nothing in the paper tests whether the idle experts could recover. |
| LIT-1515 | Optimal Payoff Functions for Members of Collectives | Wolpert & Tumer (2001). Members of a collective each maximise a private payoff utility; the design question is which payoffs make the members’ success yield a high provided world utility. Against the “team game”, which gives each member the world utility, the paper uses “a novel mathematical framework” to review the class of payoffs that are easy to maximise and aligned with the world utility, the Aristocrat Utility and a new variant of the Wonderful Life Utility, and reports experiments in which they beat earlier COIN utilities, the team game and “perhaps the most natural version” of endogenizing externalities (abstract, RePEc). | Unavailable — filed 2026-10-24 for difference rewards (the Wonderful Life Utility); a reading was attempted the same day and failed on access to a lawful copy. The version of record is closed at World Scientific; Unpaywall and OpenAlex report it closed with no repository copy. The second author’s publications page at Oregon State (web.engr.oregonstate.edu/~ktumer), which lists the paper, failed certificate verification through this client’s proxy and returned 503 to a second fetcher. CiteSeerX, which OpenAlex lists, reset every connection. It is not on arXiv (both authors’ arXiv listings checked) or NASA NTRS. A copy is posted on a third party’s course-library server (jmvidal.cse.sc.edu), not by the authors or their institution; it was not used. The closest lawful text, the authors’ NASA report of the same programme with a near-identical abstract, is filed and read as LIT-1506. It stays Unavailable until a copy is supplied or the author’s page can be reached. No NOTE is filed. |
| LIT-1516 | α-Rank: Multi-Agent Evaluation by Evolution | Omidshafiei, Papadimitriou, Piliouras, Tuyls, Rowland, Lespiau, Czarnecki, Lanctot, Pérolat & Munos (Scientific Reports 2019). Agents in a K-player meta-game are ranked by the stationary distribution of a Markov chain on pure profiles whose transitions are fixation probabilities of Fermi imitation with rare mutation. Unique at every finite ranking intensity α; as α → ∞ the transitions become those of Markov-Conley chains, the sink components of the better-response graph, which the paper ties to the chain components of the replicator flow. Applied to AlphaGo, AlphaZero, MuJoCo soccer and poker meta-games. | Active — read in full 2026-10-24 (NOTE-1156) from arXiv v4 (4 October 2019); worth reading as the evolutionary alternative to Nash for ranking agents when wins cycle. It ranks strategy profiles by the stationary distribution of a finite-population, rare-mutation imitation chain, unique at every finite ranking intensity α. As α grows its transitions become those of Markov chains on the sink strongly connected components of the better-response graph. The paper calls these sinks Markov-Conley chains and ties them to the chain components of the replicator flow. The large-α limit of the stationary distribution, and so the selection among several sinks, is not characterized. In the symmetric single-population model a copied strategy changes the ranking (checked by script), so it does not share Nash averaging’s resistance to padding. |
| LIT-1517 | Counterfactual Multi-Agent Policy Gradients | Foerster, Farquhar, Afouras, Nardelli & Whiteson train decentralised actors with one centralised critic Q(s, τ, u) and credit agent a with A^a = Q(s, τ, u) − Σ_{u′a} π^a(u′a | τ^a) Q(s, τ, (u^−a, u′a)): the value of the joint action against the critic’s average over a’s own alternatives under a’s policy, the others’ actions fixed. The baseline does not depend on a’s action, so it does not change the expected gradient (Lemma 1). On decentralised StarCraft micromanagement with a narrow field of view, it beat independent and central-V/QV critics on four maps. | Active — read in full 2026-10-24 (NOTE-1152) from arXiv v3 (11 December 2024, arXiv non-exclusive licence), the corrected version; worth reading as the standard statement of counterfactual credit in cooperative multi-agent learning: each agent’s action is scored against the critic’s average over that agent’s own alternatives, the others’ actions held fixed, and that baseline provably leaves the expected gradient unchanged. The counterfactual values are the learned critic’s estimates, which exist only for actions the agents sometimes take, so the method credits agents that all act on every step and does not recover the value of one that is never heeded. Results are StarCraft win rates on four small maps. |
| LIT-1519 | Learning with Opponent-Learning Awareness | Foerster et al. replace an agent’s objective V¹(θ¹, θ²) with its value after the opponent’s anticipated naive gradient step, V¹(θ¹, θ² + Δθ²), and differentiate through Δθ² = η∇θ²V², adding (∇θ²V¹)ᵀ∇θ¹∇θ²V² · δη to the naive gradient. In the memory-one iterated prisoner’s dilemma, two exact-gradient LOLA learners played tit-for-tat-like policies in 81% of runs (return −1.06 against −1.98 for naive learners); in iterated matching pennies they settled at the mixed Nash equilibrium. A policy-gradient version cooperated in the Coin Game, and with opponent modelling did so less reliably. | Active — read in full 2026-10-24 (NOTE-1150) from arXiv v4 (19 September 2018, the AAMAS 2018 text, arXiv non-exclusive licence); worth reading as the first learning rule that models another learner’s update and climbs its own return through it: a LOLA agent adds to its gradient the effect of its own parameters on the opponent’s next naive step. Two exact-gradient LOLA learners reached tit-for-tat-like reciprocity in the iterated prisoner’s dilemma where naive learners defected. Shaping is one step deep, assumes access to the opponent’s parameters (or a behavioural clone of them), and is shown only in two-player games; its numbers have one internal inconsistency (see the NOTE). |
| LIT-1520 | Coupled replicator equations for the dynamics of learning in multiagent systems | Sato & Crutchfield (2002; PRE 2003). Two agents choose by softmax over reward memories Q_i that decay at rate α. In the limit where learning is slow relative to play, each agent’s choice distribution follows ẋ_i/x_i = β[(Ay)_i − x·Ay] + α Σ_j x_j log(x_j/x_i): a coupled replicator dynamic plus an entropy-like term towards uniform play. With α = 0 it is the conservative two-population replicator equation; with α > 0 it is dissipative. In rock–scissors–paper, simulations show quasiperiodic tori and Hamiltonian chaos without decay, and limit cycles, intermittency and chaotic attractors with it. | Active — read in full 2026-10-24 (NOTE-1144) from arXiv:nlin/0204057v2 (18 August 2002); worth reading as a four-page derivation of the learning dynamics of softmax reinforcement learners with forgetting: two such agents follow coupled replicator equations plus a term α_X Σ_j x_j log(x_j/x_i), set by the rate at which reward memories decay, that pushes each agent towards uniform play. Without forgetting the system is conservative, Hamiltonian in zero-sum games, and in rock–scissors–paper it is chaotic for nonzero tie rewards; with forgetting it is dissipative, with limit cycles and chaotic attractors. The memory update is not Watkins’s Q-learning: every action’s memory is updated with its average reward at the same rate. |
| LIT-1521 | A Simple Adaptive Procedure Leading to Correlated Equilibrium | Hart & Mas-Colell (Econometrica 2000; handout 1996). Regret matching: each period a player who last played j switches to k with probability R_t(j, k)/μ, the positive part of its average gain had it played k whenever it played j, and otherwise repeats j. If all players do so, the empirical distribution of play converges almost surely to the set of correlated equilibria. The proof runs through Blackwell’s approachability theorem; regrets vanish exactly when the empirical distribution approaches the correlated-equilibrium set. Inertia, a positive probability of repeating, is needed for the result. | Active — read in full 2026-10-24 (NOTE-1142) from the published article as posted on Sergiu Hart’s homepage; worth reading as the theorem that correlated equilibrium is reachable by simple adaptive play: if every player switches from its last action to each alternative with probability proportional to its positive regret for not having made that switch in the past, the empirical joint distribution of play converges almost surely to the set of correlated equilibria. No device, no forecasts and no best replies are needed; the common history is the correlating device. The limits matter: convergence is to the set, not to a point; correlated equilibria that are interior and not pure Nash cannot be the limit of the empirical distribution; and the players must observe one another’s past play. |
| LIT-1522 v2 | MultiDiffusion: Fusing Diffusion Paths for Controlled Image Generation | Bar-Tal, Yariv, Lipman & Dekel (2023). MultiDiffusion is a training-free generation process for a pre-trained text-to-image diffusion model that binds several diffusion generation processes together through an optimization task with shared parameters or constraints. It gives controllable generation, such as panoramas at a chosen aspect ratio and images following spatial guidance from segmentation masks to bounding boxes, without retraining or fine-tuning (abstract). | Active — read 2026-10-25 (NOTE-1158) in full, from arXiv v1 and the PMLR proceedings PDF; worth reading for the clearest published form of several denoiser runs acting on one shared state, each with authority over part of the canvas, fused at every step by a per-pixel weighted average (THEORY-958), and for its panorama and region-layout results (THEORY-948). Read it knowing that its closed form (Eq. 5) is the minimizer of its printed loss only for 0/1 weights, that Proposition 3.1’s guarantee needs zero loss at every step and so does not describe the method as run, that most of the layout gain comes from the early masked bootstrapping phase rather than from fusion, and that “without introducing computational overhead” means one network call per view per step. |
| LIT-1524 v2 | Cleaning up the neighborhood: A full classification for adversarial partial monitoring | Lattimore & Szepesvári (2018; ALT 2019). Completes the classification of finite adversarial partial-monitoring games by minimax regret to all games, including the degenerate ones (with actions optimal only on a lower-dimensional part of the simplex) that Bartók et al. (2014) left open: 0 with no pair of neighbouring actions, Θ̃(√n) when locally observable, Θ(n^2/3) when globally but not locally observable, and Ω(n) otherwise. A new NeighborhoodWatch2 attains √n with high probability by redistributing mass onto degenerate actions in proportion to the neighbours they lie between. It corrects a slip in Bartók et al.’s T^2/3 lower-bound proof and, in the preprint’s words, an exchanged expectation and maximum in the earlier localisation argument. A second algorithm, RelExp3, gives point-locally observable games an asymptotic √n bound free of arbitrarily large game constants, with a matching linear dependence on the number of feedback symbols. | Active — read 2026-10-25 (NOTE-1162) in full, from the arXiv preprint v1 of 23 May 2018 (arxiv.org/pdf/1805.09247v1, 24 pp.); the ALT 2019 version (PMLR 98:529–556) was fetched and its theorem statements, footnote and contribution list compared, not read in full. Worth reading for the completed classification of finite adversarial partial-monitoring games, degenerate and duplicate actions included (THEORY-963); for how actions that are optimal only on a boundary enter it, through the neighbourhood set and not as neighbours, and for its correction of the lower-bound proof of Bartók et al. (2014) (THEORY-961); for NeighborhoodWatch2, a simpler algorithm with a high-probability √n bound; and for RelExp3 on point-locally observable games (THEORY-950). Read it knowing that the arXiv text says Foster and Rakhlin and Bartók et al. “inadvertently exchanged an expectation and maximum” in their localisation argument and the published text no longer says so, though both abstracts say “correct errors in previous analyses”; that it proves only the two pieces of the classification that were missing or flawed and cites the rest; that its constants are game-dependent and can be large; and that a few small typos survive in the arXiv text. |
