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.
Compositionality — how the structure and meaning of a whole come from its parts and how they are combined — the principle of compositionality and compositional semantics, the emergence of compositional structure in languages under transmission, and composition and binding in learned models (generalization to unseen combinations, attribute binding, composing distributions or concepts). A work on making a model generalize compositionally is ML practice and goes to the anthology (ADR-031).
36 of 2004 LIT documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| LIT-272 v2 | Reasoning about Meaning in Natural Language with Compact Closed Categories and Frobenius Algebras | Kartsaklis et al. (2014), ARXIV-1401.5980. The paper sets the sentence space equal to the noun space (F(s)=F(n)=W). It then uses the basis-copying map σ: v_i ↦ v_i⊗v_i of the Frobenius algebra fixed by W’s basis to lift corpus-built verb matrices Σ_i sbj_i⊗obj_i into rank-3 tensors. So a transitive sentence becomes sbj⊙(verb·obj) (“copy-subject”) or obj⊙(verbᵀ·sbj) (“copy-object”), a vector in W. On the Grefenstette–Sadrzadeh verb-disambiguation set, copy-object gets Spearman ρ=0.172, against 0.168 (Kron), 0.163 (Multp), 0.143 (copy-subject), 0.050 (additive) and an inter-annotator bound of 0.620. On a 112-term definition-matching task the best F1 is 0.21 (nouns) and 0.27 (verbs). No significance test is reported. The disambiguation numbers are identical to the authors’ COLING 2012 paper, and the definition numbers are the same data re-reported. | Active — read in full 2026-09-27 (NOTE-246); worth reading for the construction, not for its evidence. It is the citable categorical statement of the Frobenius lift of DisCoCat verbs: the functorial recasting, S=N=W, and the copy-subject and copy-object maps. Its experiments are the authors’ COLING 2012 numbers, re-reported without that paper’s caveat that copy-object’s lead is statistically insignificant. The best disambiguation ρ is 0.172, against a human bound of 0.620. The construction depends on a choice of basis, which the paper cites but never draws the consequence of. |
| LIT-273 v2 | Mathematical Foundations for a Compositional Distributional Model of Meaning | Coecke et al. (2010), ARXIV-1003.4394. The paper builds DisCoCat as a product category FVect × P of vector spaces and a free pregroup. A sentence’s meaning is f(w₁⊗…⊗wₙ), where f is the linear map got by putting vector spaces in place of the pregroup types in the reduction p₁…pₙ ≤ s. The ε maps become inner products and the η maps become Σᵢ eᵢ⊗eᵢ. So “John likes Mary” is Σ_ijk c_ijk⟨v|vᵢ⟩⟨w_k|w⟩ sⱼ ∈ S, and “not” is the 2×2 swap matrix placed on the sentence wire. There is no theorem and no experiment. The sentence space S is shown only as 1- or 2-dimensional truth values, over a toy model with one basis vector per individual, and the paper says how neither S nor the verb tensors are to be built from data. Its headline similarity numbers (3/4, 1/4, 3/8) are unnormalised inner products. Under its own Definition 5.1 they are 0.949, 0.316 and 0.6. | Active — read in full 2026-09-27 (NOTE-245); worth reading as the founding statement of DisCoCat, whose ε/η composition, S-for-every-sentence design and “not”-as-swap every later paper in the programme starts from. Read it as a specification, not a result. It proves nothing, runs no experiment, leaves S and the verb tensors unbuilt, and its categorical framing (a product with a posetal pregroup) cannot tell apart the parse ambiguities it says it can reason about. |
| 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-769 | Language Evolution by Iterated Learning With Bayesian Agents | Griffiths & Kalish (2007), Cognitive Science 31(3):441–480. When Bayesian learners sample a language from their posterior, iterated learning is a Gibbs sampler and converges to the learners’ prior; when they take the posterior mode, it behaves like a variant of EM and depends also on how much data passes between generations; in a two-language case the MAP outcome is the same for a wide range of prior strengths. Read in NOTE-599. | Active — read 2026-10-09 (NOTE-599); worth reading as the analytic result behind iterated learning: a chain of learners who share a prior and each sample a hypothesis from their posterior is a Gibbs sampler, so the probability that a learner holds a language converges to its prior probability whatever the amount of data passed on, and the data produced converge to the prior predictive. Learners who take the MAP hypothesis run stochastic EM instead: the outcome centres on the prior’s mode but depends on data amount and noise, and in the two-language case is the same for every prior strength α between 0.5 and 1 − ε, so weak biases can be amplified. Equal-fitness populations share the equilibrium. Source of THEORY-155. |