| LIT-1526 v2 | Online Learning and Blackwell Approachability with Partial Monitoring: Optimal Convergence Rates | Kwon & Perchet (2017). Blackwell approachability generalizes regret minimization (multi-criteria optimization, online optimization of a convex loss, online linear optimization under a cumulative constraint). Under partial monitoring the decision maker does not see outcomes but a random signal correlated with the decision–outcome pair, or with the outcome only. The paper gives the first approachability algorithms for polytope target sets with convergence rate O(T^-1/2) when the signal does not depend on the decision and O(T^-1/3) for general signals, and shows that these rates cannot be improved without further assumptions on the objectives or the signals (abstract). | Active — read 2026-10-25 (NOTE-1159) in full, from the PMLR copy and its supplementary material; worth reading for the reduction of approachability under partial monitoring to full-information approachability in a lifted space of flags, for the optimal worst-case rates, O(T^-1/3) for general signals and O(T^-1/2) when the signal does not depend on the decision (THEORY-947), and for the exploration floor that decision-dependent signals require (THEORY-957). Read it knowing that approachability is assumed rather than decided, that the guarantees concern the average payoff and so let violations be compensated over time, that optimality is worst case over games, that the constants grow with a simplicial subdivision, and that the proof for outcome-dependent signals is omitted. |
| LIT-1528 v3 | Partial Monitoring—Classification, Regret Bounds, and Algorithms | Bartók, Foster, Pál, Rakhlin & Szepesvári (2014). In a partial-monitoring game the learner picks an action, the environment an outcome, and the learner suffers a loss and receives a feedback signal, both fixed functions of action and outcome. For every game with finitely many actions and outcomes (against adversarial opponents, every non-degenerate one) the minimax regret is zero or scales as T^1/2, T^2/3 or T, up to constants and logarithmic factors; computationally efficient algorithms attain it within a logarithmic factor, with individual bounds on expected regret for i.i.d. outcomes (abstract). | Active — read 2026-10-25 (NOTE-1164) in full, from the authors’ preprint of March 2013 on Alexander Rakhlin’s MIT page; worth reading for the complete classification of finite partial-monitoring games by minimax regret (0, Θ̃(√T), Θ(T^2/3), Θ(T); THEORY-964), for the linear-algebra conditions, global and local observability, that decide the class (THEORY-962), and for the algorithms CBP and NeighborhoodWatch. Read it knowing that the adversarial classification covers non-degenerate games only, that Lattimore and Szepesvári (2019, LIT-1524) completed it to degenerate games, corrected the kernel in its T^2/3 lower-bound proof, and, in their preprint, an exchanged expectation and maximum in NeighborhoodWatch’s localisation (THEORY-963, THEORY-961), that the classification concerns the best policy in a fixed, known game, and that the copy read is the preprint, not the journal text. |
| LIT-1529 v2 | Multi-LoRA Composition for Image Generation | Zhong et al. (2024). Composing several LoRAs in a text-to-image model gets harder as their number grows. Taking a decoding-centric view, the paper gives two training-free methods: LoRA Switch alternates between LoRAs at each denoising step, and LoRA Composite incorporates all of them at every step to guide synthesis. On ComposLoRA, a testbed of 480 composition sets, a GPT-4V-based evaluation finds both better than the prevalent baseline, most clearly as the number of LoRAs grows (abstract). | Active — read 2026-10-25 (NOTE-1160) in full, from arXiv v2, the TMLR version; worth reading for two ways of letting separately trained low-rank operators act on one denoising trajectory without merging them, by turns (LoRA Switch) or by averaging their guided predictions (LoRA Composite), for GPT-4V and human judgements preferring both to weight merging as LoRAs are added (THEORY-955), and for its ablation in which, at equal shares of steps, the switching interval and the LoRA acting first change the result (THEORY-949). Read it knowing that the merging baseline’s summed update grows with the number of LoRAs, which the design does not separate from the methods’ idea; that the two methods are never compared head to head by the judge; that the judge has a positional bias of about ten points and correlates 0.45 with two annotators; and that §5.1.1 attributes LoRA Switch’s scores to LoRA Composite. |
| LIT-1530 v2 | Invariant Learning Dynamics of Transformers in Inductive Reasoning Tasks | Musat, Pimentel, Zucchet & Hofmann (2026), arXiv:2607.11875. For a class of block-list inductive tasks that includes associative recall, k-hop induction and in-context n-grams, it proves that population gradient descent keeps an idealized attention-only transformer’s key–query and output weights on a low-dimensional invariant manifold fixed by the data’s symmetries, whose coordinates are interpretable head actions. On it, it proves that a saturated in-weights circuit suppresses the in-context circuit’s gradient by the fraction of data it does not solve and that burstiness multiplies it, shows empirically that initialization decides which of several induction circuits wins, and detects circuits in trained models by projecting onto the coordinates. | Active — read 2026-10-26 (NOTE-1177) in full, from the arXiv copy (v2); worth reading for its proof that symmetries of the data confine population gradient descent to a subspace of interpretable head actions (THEORY-981), for its theorem that a saturated memorized shortcut starves the in-context circuit of gradient on the data it solves (THEORY-1009), and for its scan in which initialization picks one of several induction circuits, read by the authors as cooperation through shared units and competition for layers (THEORY-1001). Read it knowing that the architecture is idealized (fixed orthogonal write maps, merged key-query, orthonormal embeddings), that invariance is proved and attraction is not, that the results are for gradient descent while its circuit-detection experiment uses AdamW, and that the competition scan was run in training projected onto seven coordinates. |
| LIT-1531 v3 | AtP*: An efficient and scalable method for localizing LLM behaviour to components | Kramár, Lieberum, Shah & Nanda (2024), arXiv:2403.00745. Attribution patching (AtP; Nanda 2023), a first-order gradient estimate of every node’s activation-patching effect, is tested against exhaustive patching on Pythia models from 410M to 12B. Verifying nodes in AtP’s order finds the largest-effect nodes far more cheaply than unordered or sampling-based search. Its false negatives come from saturated attention and from cancelling direct and indirect effects; AtP* adds an exact softmax for queries and keys and layer-wise gradient dropout, and a subsampling test bounds the size of nodes still missed. On the two prompt distributions, simpler variants are cheaper than AtP*. | Active — read 2026-10-26 (NOTE-1190) in full, from arXiv v1, with the comparison figures read from page renderings; worth reading as the systematic test of attribution patching against exhaustive activation patching on models up to 12B, for the two named causes of its misses (saturated attention, cancelling direct and indirect effects) and their fixes, and for the statistical bound on what is still missed (THEORY-996). Read it knowing that the ground truth is single-node noising on one model family and five settings, that only ranks are scored, that the numbers are only in charts, and that AtP* is not the cheapest variant on its two distributions. |
| LIT-1532 v2 | Sparse Feature Circuits: Discovering and Editing Interpretable Causal Graphs in Language Models | Marks et al. (2024), arXiv:2403.19647, ICLR 2025. Circuits are found over sparse-autoencoder features and autoencoder error terms of Pythia-70M and Gemma-2-2B, scoring nodes and edges by linear approximations to their indirect effects. On subject–verb agreement, small feature circuits carry much of the behaviour where neuron circuits need many more nodes, though only with the error terms kept. A human can remove features judged task-irrelevant from a classifier’s circuit (SHIFT), and the same method is run without supervision over thousands of behaviours clustered from the Pile. | Active — read 2026-10-26 (NOTE-1187) in full, from the arXiv copy (v3, the ICLR 2025 version); worth reading as the method that turned sparse-autoencoder features into circuits, by linear approximations of indirect effects over features and autoencoder errors together, and for its own evidence that the features alone are not the circuit: with the error nodes removed, its feature circuits recover little of the behaviour (THEORY-1007). Read it knowing that faithfulness is scored on the latter two thirds of each circuit only, that the interpretability of the circuits is the authors’ annotation, that the half-performance figure for Gemma neurons lies beyond its plotted range, and that its debiasing method is ML practice, which this record does not hold. |
| LIT-1533 v2 | Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet | Templeton et al. (2024), Transformer Circuits Thread. Sparse autoencoders of about 1M, 4M and 34M features, sized by scaling laws for the autoencoder’s own loss, are trained on the middle-layer residual stream of Claude 3 Sonnet and explain at least 65% of its variance. They yield features that respond to the same concept across languages and in images, that steer the model when clamped, and that include features for unsafe code, bias, sycophancy and deception. Whether a concept has its own feature depends on its training frequency relative to the dictionary’s size. | Active — read 2026-10-26 (NOTE-1178) in full, from the article’s own page on the Transformer Circuits Thread; worth reading as the demonstration that sparse-autoencoder features of a production model are abstract (multilingual, multimodal) and can steer it, for the finding that a concept gets its own feature only once its training frequency passes about one over the number of live features (THEORY-992), and for the observation that the most active features are mostly not the ones that matter to an answer (THEORY-987). Read it knowing that its interpretability and steering evidence is shown on features the authors selected, that the model, its size and some plot units are withheld, that the reconstructions explain only at least 65% of the variance, and that its wider thesis that these features are the model’s working concepts is open (THEORY-965). |
| LIT-1534 v2 | A Mathematical Framework for Transformer Circuits | Elhage et al. (2021), Transformer Circuits Thread. Rewrites a transformer without changing it: heads act independently and add into a linear residual stream, each head is a low-rank QK matrix that sets where to attend and a low-rank OV matrix that sets what is moved, and with attention patterns frozen an attention-only model is a sum of end-to-end path terms. Zero-layer models are bigram tables, one-layer attention-only models bigram and skip-trigram tables read off the weights, and two-layer models use K-composition with a previous-token head to form induction heads, [a][b] … [a] → [b]. A correction of the composition diagram adds more composition than first reported. Models with MLPs are left open. | Active — read 2026-10-26 (NOTE-1189) in full, from the Transformer Circuits article; worth reading for the exact decomposition of an attention-only transformer into end-to-end paths and of each head into separable QK and OV circuits (THEORY-988, re-derived and checked by the record), for the three kinds of head composition, and as the paper in which induction heads are found, by K-composition with a previous-token head, in a two-layer attention-only model (THEORY-977). Read it knowing that the models are toys without MLPs, that the framework ignores layer norm and biases, that the composition diagram was corrected after publication, and that the interactive figures carry much of the evidence. |
| LIT-1535 v2 | Differentiation and Specialization of Attention Heads via the Refined Local Learning Coefficient | Wang, Hoogland, van Wingerden, Furman & Murfet (2024), arXiv:2410.02984, ICLR 2025. Refined variants of the local learning coefficient, a complexity measure from singular learning theory, are applied to individual components of a two-layer attention-only transformer, tracing how attention heads progressively differentiate into distinct functional roles during training, what data they specialize to, and a previously unidentified multigram circuit (abstract). | Active — read 2026-10-26 (NOTE-1188) in full, from the arXiv PDF of v1; worth reading as the record’s first developmental use of the learning coefficient, for per-head trajectories that separate attention heads by function (THEORY-975), and for its observations of heads changing role as circuits form (THEORY-967, Proposed). Read it knowing that it is one two-layer attention-only model with three appendix seeds, that the estimates are trusted only for their ordering, that cluster counts were chosen by eye, and that nothing is predicted before it happens. |
| LIT-1536 v2 | Locating and Editing Factual Associations in GPT | Meng et al. (2022), arXiv:2202.05262, NeurIPS 2022. Causal tracing corrupts the subject’s embeddings with noise and restores one hidden state at a time: in GPT-2 XL, recall of known facts depends most on middle-layer MLP outputs at the last subject token and on attention at the last token. Treating an MLP’s output matrix as a linear associative memory, ROME inserts a fact by a closed-form rank-one update keyed on the last subject token. On the CounterFact dataset of 21,919 counterfactuals, ROME is the only editor tested with both high generalization to paraphrases and high specificity to neighbouring subjects. | Active — read 2026-10-26 (NOTE-1166) in full, from the arXiv copy (v5, the NeurIPS 2022 version); worth reading for causal tracing, a corrupt-and-restore intervention that locates where a fact’s recall depends on hidden states, for the finding that in GPT-2 XL that dependence concentrates at the last subject token in middle-layer MLPs, and for the rank-one edit that tests it (THEORY-998). Read it knowing that the storage hypothesis it draws is wider than what it shows (THEORY-1011), that LIT-968 proves a working rank-one edit does not locate a fact, and that several numbers and figure references in v5 do not match. |
| LIT-1538 v2 | In-context Learning and Induction Heads | Olsson et al. (2022), arXiv:2209.11895. Induction heads, attention heads that complete [A][B] … [A] → [B], develop at precisely the point of a sudden sharp increase in in-context learning, visible as a bump in the training loss. Six lines of evidence argue that they may be the mechanism of most in-context learning in large transformers: strong and causal for small attention-only models, correlational for larger models with MLPs (abstract). | Active — read 2026-10-26 (NOTE-1173) in full, from the arXiv print of the Transformer Circuits article; worth reading as the first mechanistic account of a transition in training, for the in-context learning score, and for the way it grades its own evidence by model class. Read it knowing that the case is causal only for small attention-only models (THEORY-973), that the large-model thesis is correlational and by analogy (THEORY-990), that every model was trained once on the same data order, and that its dates for the window disagree between sections. |
| LIT-1539 v2 | Towards Understanding Fine-Tuning Mechanisms of LLMs via Circuit Analysis | Wang, Hu, Du, Cheng, Wang & Zou (2025), arXiv:2502.11812, ICML 2025. On five arithmetic tasks where LoRA fine-tuning of Pythia-1.4B brings large gains, EAP-IG circuits found at successive checkpoints keep most of their nodes while most of their edges change, unlike earlier reports that fine-tuning only adds components. A circuit-aware LoRA that assigns higher rank to the layers whose edges change most beats uniform and random rank allocation at similar size (the abstract’s 2.46% average gain over LoRA is not recoverable from the tables, which give about 1.8–2.4 points), and the union of two subtasks’ circuits approximates a two-step task’s circuit. | Active — read 2026-10-26 (NOTE-1183) in full, from the arXiv copy (v2, the ICML 2025 version); worth reading for its tracking of EAP-IG circuits across fine-tuning checkpoints on tasks the pretrained model fails, with node change rates of 15–24% against edge change rates of 66–80%, and for using the layers whose edges change most to allocate LoRA rank. Read it knowing that the node–edge contrast has no null comparison at the same edge budget and that a fixed edge budget produces it by itself (THEORY-997), that it concerns fine-tuning a pretrained model on arithmetic rather than development from initialization, and that the abstract’s 2.46% gain is not recoverable from its tables. |
| LIT-1540 v2 | Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small | Wang et al. (2022), arXiv:2211.00593. On sentences like “When Mary and John went to the store, John gave a drink to”, GPT-2 small predicts the indirect object through 26 attention heads in seven classes, found by path patching: duplicate-token and induction heads mark the repeated name, S-inhibition heads pass that on, and name mover heads copy the remaining name. Mean-ablating every other head keeps 87% of the logit difference. Knocking out the name movers costs only 5%, because backup heads take over, and a greedy search finds knockouts under which circuit and model differ by up to 87%. | Active — read 2026-10-26 (NOTE-1167) in full, from the arXiv copy; worth reading as the first end-to-end circuit for a natural behaviour in a pretrained language model, for path patching, for the backup heads that take over when the main ones are knocked out, and for its three tests of a circuit explanation, which its own circuit passes on average and fails under a greedy search (THEORY-980). Read it knowing that the behaviour is fifteen templates in one model, that MLPs are left out of the circuit though removing them breaks the task, that the dropout explanation of the backups is a hypothesis (THEORY-995), and that a few of its numbers and its greedy algorithm do not match their descriptions. |
| LIT-1541 v2 | On the Biology of a Large Language Model | Lindsey et al. (2025), Transformer Circuits Thread. Ten case studies of Claude 3.5 Haiku with the attribution graphs of LIT-1552, each validated by perturbing features in the model: two-hop factual recall through an intermediate “Texas” representation, rhymes planned before a line is written, addition features reused across contexts, a default “can’t answer” circuit inhibited by known entities whose misfire yields hallucination, unfaithful chains of thought, multilingual circuits, medical diagnosis, refusals, a jailbreak and a model with a hidden goal. The authors present each as an existence claim about specific prompts. | Active — read 2026-10-26 (NOTE-1181) in full, from the Transformer Circuits page; worth reading for ten case studies that show, on single prompts and with interventions, a two-hop intermediate in Claude 3.5 Haiku (THEORY-984), a rhyme planned on the newline before a line is written (THEORY-1017), a jailbreak whose target word is not represented before it is written (THEORY-1018), a trained-in goal represented in every Assistant context of a model organism (THEORY-976), and a default “can’t answer” circuit that known entities inhibit. Read it knowing that the authors find a satisfying graph for about a quarter of the prompts they try, that the graphs shown are curated and of a replacement model (LIT-1552), that several cruxes run through attention the graphs cannot see, and that its general statements about the model (THEORY-982) go beyond what the case studies show. |
| LIT-1543 v2 | Zoom In: An Introduction to Circuits | Olah et al. (2020), Distill, doi:10.23915/distill.00024.001. The essay that opened the Circuits thread. It offers three “speculative claims” about neural networks: features, which are directions in a layer’s activation space, are their fundamental unit; features connected by weights form circuits that can be studied rigorously, and meaningful algorithms can be read off the weights; and analogous features and circuits form across models and tasks. The examples are from InceptionV1: curve and high-low frequency detectors, a pose-invariant dog head detector, and a car feature spread across dog neurons, which the authors call superposition. For universality the authors call their own evidence anecdotal. | Active — read 2026-10-26 (NOTE-1182) in full, from the Distill article; worth reading as the programme statement of circuits research: features as the unit, circuits as weighted subgraphs between them that can be read off the weights, and universality across models, each offered by the authors as a deliberately speculative claim. Read it knowing that the examples are from one vision model, InceptionV1, that the detailed evidence was promised to later articles of the thread, and that the universality claim rests, in the authors’ word, on anecdote (THEORY-1013, THEORY-986). |
| LIT-1545 v2 | Towards Monosemanticity: Decomposing Language Models With Dictionary Learning | Bricken et al. (2023), Transformer Circuits Thread. Sparse autoencoders trained on 8 billion MLP activations of a one-layer transformer with 512 ReLU neurons, at widths from 512 to 131,072, yield features that are rated far more interpretable than the neurons, that mostly do not correspond to any neuron, whose downstream effects on the logits match their activations, and that recur in a second model trained from another seed. As the dictionary widens, features split into families of finer ones, and the authors conclude that there may be no single correct number of features. | Active — read 2026-10-26 (NOTE-1191) in full, from the article’s own page on the Transformer Circuits Thread; worth reading as the paper that made sparse-autoencoder features the field’s unit of analysis below the neuron, for its case that a feature is a causal unit and not only a pattern in the data (downstream effects the autoencoder never saw), and for feature splitting, which makes the feature set a resolution chosen by dictionary size (THEORY-1016). Read it knowing that everything is one 512-neuron MLP layer of a one-layer model, that the four deep-dive features are cherry-picked by the authors’ own account, that the human ratings are one author’s, and that the 79% of loss recovered is measured against zero-ablation, a baseline the authors call an overestimate (THEORY-991). |
| LIT-1547 v2 | Towards Automated Circuit Discovery for Mechanistic Interpretability | Conmy, Mavor-Parker, Lynch, Heimersheim & Garriga-Alonso (2023), arXiv:2304.14997, NeurIPS 2023. ACDC automates the patching step of circuit discovery: from the output back, it removes each edge of a model’s computational graph whose replacement by a corrupted activation raises the KL divergence from the full model by less than a threshold. Against circuits found by hand it ranks edges with AUCs of 0.85–0.98 on IOI, Greater-Than and Docstring, above two adapted pruning methods, and on Greater-Than keeps 68 of about 32,000 edges, all in the reported circuit. It is sensitive to metric, ablation and visiting order, misses negative components at sparse thresholds, and recovers compiled tracr circuits only with zero activations. | Active — read 2026-10-26 (NOTE-1170) in full, from arXiv v4, the NeurIPS 2023 version; worth reading for the workflow it names behind manual circuit discovery, for ACDC as that workflow’s patching step made into an algorithm, and for the first scoring of circuit-finding methods against circuits found by hand (THEORY-970). Read it knowing that the hand-found circuits are themselves partial, that the results move with the metric, the ablation and the order of visiting edges, that sparse runs miss negative components, that an OR gate keeps one input (THEORY-966), and that §4.1 claims a tracr-reverse lead its own Table 2 does not show. |
| LIT-1549 v3 | Causal Scrubbing: a method for rigorously testing interpretability hypotheses | Chan, Garriga-Alonso, Goldowsky-Dill, Greenblatt et al. (2022), Alignment Forum (Redwood Research). Causal scrubbing tests a mechanistic hypothesis, stated as a correspondence from an interpretable computational graph into the model’s, by replacing every activation with its value on another input the hypothesis treats as equivalent and measuring how much of the behaviour survives. On a parenthesis-balance classifier, hypotheses recover 93% down to 72% of the loss as they are made more specific; on induction in a two-layer attention-only model, a naive hypothesis recovers 35% and a refined one 86%. | Active — read 2026-10-26 (NOTE-1168) in full, the main post of the sequence from the Alignment Forum; worth reading for the method that turns a circuit hypothesis into a mechanical test, for the argument for resampling ablation that later circuit work adopts, and for its own list of the ways a hypothesis can pass while wrong. Read it knowing that the test sees only the partitions of the data a hypothesis induces (THEORY-978), that its two applications are summarized here and detailed in companion posts the record has not read, and that hypotheses recovering 72% to 86% are accepted under a test that permits only exact swaps. |
| LIT-1550 v2 | MIB: A Mechanistic Interpretability Benchmark | Mueller et al. (2025), arXiv:2504.13151, ICML 2025. A benchmark with a circuit-localization track over IOI, arithmetic, MCQA and ARC, and a causal-variable-localization track that adds RAVEL, on four language models plus an InterpBench model with a known circuit. In circuit localization, edge attribution patching with integrated gradients does best on average; in causal-variable localization, supervised distributed alignment search (DAS) does best, and sparse-autoencoder features selected by a learned mask are never better than the model’s own hidden dimensions. | Active — read 2026-10-26 (NOTE-1169) in full, from the arXiv copy (v2, the ICML 2025 version); worth reading for its fixed-counterfactual design of two localization tracks, for the finding that public sparse-autoencoder features were worse units than neurons for known causal variables in every cell tested (THEORY-968), and for the carry-the-one case, where a representation that looked like the variable was the correlated output digit. Read it knowing that the best causal-variable method, DAS, is supervised by the causal model it is scored against, that the SAE comparison is on residual streams of two models under one selection method, and that the method rankings are interpretability practice. |