| 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-771 | Cumulative cultural evolution in the laboratory: An experimental approach to the origins of structure in human language | Kirby, Cornish & Smith (2008), PNAS 105(31):10681–10686. Artificial languages passed down chains of human learners become easier to learn and more structured, without any learner intending it: cultural transmission produces design without a designer. Without a pressure against ambiguity the languages collapse into a few words for classes of meanings; with homonyms filtered from training they become compositional. Skimmed in NOTE-596. | Active — skimmed 2026-10-09 (NOTE-596): every section of the main text was covered, but through a fetch tool’s section-by-section account of the PubMed Central page, not a verbatim copy, and the SI was not reached; worth reading as the laboratory demonstration of iterated learning: four chains of ten adults, each learning a 27-item artificial language from 14 items and passing on its test output, made the languages steadily easier to learn and more structured while believing they were copying. Unfiltered, the languages collapsed to 2–5 words that systematically underspecified meaning (colour lost in every chain); with homonyms filtered out of training, they became compositional. Consistent with THEORY-155, not a test of it: no prior was measured, and the filter is a selection the theory excludes. |
| 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-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-803 | Seven Sketches in Compositionality: An Invitation to Applied Category Theory | Fong & Spivak (2018; Cambridge University Press 2019), ARXIV-1803.05316. A textbook invitation to applied category theory in seven chapters, each pairing an application with a structure: generative effects with orders and Galois connections, resources with monoidal preorders and enrichment, databases with functors and adjunctions, co-design with profunctors, signal-flow graphs with props, circuits with hypergraph categories and operads, behaviour with sheaves and toposes. Its first chapter defines a generative effect as an observation that preserves order but not joins, and shows that such maps are exactly the non-left-adjoints. | Active — skimmed 2026-10-09 (NOTE-601): the preface and Chapter 1 read closely, Chapter 3’s opening and its section on adjunctions and data migration read closely, the openings and summaries of Chapters 2 and 4–7 read; Chapters 2 and 4–7 otherwise not read. Worth reading as an introduction to applied category theory organised around compositionality: an observation of systems is a monotone map, a generative effect is its failure to preserve joins, and a monotone map out of a preorder with all joins has no generative effect exactly when it is a left adjoint. Later chapters treat resource theories, databases as functors, co-design, signal-flow graphs, circuits and temporal logic, each through a categorical structure. |
| 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-825 | Larger communities create more systematic languages | Raviv, Meyer & Lev-Ari (2019), Proc. R. Soc. B 286:20191262. In a laboratory group-communication game, groups of eight built more systematic (compositional) languages than groups of four, faster and more consistently, with the same total interaction; small groups varied more, as drift would predict. Input variability, greater in larger groups, predicted gains in structure. Group size was isolated from network structure and second-language learners, the confounds of cross-linguistic correlations. | Active — read 2026-10-09 (NOTE-628), the main text in full from the publisher’s version in the Radboud University repository; the electronic supplementary material was not read. Worth reading as the experiment that isolates community size: 12 groups of four and 12 groups of eight Dutch adults, fully connected, invented a typed language to describe moving shapes over 16 rounds with the same total amount of interaction. Larger groups’ languages became more systematic (string distance tracking meaning distance), faster, and ended more systematic; larger groups also varied far less from one another, while some small groups never built structure. Larger groups faced more input variability, and input variability predicted the next round’s gain in structure better than shared history did. Communicative success and convergence ended equal. It is a group-interaction paradigm, not a transmission chain: no generation turnover. Source of THEORY-164. |
| 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-828 | Морфология сказки | Propp (1928), Leningrad, Academia (Morphology of the Folktale, English 1958, 2nd ed. 1968). In a hundred Russian wondertales the constants are the characters’ functions, defined by their place in the action, not the characters or motifs: 31 functions, always in one order, distributed over seven spheres of action, so all wondertales are of one type and differ in what they omit, how functions are realised and who performs them. Classification by plot is impossible; the wondertale may be morphologically a myth. | Active — read 2026-10-09 (NOTE-615), in Russian, the preface and all nine chapters of the 1928 first edition from a library scan in full view at the Internet Archive; the appendices skimmed. Worth reading as the founding structural analysis of narrative: in the wondertales of Afanasyev’s collection (nos. 50–151), what stays constant is not the characters or motifs but the functions of the characters, actions defined by their consequence for the plot; there are only 31, they follow one order, and so all wondertales are of one type, differing in which functions they omit, how each is realised and who performs it. Functions fall to seven spheres of action (villain, donor, helper, princess and her father, dispatcher, hero, false hero). Propp holds that a tale can be generated from the scheme, that plot and variant cannot be told apart, and that wondertales in their morphological basis are myth. Source of THEORY-167. |