| LIT-1551 v2 | Finding Alignments Between Interpretable Causal Variables and Distributed Neural Representations | Geiger, Wu, Potts, Icard & Goodman (2024; arXiv 2023), CLeaR 2024. Distributed alignment search (DAS) learns, by gradient descent, an orthogonal rotation of a layer such that swapping rotated subspaces between inputs reproduces a high-level causal model’s counterfactuals, replacing brute-force search over disjoint neuron sets. In a network trained on hierarchical equality and in BERT fine-tuned on monotonicity NLI it finds alignments with interchange intervention accuracy 1.00 where axis-aligned ones reach 0.73 and 0.51; BERT’s entailment subspace decomposes into the identities of the two words, the equality network’s relation subspace does not. | Active — read 2026-10-26 (NOTE-1192) in full, from arXiv v4, the CLeaR 2024 version; worth reading for distributed interchange interventions and their training objective, for the exact example in which an algorithm fails in the neuron basis and holds after a rotation, and for the finding that BERT’s entailment subspace decomposes into the two words’ identities while an equality network’s relation does not (THEORY-983). Read it knowing that IIA is reported on the rotation’s training data, that Table 2’s brute-force row is a copy of Table 1’s, that the learned subspaces are fixed only up to rotations within them, and that its random-network control reaches 0.64 at width 4,096 (THEORY-1002). |
| LIT-1552 v2 | Circuit Tracing: Revealing Computational Graphs in Language Models | Ameisen et al. (2025), Transformer Circuits Thread. A cross-layer transcoder, whose features each read the residual stream at one layer and write to every later MLP output, stands in for a model’s MLPs. On one prompt, with attention patterns and normalization denominators frozen and error terms added back, this local replacement model reproduces the model’s activations exactly, and its attribution graph decomposes each feature’s pre-activation into linear edges. The graphs, pruned and grouped into supernodes, give hypotheses about an 18-layer model and Claude 3.5 Haiku that are then tested by perturbing features in the underlying model. | Active — read 2026-10-26 (NOTE-1185), main text in full and five appendices, from the Transformer Circuits page; worth reading for the attribution graph, a prompt’s computation rewritten exactly in sparse features of a local replacement model (THEORY-979), for its own measurements of how far that replacement responds to perturbation like the model (THEORY-994), for the parallelogram constraint on one-step sums (THEORY-971), and for a limitations section that names where the method fails: attention, error nodes, inactive features. Read it knowing that the graphs explain the replacement, not the model, that intervening confirms a direct edge almost by construction, that the replacement matches the model’s top token on about half of prompts, and that its faithfulness figures differ between the main text and the appendix. |
| LIT-1554 v2 | Transcoders Find Interpretable LLM Feature Circuits | Dunefsky, Chlenski and Nanda (2024), arXiv:2406.11944, NeurIPS 2024. A transcoder is a wide, sparsely activating ReLU MLP trained to imitate one MLP sublayer’s input–output map. With attention patterns frozen, the attribution of one transcoder feature to a later one factors into the earlier feature’s activation times an input-invariant dot product of decoder and encoder vectors, which allows weights-based circuit analysis through MLPs. On GPT2-small, Pythia-410M and Pythia-1.4B transcoders are at least on par with MLP SAEs in sparsity, faithfulness and human-rated interpretability, and they give a sparse account of MLP10 in GPT2-small’s greater-than circuit. | Active — read 2026-10-26 (NOTE-1174) in full, from the arXiv copy (v2, the NeurIPS 2024 version); worth reading as the source of transcoder circuit analysis, for the factorization of a feature-to-feature attribution into an input-dependent activation times an input-invariant weight (THEORY-974), on which the attribution graphs of LIT-1552 are built, and for its blind case studies, which grade the method against hidden prompts and report the failures. Read it knowing that the comparison with SAEs is one learning rate, one seed and three models (THEORY-1005), that Gemma Scope found the reverse on Gemma 2 2B (NOTE-482), that attention patterns are frozen and QK circuits are not explained, and that its counts of case studies disagree between sections. |
| LIT-1555 v2 | Crosscoding Through Time: Tracking Emergence & Consolidation Of Linguistic Representations Throughout LLM Pretraining | Bayazit, Mueller & Bosselut (2025), arXiv:2509.05291, ACL 2026. Sparse crosscoders trained between pairs and triplets of pretraining checkpoints with large performance and representation shifts discover and align features across checkpoints; a metric, Relative Indirect Effects (RelIE), traces the stage at which a feature becomes causally important for a task. Crosscoders detect feature emergence, maintenance and discontinuation during pretraining (abstract). | Active — read 2026-10-26 (NOTE-1186) in full, from the arXiv PDF of v2 (30 April 2026); worth reading for the method of tracking features across pretraining checkpoints with one joint dictionary, and for separating a feature’s presence (decoder norm) from its causal use (RelIE). Read it knowing that its developmental trends rest on the authors’ annotation of a few top features (THEORY-1004, Proposed), that it covers one layer and two or three checkpoints at a time, and that a feature shared across checkpoints is shared by construction of the dictionary, not shown to be carried forward. |
| LIT-1575 v3 | Curve Detectors | Cammarata et al. (2020), Distill, doi:10.23915/distill.00024.003. The first of a planned three-part study of curve detectors in InceptionV1, on their behaviour. The ten curve units of layer mixed3b each prefer a curve of one orientation and jointly span all orientations, and are argued to detect curves causally, generally and purely: by feature visualization, a blind hand-labelling of over 800 dataset examples of unit 3b:379, attribution, rotated dataset examples and synthetic curves that drive the unit up to 24 standard deviations above its dataset mean. Curve families are also found in three neighbouring layers. How the units are built from earlier ones is left to the next article. | Active — read 2026-10-27 (NOTE-1198) in full, from the Distill article; worth reading as the first detailed case for one neuron family in a trained vision model, the ten curve units of InceptionV1’s mixed3b, argued by feature visualization, blind hand-labelling of dataset examples, attribution, rotation tuning and synthetic stimuli (THEORY-1028). Read it knowing that the dataset analysis is of one unit, 3b:379, that the circuit, the downstream use and the hand-written reimplementation that Zoom In counts among its seven arguments are deferred again to a later article of the thread, and that its claim of purity is not settled here (THEORY-1022). |
| LIT-1576 v2 | Transformer Feed-Forward Layers Are Key-Value Memories | Geva et al. (2021), arXiv:2012.14913, EMNLP 2021. A transformer feed-forward layer, f(x·Kᵀ)·V, is an unnormalized key–value memory: each key correlates with textual patterns in training prefixes and each value induces a distribution over the output vocabulary. In a 16-layer WikiText-103 model, keys’ patterns are shallow in lower layers and semantic in upper ones; upper-layer values favour the token that follows their key’s patterns; a layer’s output composes hundreds of memories, and the residual stream refines the prediction from layer to layer. | Active — read 2026-10-27 (NOTE-1200) in full, from the arXiv copy (v2, the EMNLP 2021 version); worth reading as the origin of the reading of a transformer’s MLP as a key–value memory, with each hidden unit a key over input patterns and each output row a value over the vocabulary, and for its evidence in one 16-layer language model: keys whose top training prefixes share recognizable patterns, shallow below and semantic above, and upper-layer values that agree, weakly, with what follows those patterns (THEORY-1033). Read it knowing that the general reading is wider than the evidence (THEORY-1030), that the agreement it reports peaks at about 3.5%, that the patterns were labelled by one annotator per key, and that ROME and MEMIT took the matrix-level version of the idea, not this unit-level one. |
| LIT-1578 v2 | Does Localization Inform Editing? Surprising Differences in Causality-Based Localization vs. Knowledge Editing in Language Models | Hase et al. (2023), arXiv:2301.04213, NeurIPS 2023. On 652 CounterFact facts GPT-J knows, the layer where causal tracing places a fact does not predict at which MLP layer an edit of that fact succeeds: regressing edit success on the edit layer explains most of its variance, and adding the tracing effect adds at most 3.2 points of R², across ROME, MEMIT, constrained fine-tuning and five editing problems, and again on GPT-2 XL, zsRE and representation zeroing. Only Fact Forcing, which edits on the same noised input tracing uses, shows a small reliable relation. | Active — read 2026-10-27 (NOTE-1194) in full, from the arXiv copy (v2, the NeurIPS 2023 version); worth reading as the direct test of whether causal tracing tells where to edit a fact: in GPT-J, fact by fact, it does not. The tracing effect at a layer adds at most 3.2 points of explained variance in edit success once the edit layer is known, and ROME’s rewrite score is above 96% at every layer but the last (THEORY-1024). Read it knowing that its own account of why, that tracing and editing answer different questions, is offered and not tested (THEORY-1026), that “where a fact is stored” means where causal tracing peaks, and that its Table 3 and Figure 6 disagree on one cell. |
| LIT-1579 v2 | Causal scrubbing: results on induction heads | Chan, Garriga-Alonso, Goldowsky-Dill, Greenblatt, Lin et al. (2022), Alignment Forum (Redwood Research). In a two-layer attention-only language model with two induction heads, the standard induction story (queries and values from the token embedding, keys from the previous-token head) recovers 35% of the loss the induction heads contribute on tokens where induction helps. Letting the heads match the last three tokens, read layer 0’s average effect, and treat three layer-0 heads that attend to copies of the current token as attending to itself raises it to 86%, and to 89% with a crude model of where the previous-token head deviates. | Active — read 2026-10-27 (NOTE-1196) in full, from the Alignment Forum’s own API; worth reading as the quantified test of the textbook induction circuit in a two-layer attention-only model, pathway by pathway, and for what the refinements found: three-token matching, layer-0 heads that attend to copies of the current token, and a previous-token head that deviates on quotes, articles and punctuation (THEORY-1020). Read it knowing that its percentages are of the 0.053 nats the induction heads contribute on a filtered token subset, not of a label-shuffled loss like the paren post’s, that its headline 89% is the bonus hypothesis and the main post’s 86% the fourth, and that the model is not published. |
| LIT-1580 v2 | Causal scrubbing: results on a paren balance checker | Chan, Garriga-Alonso, Goldowsky-Dill, Greenblatt, Lin et al. (2022), Alignment Forum (Redwood Research). Causal scrubbing applied to a three-layer, two-head bidirectional transformer that classifies parenthesis strings as balanced. The high-level hypothesis (two heads carry an equal-count-and-starts-open test, one head the horizon test) recovers 88–93% of the loss; making it specific about how the heads compute their tests brings it to 72%. Two still more specific hypotheses score higher than the one they refine, which the authors judge illegitimate. | Active — read 2026-10-27 (NOTE-1195) in full, from the Alignment Forum’s own API; worth reading as the worked case of causal scrubbing on a model whose algorithm is known, with the full table of hypotheses, their losses and standard errors, and for the authors’ own finding that two more specific hypotheses scored higher for reasons they judge illegitimate (THEORY-1029). Read it knowing that the model and the prior dissection are unpublished, that every number is one run of 2,000 scrubbed samples, and that its prose states the horizon test backwards in two places. |
| LIT-1581 v2 | Attribution Patching: Activation Patching At Industrial Scale | Nanda (2023), blog post, neelnanda.io. Introduces attribution patching: a first-order estimate of every activation-patching effect at once, the difference between clean and corrupted activations dotted with the metric’s gradient on the corrupted run, from two forward passes and one backward pass instead of one forward pass per patch. On the IOI task in GPT-2 Small it approximates head-level patches well and residual-stream and first-MLP patches badly. The author offers it as an exploratory heuristic for finding what to patch, not a replacement for patching. | Active — read 2026-10-27 (NOTE-1201) in full, from the author’s own page; worth reading as the origin of attribution patching, for its derivation of the estimate and the identities that make it cheap and additive (THEORY-1023), for the account of where a first-order estimate should fail (THEORY-1021), and for its framework of what patching can and cannot show: sufficiency against necessity, specific against general understanding, and parallel against serial redundancy. Read it knowing that the author calls it an incomplete project, that its empirical evidence is one task in GPT-2 Small shown only in plots and a notebook, and that the systematic test is Kramár et al. (LIT-1531). |
| LIT-1582 v2 | Mass-Editing Memory in a Transformer | Meng et al. (2022), arXiv:2210.07229, ICLR 2023. MEMIT inserts many facts at once by treating each MLP output matrix in a range of layers implicated by causal tracing as a linear associative memory, solving a least-squares update for the whole batch, and spreading each fact’s required change over those layers. On GPT-J it inserts 10,000 CounterFact counterfactuals with score 85.8, against 50.3 for ROME applied one fact at a time, and does likewise on GPT-NeoX (82.0) and for 10,000 true zsRE facts. | Active — read 2026-10-27 (NOTE-1197) in full, from the arXiv copy (v2, the ICLR 2023 version); worth reading as ROME’s extension from one edit to thousands, by a closed-form least-squares update spread over the range of MLP layers that causal tracing implicates, and for the scaling curves that show sequential single edits failing where it holds (THEORY-1031). Read it knowing that it inherits ROME’s storage reading (THEORY-1011) rather than testing it, that its layer range is chosen from traces averaged over facts, which Hase et al. (LIT-1578) show do not predict per fact where an edit works, and that its zsRE specificity hardly moves for any method. |
| LIT-1583 v2 | Causal scrubbing: Appendix | Chan, Garriga-Alonso, Goldowsky-Dill, Greenblatt et al. (2022), Alignment Forum (Redwood Research). The technical appendix to causal scrubbing. It defines ”% loss recovered” against a label-shuffled baseline and argues against zero and mean ablation with three examples. It proves that the inputs the algorithm samples are the maximum-entropy distribution preserving the joint inputs of every interpretation node, shows how a superposed toy model is explained as monosemantic terms plus an unexplained interference term, and builds a false hypothesis that scores far above the model. It proposes, without solving, adversarial validation. | Active — read 2026-10-27 (NOTE-1202) in full, from the Alignment Forum’s own API; worth reading for the details the main post defers: the definition of the metric, the full case against zero and mean ablation, how unimportant inputs are grouped, a proof that the scrubbed inputs are the maximum-entropy distribution consistent with the hypothesis (THEORY-1025), how superposition is handled, and a second constructed false hypothesis that scores above the model. Read it knowing that its Lemma 1 says “product distribution” where its proof shows the joint, and that it says both results posts use a randomized-loss baseline when the induction post does not. |
| LIT-1584 v2 | Auditing language models for hidden objectives | Marks, Treutlein et al. (2025), arXiv:2503.10965. Claude 3.5 Haiku is given continued pretraining on synthetic documents describing 52 fictional reward-model biases, then trained by SFT and RL to exploit 47 of them and to hide why. It comes to exploit the 5 held-out biases, and new ones described to it in context, more often, which the authors take as a learned objective of reward-model sycophancy. In a blind auditing game the three teams with weights and training data found the objective, by sparse-autoencoder features, prefill attacks and data search, and the API-only team did not. An unblinded study of eight techniques finds sparse-autoencoder features matched by simple semantic search in some uses and better in others. | Active — read 2026-10-27 (NOTE-1193) in full, from arXiv v2; worth reading as the construction and validation of the model organism on which the Biology paper’s hidden-goal case (THEORY-976) is run, for its behavioural definition of an objective tested by generalization to behaviours never reinforced (THEORY-1032), and for a blind auditing game and an unblinded comparison of eight auditing techniques, in which sparse-autoencoder features were matched by semantic search on the fastest team’s query but succeeded where it failed on others (THEORY-1027). Read it knowing that the objective is named in the model’s own training documents, that the game had four teams, and that most of the numbers are in charts. |
| LIT-1624 v2 | An Overview of Early Vision in InceptionV1 | Olah et al. (2020), Distill (printed DOI 10.23915/distill.00024.002, unregistered on 2026-10-27). A guided tour of InceptionV1’s first five layers (conv2d0, conv2d1, conv2d2, mixed3a, mixed3b; 1,056 units), every unit sorted into one of 77 “neuron groups” with its share of the layer, and selected circuits (complex Gabors, a black-and-white detector, small circles, triangles, boundary, curve-based and oriented-fur detectors). The authors call the groups ad hoc and many of them low-confidence; by their own captions 37% of the units are in catch-all, broad, low-confidence or unstudied groups. The curve family is 10 of mixed3b’s 480 units. | Active — read 2026-10-27 (NOTE-1206) in full, from the Distill article; worth reading as the only place the Circuits thread gives a denominator: it sorts every one of the 1,056 units of InceptionV1’s first five layers into 77 named groups, with each group’s share of its layer (THEORY-1042). Read it knowing that the groups are, in the authors’ words, “ad-hoc, human defined”, that 391 of the 1,056 units (37%) sit in groups the authors’ own captions call catch-all, broad and not well defined, low-confidence, unstudied or not understood, and that being sorted into a group is not the same as being understood in Zoom In’s sense. The authors’ wider claim, that neuron families are real and recur across models, is THEORY-1040. |
| LIT-1625 v2 | Polysemanticity and Capacity in Neural Networks | Scherlis, Sachan, Jermyn, Benton and Shlegeris (2022), arXiv 2210.01892. Defines a feature’s capacity, C_i = (w_i·w_i)²/Σ_j(w_i·w_j)², in [0, 1] with total at most the embedding dimension, and treats superposition as the optimal allocation of capacity. In a model computing Σ(W_i·x)² + b to fit y = Σ v_i x_i², the expected loss depends only on capacities and embedding lengths, and the optimum is in closed form: with fourth moment k ≤ 3 features are ignored or given a whole dimension; with k > 3 a feature is shared when its importance lies in an interval set by k. Efficient embeddings are exactly the block-semiorthogonal ones. | Active — read 2026-10-27 (NOTE-1205) in full, from arXiv v4; worth reading as the source of the toy model the causal scrubbing appendix (LIT-1583, §6) analyses, and as an exactly solvable case of superposition: in a one-hidden-layer model with quadratic activations, the expected loss depends on the weights only through each feature’s capacity and embedding length, and the optimal allocation has a closed form in which features are ignored, shared or given a whole dimension according to their importance and the inputs’ fourth moment (THEORY-1036, re-derived and checked numerically). Read it knowing that its other models are shown only as phase-diagram figures, and that its bearing on trained networks is the authors’ hope, not a result (THEORY-1035). |
| LIT-1629 v2 | Curve Circuits | Cammarata et al. (2021), Distill (printed DOI 10.23915/distill.00024.006, unregistered on 2026-10-27). The second of the curve-detector articles, on how the ten curve units of InceptionV1’s mixed3b are built: from conv2d0 Gabors through conv2d2 and mixed3a lines and early curves, each family tiled along the curve’s tangent, with rotational and scale equivariance said to cut the circuit’s 50,000-plus parameters about 50-fold. Negative weights from opposite early curves remove the 180° echo, and deleting them by circuit editing brings it back. A hand-set network written from the description approximates the curve units on synthetic curves, and substituting it for them recovers 26 of the 46 top-5 classifications lost by removing them, on 2,000 images. | Active — read 2026-10-27 (NOTE-1207) in full, from the Distill article; worth reading as the second part of Curve Detectors (LIT-1575) and the one that supplies the three arguments that article deferred: the circuit that builds the ten mixed3b curve units from earlier families, a downstream use in mixed4a, and a hand-written “artificial artificial neural network” whose curve units approximate the natural ones (THEORY-1038). Read it knowing that every comparison but one is qualitative (figures), that the one number, the accuracy recovered by substituting the hand-written units, is on 2,000 validation images and called preliminary by the authors, that the hand-written units are grayscale, and that the claim that the circuit is “a simple algorithm” is the authors’ (THEORY-1047). |
| LIT-1649 v2 | Compositional Dynamics in Learning and Mechanics | Spivak (2026). An operad ArrSm of “smooth adaptive arrangements” (a responsive parameter space, a lens of output and input maps between manifolds, and a real potential) with two operad functors into polynomial coalgebras, discrete-time input-output machines, built by lens internalization (Corollary 3.3.4). The functors differ only in the integrator: Φconf stores position and steps down the sharpened gradient, Φphase stores position and momentum and takes a symplectic-Euler step. Φconf gives gradient descent with backpropagation, weight decay and Newton’s method. One arrangement of harmonic particles on a finite directed graph gives the discrete wave equation under Φphase and the discrete heat equation under Φconf, both governed by the symmetrized graph Laplacian, and for harmonic closed systems the phase step preserves the symplectic pairing exactly. | Active — read in full 2026-10-27 (NOTE-1225), from arXiv v2; worth reading as a compositional account in which the coupling data (wiring, potential, response) and the dynamical regime are separate choices: one arrangement of harmonic particles gives a discrete wave equation or a discrete heat equation depending only on whether the integrator stores momentum, and gradient descent with backpropagation is the same construction applied to a parameterized map. Rigorous for its categorical core and its worked examples; many further examples are reported from Claude-written code the author did not audit. |
| LIT-1800 v2 | CycleGAN, a Master of Steganography | Chu, Zhmoginov & Sandler (2017). A CycleGAN trained between aerial photographs and maps reconstructs details of a photograph that its generated map does not show, because it hides them in a low-amplitude, high-frequency signal: noise of three 8-bit levels or JPEG compression destroys the reconstruction, low-frequency change does not. The same sensitivity lets an imperceptible perturbation of almost any map make the reverse generator produce any chosen photograph. The authors trace it to the cycle-consistency loss between domains of unequal complexity, by argument; the evidence is one model, mostly shown by example. | Active — read in full 2026-10-29 (NOTE-1346) from arXiv v2; worth reading as the plainest demonstration that a perfect round trip can rest on a channel the intermediate’s reader never sees, so that round-trip agreement certifies invertibility, not content. One model, mostly shown by example. |
| LIT-1801 v2 | Word Embeddings as Metric Recovery in Semantic Spaces | Hashimoto, Alvarez-Melis & Jaakkola (2016). If text were a random walk whose steps depend on distance in a latent space, negative log co-occurrence over a long window, less per-word terms, would converge to squared latent (geodesic) distance whatever the one-step link. Under that toy model, which the authors say is not a model of language, GloVe, softmax word2vec and PMI-SVD have the true embedding among their optima, up to isometry; a negative-binomial “metric regression” matches them on analogy, series and classification tasks, and word embedding of graph random walks does manifold learning on MNIST. Evidence that real co-occurrences form a Euclidean space is a correlation of 0.75 with human similarity and two nearest-neighbour statistics. | Active — read in full 2026-10-29 (NOTE-1329) from the ACL Anthology PDF; worth reading for the proof that, under a metric random walk, log co-occurrence over long windows converges to squared latent distance whatever the one-step link, so embeddings identify a metric and coordinates only up to isometry; its evidence that real corpora have a Euclidean semantic space is weaker than its theorem. |