| LIT-830 v2 | Morphology of the Folktale | Propp (1958), Morphology of the Folktale, trans. Laurence Scott, ed. Svatava Pirkova-Jakobson, Indiana University Research Center in Anthropology, Folklore and Linguistics, Publication 10 (IJAL 24(4), pt. 3); 2nd ed., revised and edited by Louis A. Wagner, introduction by Alan Dundes, University of Texas Press, 1968. The English translation of Морфология сказки (1928), held as LIT-828 and read there in Russian. Skimmed in the 1968 edition (chs. II–III and Dundes’s introduction): the four theses and 31 functions match the Russian reading; the genre and character terms shift (“folktale”, “fairy tale”, “dramatis personae”); and Wagner’s notes correct Propp’s tale references and renumber the tales. | Active — skimmed 2026-10-09 (NOTE-642) in the 1968 edition: Dundes’s introduction and chs. II–III (pp. 19–65), from copies on University at Buffalo and MIT web pages; the 1958 edition was not seen. Worth reading as the text through which most readers outside Russia met Propp. Chs. II–III carry the four theses and all 31 functions with their definitions and signs, matching the record’s reading of the Russian (NOTE-615), but with Latin and Greek signs that differ from the Cyrillic. It names the class “fairy tales” under a title its own introducer calls misleading, and alternates “dramatis personae” with “characters”. Wagner’s 41 notes in those two chapters correct Propp’s tale citations and renumber the tales to the later Afanas’ev numbering. The seven roles (ch. VI) were not read. |
| LIT-839 | Classifier-Free Diffusion Guidance | Ho & Salimans (2022). One diffusion network learns both the conditional and the unconditional score by dropping the condition during training; sampling with (1 + w)ε(z, c) − wε(z) reproduces the FID/Inception-score trade-off of classifier guidance on ImageNet without any classifier. The combination is motivated by an implicit Bayes classifier, but the paper itself shows that learned scores need not be conservative, so the guided field is in general the score of no density. | Active — read 2026-10-09 (NOTE-603); worth reading here as the paper that makes guidance a combination of two scores from one network: train with the condition dropped at random, then sample with (1 + w)ε(z, c) − wε(z), extrapolating away from the unconditional estimate. Its own analysis is the point for this record: the combination is inspired by Bayes’ rule (an implicit classifier p(c|z) ∝ p(z|c)/p(z)) but, because learned scores need not be gradients of anything, it is in general the score of no density and the gradient of no classifier. The evidence is one proof-of-concept on class-conditional ImageNet: the guidance weight trades FID against Inception score as classifier guidance does, best FID at w = 0.1–0.3, best IS at w ≥ 4. |
| LIT-841 | Semantic Unification: A Sheaf Theoretic Approach to Natural Language | Abramsky & Sadrzadeh (2014), ARXIV-1403.3351, in the Lambek Festschrift (LNCS 8222, pp. 1–13). Models basic Discourse Representation Structures as sections of a presheaf of consistent literals over (vocabulary, variables), an anaphoric resolution as a cover by variable maps, and the meaning of a discourse as the gluing of its sentences’ sections, unique when it exists. Failure to glue is inconsistency, not contextuality in the Abramsky–Brandenburger sense; competing resolutions are competing covers, ranked in the probabilistic section by corpus counts. | Active — read in full 2026-10-09 (NOTE-644); worth reading as the first paper to model discourse meaning as sheaf-theoretic gluing, and as the record’s earliest bridge between the Abramsky–Brandenburger framework and natural-language semantics. A basic Discourse Representation Structure is a section of a presheaf of consistent literals; a choice of anaphoric resolution is a cover, a jointly surjective family of variable maps; and the meaning of the discourse is the gluing, which is unique when it exists (Proposition 1, its only result). Its “contextuality” is context dependence, not the failure of a compatible family to glue: the only obstruction it exhibits is inconsistency of literals. Its probabilistic section ranks candidate covers by summed corpus counts rather than gluing distributions. Short and preliminary by its own account. |
| LIT-842 | On the Quantum-like Contextuality of Ambiguous Phrases | Wang, Sadrzadeh, Abramsky & Cervantes (2021), ARXIV-2107.14589, SemSpace 2021 (ACL Anthology, pp. 42–52). Treats each word of a two-word ambiguous phrase as a measurement whose outcome is the selected meaning. One hand-specified support model is possibilistically contextual; its BNC/ukWaC probabilities, like all the corpus models, are signalling. Under Contextuality-by-Default two subject–verb/verb–object pairs come out contextual, but the paper’s own bootstrap leaves both plausibly noncontextual. | Active — read in full 2026-10-09 (NOTE-645); worth reading as the first paper to test natural-language data for contextuality in the formal sense rather than the loose one, and as the paper that names the obstacle the test meets: corpus estimates of meaning selection are almost always signalling. In the sheaf-theoretic framework one hand-specified support model ({tap, box} × {pitcher, cabinet}) is possibilistically contextual (a Hardy-type failure), but its corpus probabilities are signalling, so the framework cannot judge them. Under Contextuality-by-Default the Bell-type corpus systems are noncontextual, and two noun–verb pairs read as subject–verb and verb–object (adopt/boxer, throw/pitcher) are CbD-contextual, by 1/30 and 7/30. Its own parametric bootstrap then gives those two a probability above .56 and above .08 of being noncontextual, on fractions with denominators between 3 and 30, so the contextual verdicts are not statistically secured. |