| LIT-1803 | Relational Composition as Parallel Transport: What Per-Entity Frames Force in Attention | Sarkar, Mukherjee & Sato (2026), ARXIV-2610.34945. Under per-entity O(d) frames, an invariant attention score depends on two entities only through norms and ⟨xᵢ, Tᵢⱼxⱼ⟩, and a transport computed from one vector per endpoint has rank at most one, so relations must be supplied. On parsed CLUTRR unconstrained transports score 0.848 out of distribution, a transformer 0.436, and orthogonal ones 0.202; on StepGame’s commutative algebra rotations help. The paper calls the premise “a chosen inductive bias, not a consequence of fresh names”. | Active — skimmed 2026-10-29 (NOTE-1326) as the companion LIT-1820 cites for its premise. Worth keeping for its own statement that per-entity frame freedom is an inductive bias, and for the parsed-CLUTRR control in which forcing orthogonal transports costs 64.6 points out of distribution. |
| LIT-1804 v2 | CO-Optimal Transport | Redko, Vayer, Flamary & Courty (2020). Co-optimal transport matches two data matrices of different shapes by fitting a coupling of rows (samples) and a coupling of columns (features) together, under an entry-wise bilinear cost. With equal sizes and uniform weights it is zero exactly when the matrices agree up to separate row and column permutations, and it satisfies the triangle inequality; it proves no uniqueness of the plans, which are fixed only up to joint row–column symmetries. For symmetric matrices it lower-bounds Gromov–Wasserstein, with equality and shared optimal couplings for squared Euclidean distances. Exact solution is NP-hard in general; block coordinate descent reaches block-wise optima. It reports the best accuracy among the baselines on heterogeneous domain adaptation and co-clustering. | Active — read in full 2026-10-29 (NOTE-1335), from arXiv v3; worth reading for the exact relation between co-optimal transport and Gromov–Wasserstein (a lower bound, equal for squared Euclidean distances) and for what its zero theorem implies: joint sample and feature plans are fixed only up to the matrix’s joint row–column symmetries. |
| LIT-1805 v2 | Towards Optimal Transport with Global Invariances | Alvarez-Melis, Jegelka & Jaakkola (2019). Optimal transport between two point sets whose spaces differ by one unknown global linear map, drawn from a Schatten ℓp ball, posed as a joint optimisation over the coupling and the map. When the maps preserve angles or the target is whitened, the map can be eliminated in closed form: orthogonal maps leave the nuclear norm of the transported cross-covariance XΓYᵀ to maximise, and Frobenius-sphere maps its Frobenius norm, which is exactly Gromov–Wasserstein with cosine similarity. Alternating SVD and annealed Sinkhorn steps solve it without adversarial initialisation; on word translation it is level with GW and behind self-learning. | Active — read in full 2026-10-29 (NOTE-1333); worth reading for the exact statement of what a global-invariance alignment treats as arbitrary, one transformation from a stated class acting on the whole space, and for Lemma 4.3, which shows that for one class this is the same problem as Gromov–Wasserstein (THEORY-1135). |
| LIT-1807 v2 | AvAtar: Learning to Align via Active Optimal Transport | Yu et al. (2026). Active learning for alignment by entropic optimal transport. The oracle answers one fixed question, which target matches a given source object, and an answer sets that pair’s cost to zero. AvAtar ranks source objects by the derivative of a proxy utility of the transport plan (its squared norm, entropy, or graph consistency) with respect to supervising each candidate match, averaged over matches by the current plan. The derivative is exact, from one singular positive semidefinite (n + m) linear system solved by conjugate gradients. With 20% of true pairs given and 20% more chosen in ten rounds, it beats uncertainty, centrality and matching-sampling rules on network alignment and image–text tasks by up to 2.8 MRR points, without reported variances; weighting by the current plan is essential. | Active — read in full 2026-10-29 (NOTE-1327), from the arXiv v2; worth reading as a precise case of budgeted question choice within a fixed vocabulary: which object’s correspondence to ask about, scored by an exact derivative through entropic transport of a coherence proxy, and evaluated on accuracy. |
| LIT-1808 v2 | Unsupervised Hierarchy Matching with Optimal Transport over Hyperbolic Spaces | Alvarez-Melis, Mroueh & Jaakkola (2020). Matches two hierarchies, such as WordNets in two languages or two ontologies, from their Poincaré embeddings alone. Embeddings of one hierarchy from different seeds place branches differently and change pairwise distances, so Euclidean aligners and hyperbolic OT with an orthogonal map fail (P@10 at most 11%). Registering one space with a hyperbolic neural network trained through a Sinkhorn divergence reaches P@10 of 38–60% across languages, but P@1 of only 22% when a hierarchy is matched to a second embedding of itself, and fails without a pretraining step. GW is dismissed, not tested. | Active — read in full 2026-10-29 (NOTE-1323); worth reading as a case where the arbitrariness between two learned representations is not a group action at all: embeddings of one hierarchy from different seeds differ non-isometrically, and only a non-linear registration recovers part of the correspondence (THEORY-1136). Its matching uses a symmetric distance and does not preserve the hierarchy’s direction. |
| LIT-1809 v2 | POT: Python Optimal Transport | Flamary et al. (2021). A software paper for POT, an MIT-licensed Python library of optimal transport solvers as of 2021: exact transport by a C++ solver, entropic transport by Sinkhorn and stabilized variants, Wasserstein barycenters, partial and unbalanced transport, regularized transport with generic solvers, Gromov–Wasserstein, sliced Wasserstein and domain adaptation, tested to 92% line coverage. It compares features with OTT, OTJulia and GeomLoss and reports no benchmarks. Fused and partial Gromov–Wasserstein and co-optimal transport are not in it. | Active — read in full 2026-10-29 (NOTE-1325); the citable description of the POT library as of 2021. Worth citing for the library, with the version used, but not for fused or partial Gromov–Wasserstein or co-optimal transport, which this paper does not describe. |
| LIT-1811 v2 | Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs | Bodnar et al. (2022). In the time limit, diffusion with a sheaf Laplacian projects node features onto its harmonic space, and the class of restriction maps decides which labellings that space can linearly separate. Weighted graph Laplacians cannot separate the sides of a balanced bipartite graph; signed one-dimensional maps separate any two classes and no one-dimensional sheaf separates three; diagonal maps with d ≥ C stalks and orthogonal bundles with C ≤ 2d separate any number. Sheaf convolutions with non-symmetric maps can raise the Dirichlet energy that graph convolutions must lower. A model that learns the maps from node features at each layer is competitive on small heterophilic benchmarks. The separating sheaves are built from the labels, and the learned ones are inspected only on synthetic data. | Active — read in full 2026-10-29 (NOTE-1332) from arXiv v4; worth reading for the cleanest statement of what a sheaf’s restriction maps decide about diffusion: which labellings its harmonic space can separate, class by class of maps, with proofs. The separating sheaves are all built from the labels, and the orthogonal construction works at d = 8, which the paper denies. |
| LIT-1812 v2 | A Collection of Pragmatic-Similarity Judgments over Spoken Dialog Utterances | Ward & Marco (2024). The first collection of human pragmatic-similarity ratings: 693 pairs of a spoken dialog utterance and a re-enactment of it, made under six conditions so that similarity varies, rated on a 1–5 slider for “overall feeling, tone, and intent” by 6–9 judges in three sessions, two English and one Spanish, 5,532 ratings in all. Mean pairwise Pearson agreement between judges was 0.45, 0.72 and 0.66 by session, rising with experience and falling for lexically different pairs (0.80 in Session 2 without them). Judges discussed ratings during the sessions, and all pairs are within one language. | Active — read in full 2026-10-29 (NOTE-1338) from the LREC-COLING 2024 PDF; worth reading as the first collection of human pragmatic-similarity ratings and for its account of what lowers agreement. The 0.45 the record cites is one session’s single-judge figure; the others are 0.72 and 0.66. |
| LIT-1813 v2 | Optimal Transport for structured data with application on graphs | Vayer, Chapel, Flamary, Tavenard & Courty (2019). Fused Gromov–Wasserstein compares two attributed graphs with one coupling whose cost mixes, with weight α, a Wasserstein term on node features and a Gromov–Wasserstein term on structure. It tends to W as α → 0 and to GW as α → 1, and for q = 1, interior α and distance-matrix structures it is a metric that vanishes exactly when a bijection preserves weights, features and structure (a 2^{q−1}-relaxed semi-metric for q > 1). A conditional-gradient solver reaches stationary points, and a block coordinate barycenter supports graph k-means. On graph classification it is competitive with kernels and graph CNNs, though the supplement’s Wasserstein-only scores match or beat it on several datasets. | Active — read in full 2026-10-29 (NOTE-1328), from arXiv v3; worth reading for the metric and interpolation theorems of fused Gromov–Wasserstein, from which it follows that anchors as features break exactly the structural symmetries that move them. |
| LIT-1815 v2 | Structured Optimal Transport | Alvarez-Melis, Jaakkola & Jegelka (2017). Optimal transport whose cost is the Lovász extension of a submodular set function on source–target assignments, so that assignments made together, such as one labelled group sent to one region, can cost less than the sum of their parts. The problem stays convex and is a zero-sum game against cost matrices in the base polytope; it is ordinary OT when the set function is modular, and is solved by saddle-point mirror prox with Sinkhorn and base-polytope projections. The groups are supplied in advance. Single-run experiments on point matching, colour transfer, 100-sample digit adaptation and sentence similarity favour the structured couplings. | Active — read in full 2026-10-29 (NOTE-1324), from arXiv v1; worth reading as a convex way to make a transport cost depend on sets of assignments rather than single ones, and for its limits: the structure is supplied, only submodular interactions are tractable, and its own threshold cost is not submodular. |
| LIT-1816 v2 | xCOMET: Transparent Machine Translation Evaluation through Fine-grained Error Detection | Guerreiro et al. (2023; TACL 2024). xCOMET, an open learned metric for machine translation, predicts from one XLM-R encoder a sentence-level score and a severity tag for each token, with or without a reference or source, and folds an MQM score computed from its own error spans into its final score. On WMT22 expert MQM it reaches segment-level Kendall τ of 0.42–0.43 and span F1 of 0.27–0.28, at or above the open metrics and GPT-4 prompting of its time, and it would have won the WMT23 metrics task. Its spans and its score agree at Pearson 0.73–0.92, far above either’s agreement with experts. Trained with synthetic hallucinations, it flags most injected critical errors and ranks fully detached hallucinations low. | Active — read in full 2026-10-29 (NOTE-1341) from the TACL version of record, with the arXiv v1 preprint compared; worth reading as the reference description of a strong open MT metric and its limits: segment-level Kendall τ near 0.43 against expert MQM, span F1 near 0.28, and score and spans that agree with each other far better than with the experts. |
| LIT-1817 v2 | VakyArth: Evaluating Pragmatic Competence in LLMs across Indic Languages | Singh et al. (2026). 578 items written by four native speakers, one per language, in Hindi, Punjabi, Tamil and Malayalam, over deixis, speech acts, implicature, social pragmatics and coherence, as disjoint MCQ, NLI and translation sets. Five LLMs score higher on MCQ than NLI in every model–language pair, and when wrong they pick the literal reading 59.1% of the time against 33.3% by chance. On 416 model translations, COMET correlates with one native rater per language at r = 0.81, better than XCOMET or MetricX, though hand-picked inverted translations score highly. The finding that translation “does not track” pragmatic understanding compares different items, and no inter-rater agreement is given. | Active — read in full 2026-10-29 (NOTE-1345) from the Findings of EMNLP 2026 PDF, with the released items counted; worth reading for its native-authored, culturally specific pragmatic items in four Indic languages and its evidence that LLM errors lean literal. Its translation and inference tasks use different items, and no agreement between people is reported. |
| LIT-1818 v2 | Gromov-Wasserstein Alignment of Word Embedding Spaces | Alvarez-Melis & Jaakkola (2018). Casts unsupervised word translation as a Gromov–Wasserstein problem between the two languages’ within-language cosine-similarity matrices, solved by entropic projected gradient with no anchors, and reads the optimal coupling as the translation. It matches adversarial-plus-refinement methods on related languages trained on comparable Wikipedia corpora (P@1 72–82) at about 4% of their CPU time, and does markedly worse on English–Russian (45) and English–Finnish (18). The objective sees only within-space relations, so it is unchanged by a rotation of either space and cannot distinguish couplings that differ by a symmetry of the data. | Active — read in full 2026-10-29 (NOTE-1336); worth reading as the cleanest statement of correspondence recovered from within-space relations alone, and for what that costs: the objective cannot tell a correspondence from one composed with a symmetry of either space (THEORY-1133), which the paper does not say. |
| LIT-1819 v2 | On the Limitations of Unsupervised Bilingual Dictionary Induction | Søgaard, Ruder & Vulić (2018). Tests the adversarial unsupervised dictionary induction of Conneau et al. (2018) outside its reported conditions. It fails (P@1 ≈ 0) for English paired with Estonian or Finnish, also with a 1.7B-word Finnish corpus; across corpus domains, even for English–Spanish; and between a skip-gram and a CBOW space. A seed of identically spelled words does as well or better. A Laplacian-spectrum dissimilarity of 10-node nearest-neighbour graphs tracks success across eight pairs (|ρ| = 0.89; negative, since it is a dissimilarity), but is computed with the gold translations. About one seed in ten succeeds for Estonian, Finnish and Greek, so part of the failure is the optimiser. | Active — read in full 2026-10-29 (NOTE-1337); worth reading as the first systematic test of when unsupervised dictionary induction fails, and for its finding that whether two embedding spaces align depends on the corpus domain and the embedding algorithm as much as on the languages (THEORY-1145). The method tested is adversarial (Conneau et al. 2018), not GW. |
| LIT-1820 | Attention as Lattice Gauge Theory: Relations Interact Only Through Wilson Loops | Sarkar, Mukherjee & Sato (2026), ARXIV-2610.37656. If every entity’s coordinate frame may be rotated independently, so that relation transports Tᵢⱼ ∈ O(d) change as gᵢTᵢⱼgⱼ⁻¹, then relations with no cycle between them cannot interact (in the pure-gauge sector), a triangle’s only invariants are class functions of its holonomy, and an equivariant edge update has the form Ψ(H)Tᵢⱼ with Ψ conjugation-equivariant. An untrained instance on S4 synchronization with outliers gains 26–45 points over a single path by composition, and 0.6–7.8 more by Wilson-loop routing; it does not beat spectral synchronization. | Active — read in full 2026-10-29 (NOTE-1331), after the owner brought it to the exchange “Paper assessment” asking whether it is “on to something or if this is AI slop nonsense”. Worth keeping for its conditional theorem, which is sound and is QUESTION-130’s construction carried further, and for its measured caution about its own experiments. Its framing outruns both. |
| LIT-1833 v2 | Multi-Level Cause-Effect Systems | Chalupka, Perona & Eberhardt (2015; AISTATS 2016). A micro cause I and micro effect J, both discrete and possibly huge, are confounded by a hidden H. The fundamental causal partition groups i by P(J | man(i)) and j by P(j | man(i)) across all i; its classes are the fundamental cause C and effect E, and do(C = c) is any manipulation into the class. E is claimed to be the minimal sufficient partition for P(J | man(i)), and (C, E) the coarsest pair that recovers it losslessly. The fundamental causal coarsening theorem extends the UAI 2015 one to the effect side: given the observational partitions, both causal partitions fail to coarsen them only on a null set. Subsidiary variables are coarsenings of (C, E) with unambiguous effects; two that do not interact compose into the fundamental partition. A simulated 100-neuron population responding to bars in images is recovered. No distribution over interventions is assumed. | Active — read 2026-10-29 (NOTE-1420) in full, from arXiv 1512.07942v1 with its supplementary material; worth reading as the source of the two-sided form of causal feature learning that the overview (LIT-1834) restates: the fundamental cause and effect are the partitions of micro causes by P(J | man(i)) and of micro effects by P(j | man(·)), learned jointly; the fundamental causal coarsening theorem says both generically coarsen the observational partitions; and the authors decline, “following Pearl”, to put any distribution on the interventions. Read it knowing that its one demonstration is a simulated neural population, built with the macro structure it recovers; that its claim that the effect partition is the minimal sufficient partition is wrong where two effect states have proportional but unequal columns (NOTE-1420); and that it compares itself with Hoel et al. 2013 in one sentence and computes nothing on their systems. |
| LIT-1834 v2 | Causal feature learning: an overview | Chalupka, Eberhardt & Perona (2016; Behaviormetrika 2017). A review of causal feature learning (CFL). Macro variables are partitions of a micro state space. The observational partition groups micro causes h with equal p(e | h); the causal partition groups those with equal p(e | man(h)), where man() sets the micro state without touching a confounder; effect states are grouped the same way by their columns. A macro intervention do(C = c) is a manipulation to any micro state in the class, which is unambiguous because all of them have the same effect. The causal coarsening theorem: among distributions with given causal and confounding partitions, the causal partition fails to coarsen the observational one only on a Lebesgue-null set, so the observational partition is learned from data and one experiment per observational class merges it into the causal one. A toy with confounded hue and skin conductance shows the steps; open problems are continuous macrovariables, cycles and learning without experiments. | Active — read 2026-10-29 (NOTE-1415) in full, from the publisher’s PDF as posted on Frederick Eberhardt’s Caltech publications page (https://www.its.caltech.edu/~fehardt/papers/CEP2017.pdf); worth reading as the plainest statement of causal feature learning: a macro cause is a class of micro states with the same P(effect | man(state)), a macro intervention is any micro manipulation into the class, and the causal coarsening theorem says that, except on a measure-zero set, the causal partition is a coarsening of the observational one, so one experiment per observational class finds it. Read it knowing that its example is a one-dimensional toy with four hue bands; that the theorem is a genericity result, not a test, and holds only for discrete macrovariables and an acyclic micro system; that the effect classes are defined by equal likelihoods while the merging step normalizes them; that two of its displayed equations carry slips (NOTE-1415); and that it gives no measure of how much causal information a coarsening keeps. |
| LIT-1879 v2 | Almost Bayesian: The Fractal Dynamics of Stochastic Gradient Descent | Hennick & De Baerdemacker (2025; ICLR 2026). Models late-training SGD as sub-diffusion on a porous landscape: a time-fractional Fokker–Planck equation, the local learning coefficient as the medium’s mass dimension, and a spectral dimension d_s fitted from weight displacement, R(t) ∝ t^(d_s/2λ) with d_s ≤ λ. Where an effective scalar diffusion D_ξ is constant on a region, the steady state is exp(−L/D_ξ), the posterior tempered to effective sample size 1/D_ξ. How mass is shared between regions is not derived; in one toy test the cluster masses of SGD solutions match an SGLD posterior after a fitted tempering scale. Displacement fits cover MNIST MLPs, TinyStories models and fine-tuned ImageNet convnets. | Active — read in full 2026-10-31 (NOTE-1467); its “tempered posterior” is a temperature, the posterior at effective sample size 1/D_ξ within a region of constant diffusion, and the “accessibility” that would make SGD differ from Bayes across regions is not derived but fitted (THEORY-1258); its sub-diffusion fits are filed as Proposed (THEORY-1272). |
| LIT-1882 v2 | Correspondence between thermodynamics and inference | LaMont & Wiggins (2019). Treats the Bayesian evidence as a partition function with sample size N as inverse temperature and derives the thermodynamic quantities. The internal energy is the leave-one-out predictive loss, whose excess K/2N is equipartition. The heat capacity becomes a learning capacity, K/2 in a regular model and the learning coefficient in a singular one, lower where the prior or discreteness fixes parameters (“freeze-out”), proposed as the definition of sloppiness. A prior chosen to make the Gibbs entropy zero is Jeffreys-like, depends on N, reproduces AIC and avoids the Lindley–Bartlett paradox in one simulation. | Active — read 2026-10-31 (NOTE-1481; appendices B, C1, C3, C4 skimmed); worth reading as the fullest working of the evidence-as-partition-function dictionary with sample size as inverse temperature: generalization error as equipartition, a “learning capacity” equal to K/2 in regular and to the RLCT in singular models (THEORY-1253), and a sample-size-dependent prior that reproduces AIC. The dictionary is formal; it has no physical currents or dissipation. |
| LIT-1883 v2 | Overcoming Scaling Limits in On-Policy Self-Distillation for LLM Reasoning | Hossain, Kousar & Tourni (2026). In on-policy self-distillation the frozen initial model, given a reference solution, teaches the student along the student’s own rollouts. Under forward KL the best student is that teacher averaged over the contexts the student cannot see, with the conditional mutual information as irreducible loss and higher entropy (Proposition 1). The teacher’s advantage lies mostly on failed trajectories, where standard OPSD puts about 70% of its supervision, and is smaller at 4B than at 1.7B. OASIS supervises only verified rollouts and gives the teacher the student’s own failed attempt in place of the reference. It keeps a 3.2–3.8-point gain over the base model on AIME and HMMT from Qwen3-1.7B to 8B, where OPSD’s gain falls to 0.14, in single runs. | Active — read in full 2026-10-31 (NOTE-1479); worth reading as the published statement of the forward-KL limit on distilling from a better-informed teacher, with its entropy consequence (THEORY-1259), and for the finding that such supervision helps only where it is imitable (THEORY-1263). The trained objective clips each entry and is not a forward KL. |
| LIT-1884 v2 | Why Self-Training Helps and Hurts: Denoising vs. Signal Forgetting | Wu, Yang & Sun (2026). In overparameterized linear regression, iterated self-training with fresh covariates, noiseless pseudo-labels and minimum-norm students projects the estimate each round onto the row space of a new design. Noise from the first fit decays at (1 + τ)⁻¹ per round, and signal along a direction of variance s at s/(s + τ). Risk is U-shaped under anisotropy, monotone under isotropy, and tends to that of predicting zero. Deterministic equivalents concentrate uniformly over ridge parameters, and an iterated GCV picks the stopping round. One CIFAR-10 experiment shows the U-shape. | Active — read 2026-10-31 (NOTE-1471), body and appendices, with the proofs skimmed; worth reading for an exact account of iterated reconstruction through fresh low-rank views, in which forgetting the ancestor first improves and then destroys fidelity to the world, and every student fits its teacher exactly on its own sample (THEORY-1269). |
| LIT-1891 v2 | General agents contain world models | Richens, Abel, Bellot & Everitt (2025), ICML. In a fully observed, finite, communicating, stationary controlled Markov process, a deterministic goal-conditioned policy whose success probability is within a factor 1 − δ of optimal on every composite goal of depth up to n determines each transition probability to within √(2P(1−P)/((n−1)(1−δ))). An algorithm reads the estimate off the policy alone, from its first action on either-or goals. A policy competent only at one-step goals determines nothing. “Contains a world model” means this recoverability from the input-output map, “not its specific use”, so it holds of any implementation of the same map. The experiments use a model-based agent in one 20-state environment. | Active — read in full 2026-10-31 (NOTE-1469); worth reading for the cleanest theorem that competence across a family of goals fixes the environment’s dynamics, and for exactly what “contains a world model” means in it: the transition function is computable from the goal-conditioned input-output map (THEORY-1249), which holds of a lookup table as much as of a planner (THEORY-1260). It is the objection to the functional reading of CLAIM-519, and it does not meet that claim’s defeated_if. |
| LIT-1894 v2 | Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition | Chen, Lau, Mendel, Wei & Murfet (2023). In Elhage et al.’s toy model of superposition with two hidden dimensions, in the high-sparsity limit, the population loss has a closed form and regular k-gons are critical points with derived local learning coefficients. The free energy formula predicts a Bayesian phase transition from the 5-gon to the 6-gon near n = 600, which MCMC confirms. SGD trajectories plateau at the same critical points and move to lower loss and higher estimated complexity, but the 5 → 6 transition was not observed in training. The paper states that there is no necessary relation between the two kinds of transition, and conjectures only that training’s transitions have Bayesian antecedents. | Active — read 2026-10-31, the derivation appendices skimmed (NOTE-1462); it does not argue that SGD and the posterior trace one journey: it states “no necessary relation” between their transitions, proposes only that training’s transitions have Bayesian antecedents, and its main Bayesian transition, 5-gon to 6-gon, was not seen in training (THEORY-1275). |
| LIT-1895 v2 | Self-Distillation Amplifies Regularization in Hilbert Space | Mobahi, Farajtabar & Bartlett (2020). In ℓ2-regularized regression in a Hilbert space held to a fixed training-error tolerance, refitting to one’s own predictions on the same inputs shrinks each eigencomponent of the labels by d_k/(c_t + d_k). The multiplier c_t must rise as the labels shrink. Rounds therefore amplify regularization, use progressively fewer basis functions (Theorem 5) and collapse to the zero function after finitely many. A few rounds can reduce overfitting and more underfit, in a spline example and, by fractions of a point, in ResNet and VGG runs on CIFAR. | Active — read 2026-10-31 (NOTE-1478), body and appendices, with the proofs of Appendix C skimmed; worth reading as the clean case of iterated self-distillation losing everything through the regularizer alone, on the same inputs with nothing withheld (THEORY-1252). |
| LIT-1896 v2 | Multi-Token Prediction Done Right | Teoh (2026). With a shared linear output head, the cross-entropy gradient at each multi-token-prediction hidden state is exactly the gradient of a forward KL toward the model’s own position-aligned next-token prediction, held fixed, plus the next-token cross-entropy gradient at the aligned state. A one-line straight-through routing sends only the KL part to the MTP heads and trunk, keeping cross-entropy at the head. The residual carries about 80% of the MTP gradient reaching the trunk at about a quarter of the KL part’s signal-to-noise ratio. Removing it (“Soft-MTP”) improves next-token loss, speculative decoding and zero-shot accuracy at 167M–1.2B by small margins in single runs. | Active — read in full 2026-10-31 (NOTE-1470); worth reading for its exact split of the multi-token-prediction gradient into self-distillation toward the model’s own next-token head and a data-target residual (THEORY-1274), and for what the split does not carry: in the DeepSeek-style heads of its main results the student sees the same tokens as its teacher, so it is not the information projection the exchange read into it. |
| LIT-1898 v2 | Stochastic Gradient Descent as Approximate Bayesian Inference | Mandt, Hoffman & Blei (2017). Constant-rate SGD near a minimum with positive-definite Hessian A and constant gradient-noise covariance BBᵀ is modelled as an Ornstein–Uhlenbeck process whose Gaussian stationary covariance solves AΣ + ΣA = (ε/S)BBᵀ. Minimising KL to the Gaussian posterior gives the optimal constant rate 2SD/(N tr BBᵀ) and the preconditioner (2S/εN)(BBᵀ)⁻¹, which matches the posterior exactly; momentum only rescales the covariance; constant-rate SGLD overestimates the posterior covariance; and averaging T iterates gives covariance A⁻¹BBᵀA⁻¹/(TS), the posterior’s for TS observations in a well-specified model. Tested on linear, logistic and softmax regression. The search phase, singular minima and multiple basins are outside the analysis. | Active — read in full 2026-10-31 (NOTE-1475); the paper the owner recalled at U4, and the right one, but what it shows is narrower than “the same process”: near one non-degenerate minimum, the optimiser’s noise scale ε/S plays the part of 1/N, and training time enters only as the wait for stationarity (THEORY-1255, THEORY-1257). |
| LIT-1899 v2 | A Variational Analysis of Stochastic Gradient Algorithms | Mandt, Hoffman & Blei (2016). The first version of the 2017 JMLR paper. Constant-rate SGD near a quadratic optimum is modelled as an Ornstein–Uhlenbeck process; its Gaussian stationary law is fitted to the Gaussian posterior by minimising KL over the learning rate, batch size and preconditioner, giving the optimal rate 2DS/(N tr BBᵀ) and the exact preconditioner (2S/εN)(BBᵀ)⁻¹. It shows SGFS’s preconditioner is KL-optimal, quantifies its diagonal truncation, and uses constant SGD for variational EM over hyperparameters. The later paper adds momentum, constant-rate SGLD and iterate averaging. | Active — read in full 2026-10-31 (NOTE-1473); every result reappears in the 2017 JMLR paper (LIT-1898), which is the one to cite. Worth keeping as the first statement and as the version Hennick and De Baerdemacker cite. |
| LIT-1951 v2 | Keep Your Options Open: An Information-Based Driving Principle for Sensorimotor Systems | Klyubin, Polani and Nehaniv (2008), PLoS ONE 3(12): e4018. Empowerment, the n-step channel capacity from an agent’s actions to its later sensor state, set in an interventional causal-network framework, with contextual empowerment, the capacity given side information, never below the context-free value and bounded by what the agent’s history can tell it. In a cart-pole, a controller greedily maximizing 10-step empowerment balances the pole with no task given, although the most empowered states are unreachable; empowerment as fitness switches a sensor’s layout between niches; and on a Sony AIBO an evolved context automaton comes to track whether a book is present. | Active — read in full 2026-11-03 (NOTE-1514), from the PLoS ONE version of record; worth reading for contextual empowerment, ordered by what the agent can read, and for the cart-pole, where greedy empowerment maximization balances the pole though the most empowered states are unreachable, a case of CLAIM-354’s distinction between reach and what can be sustained. |
| LIT-1952 v2 | The Persona Selection Model: Why AI Assistants might Behave like Humans | Marks, Lindsey and Olah (2026), Anthropic Alignment Science blog. The persona selection model, named and not claimed as original: pre-training teaches a language model a distribution over personas, post-training updates it into a posterior over Assistant personas, runtime context conditions it further, and the Assistant’s traits are a key determiner of behaviour. Surveyed evidence from generalization, behaviour and interpretability, and one new coin-flip experiment. The open question is exhaustiveness, set out as a spectrum from a shoggoth through a router that swaps personas to an operating system. “PSM” in the record already abbreviates Metzinger’s phenomenal self-model (NOTE-460), so this work is named in full. | Active — read in full 2026-11-03 (NOTE-1512), from the page at alignment.anthropic.com; worth reading as the plainest statement of the persona-distribution picture the record’s U6 to U12 build on, and for its spectrum of views on exhaustiveness, whose router is the record’s question of whether persona components condition one another. It is a named mental model with surveyed evidence, which its authors say they are unsure is complete. |
| LIT-1959 v2 | Attention Is All You Need | Vaswani et al. (2017), arXiv:1706.03762; NeurIPS 2017. The Transformer, an encoder–decoder built only from multi-head scaled dot-product attention, position-wise feed-forward layers and sinusoidal position encodings. Attention maps a query and key–value pairs to a weighted sum of the values, the weights set by query–key compatibility, softmax(QKᵀ/√d_k)V; several heads attend to different subspaces. It reached 28.4 BLEU on WMT 2014 English–German and 41.8 on English–French at a fraction of earlier training cost, and transferred to constituency parsing. | Active — read in full 2026-11-03 (NOTE-1501), from arXiv v7; worth reading as the source of query–key–value attention, whose factorization into compatibility weights and transported values is the structure TERM-064’s note of 2026-11-02 compares to legibility. The QK/OV vocabulary is Elhage et al.’s, not this paper’s. |
| LIT-1965 v2 | Empowerment: A Universal Agent-Centric Measure of Control | Klyubin, Polani and Nehaniv (2005), IEEE CEC 2005, pp. 128–135. Empowerment is defined as the Shannon capacity of an agent’s actuation channel, max over action distributions of I(Aⁿ; S_{t+n}), counted only as far as the agent’s own sensors register it. Knowing the channel’s state can raise it. As a fitness function in a grid world with a signal source it evolves sensors that switch from displacement blobs near the source to bearing arcs far from it, and a wider variety of actuators, with empowerment capped by the sensor’s capacity. | Active — read in full 2026-11-03 (NOTE-1511), from the University of Hertfordshire archive copy; worth reading as the first definition of empowerment, the capacity of the channel from an agent’s actions to its own later sensor, which counts only the control the agent can perceive, a legibility-limited measure of reach for CLAIM-354. |
| LIT-1981 v2 | The geometric minimum action method: A least action principle on the space of curves | Heymann and Vanden-Eijnden (2008), CPAM 61: 1052–1117. The Freidlin–Wentzell quasi-potential, a minimization over paths and their durations that has no minimizer between equilibria, is rewritten as a minimization of a parametrization-free action over curves, which has one; for a diffusion it is 2 inf ∫ |b| sin²(θ/2) ds. The minimizer is the maximum-likelihood transition curve in the small-noise limit. The geometric minimum action method computes it for nongradient SDEs, SPDEs and Markov jump processes. With gradient drift and identity noise transition paths run parallel to the drift; in the Maier–Stein model they leave the heteroclinic orbit for β > 4. | Active — read in part 2026-11-03 (NOTE-1504, Skimmed: §§1, 2.1–2.2, 4.1 and 5), from the published text on a Peking University course page; worth reading as the standard computation of most likely transition paths in small-noise systems, which confirms QUESTION-071’s note of 2026-11-02 for gradient drift with identity noise and shows, in the Maier–Stein model, that the noise path stays the reversed relaxation path until the drift is far from gradient. |
| LIT-1991 v2 | The Assistant Axis: Situating and Stabilizing the Default Persona of Language Models | Lu, Gallagher, Michala, Fish and Lindsey (2026), arXiv:2601.10387. Activation vectors for 275 prompted roles in Gemma 2 27B, Qwen 3 32B and Llama 3.3 70B span a low-dimensional persona space whose first component, shared across models, runs from fantastical roles to Assistant-like ones, with the default Assistant at its extreme; the Assistant Axis, the default minus the role mean, aligns with it and is largely present in base models. Steering along it modulates role-taking and persona-based jailbreaks. In synthetic multi-turn conversations the persona’s position follows the latest user message, drifting away in therapy-like and AI-philosophy talk; clamping the projection (activation capping) cuts harmful responses to persona jailbreaks by about 60% without measured capability loss. | Active — read 2026-11-03 (NOTE-1505, Skimmed: main text in full, appendices not read), from arXiv v1; worth reading for the geometry of persona space in three open-weight models, the Assistant at one end of a pre-existing first axis, and for the finding that within a conversation the persona’s position follows the latest user message, which bears on any episode-level model of personas (CLAIM-536, QUESTION-370). |
| LIT-1998 v2 | Persona Vectors: Monitoring and Controlling Character Traits in Language Models | Chen, Arditi, Sleight, Evans and Lindsey (2025), arXiv:2507.21509. An automated pipeline turns a trait name and description into a persona vector, a difference of mean residual activations between trait-exhibiting and non-exhibiting responses, in Qwen2.5-7B and Llama-3.1-8B. The vectors steer evil, sycophancy and hallucination; last-prompt-token projections predict prompt-induced shifts, mainly between prompt types; finetuning shifts along them correlate with post-finetuning trait expression (r = 0.76–0.97); steering against them after finetuning, or towards them during it, limits the shift; and projecting training data onto them flags data that will induce the trait, including in real chat data. | Active — read 2026-11-03 (NOTE-1524, Skimmed: main text in full, appendices C.2, G.2 and M only), from arXiv v3; worth reading as the method behind PSM’s interpretability evidence: one difference-of-means trait direction tracks prompt-induced and finetuning-induced persona shifts, with r = 0.76–0.97 for the latter in two 7–8B models. It is a handle on movement along named traits, not a fitted persona distribution. |