| 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-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-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-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-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-967 v2 | Evidence for the representation of non-hierarchical structures in language | Nielsen et al. (2026). The paper argues that language is represented partly through linear chunks akin to part-of-speech sequences, not only through hierarchical constituents. Four preregistered phrasal decision experiments (N = 497) show such linear structures can be primed without constituents. Corpus analyses of eye-tracked reading and conversation support the effect outside the lab. Not read. | Unavailable — searched 2026-10-14 for a lawful full text and none was reached, so it is not read and no NOTE is filed. OpenAlex: closed (PubMed and Scopus records only); Unpaywall: not open access; Europe PMC: no PMC copy, “Subscription required”; Semantic Scholar: no open PDF; nature.com: subscription article (exclusive licence, “Access through your institution”); OSF: the authors’ project osf.io/ymndq is public but holds materials, data, analyses and four preregistrations, not the manuscript, and no OSF or PsyArXiv preprint was found by title search; the Cornell lab pages failed on the proxy (502), failing on access, not availability; Aarhus Pure search returned no matching item; a web search found only news coverage. It stays Unavailable until a copy is supplied. |
| 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-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-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-1639 v2 | Interconnection of port-Hamiltonian systems and composition of Dirac structures | Cervera, van der Schaft & Baños (2007). Power-conserving interconnections of port-Hamiltonian systems are described by Dirac structures, and composing Dirac structures gives again a Dirac structure, so interconnected port-Hamiltonian systems stay port-Hamiltonian; the paper characterizes when the composition is well defined. | Unfetched — searched 2026-10-27 for a lawful full text; one exists but this client could not fetch it, so it is not read and no NOTE is filed (ADR-053). Unpaywall lists the submitted version in the University of Groningen repository (research.rug.nl/files/2818803/2007AutomaticaCervera.pdf) and a bronze copy on ScienceDirect; the repository file answered curl with a bot challenge (“Just a moment…”), and a headless-browser fetch was not attempted, since it would be working around the challenge. The failure is on access, not availability. A browser fetch of that copy is the cheapest way to read it. It stays Unfetched until a copy is supplied. |
| LIT-1644 v2 | A Compositional Framework for Passive Linear Networks | Baez & Fong (2018), TAC 33. Open circuits of passive linear two-terminal components (resistors, and inductors and capacitors through the Laplace transform over ℝ(s)) are morphisms of a hypergraph category Circ built from decorated cospans, composed by gluing terminals. The black box functor Circ → LagRel, a hypergraph functor (Thm 1.1, Thm 7.7), sends a circuit to the Lagrangian relation between its terminals’ potentials and currents, the graph of the differential of the power functional left when the interior is eliminated by the principle of minimum power. The external behaviour of a composite is the composite of the parts’ behaviours. Dirichlet forms alone compose by that elimination but lack identities, since an ideal wire has none. | Active — read in full 2026-10-27 (NOTE-1223), from arXiv v6; worth reading as the theorem that black-boxing passive linear circuits is functorial: the Lagrangian relation a circuit imposes on its terminals’ potentials and currents composes as the circuits do. It is also a clean account of eliminating an interior by minimum power, and of why that elimination alone has no identities. Its scope is narrow: passive, linear, two-terminal components, time only through the Laplace transform with zero initial conditions. |
| 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-1660 v2 | The Behavioral Approach to Open and Interconnected Systems | Willems (2007). A dynamical system is its behavior, the set of trajectories its laws allow; inputs and outputs are not assumed; systems are interconnected by sharing variables and imposing constraints on them, and modelling proceeds by tearing a system into parts, zooming into each and linking them again. | Unavailable — filed 2026-10-27, not read. A reading was attempted on 2026-10-27 and no lawful full text was reached: Unpaywall and OpenAlex both give the article as closed; OpenAlex lists a CiteSeerX cache, not used, and a University of Groningen record, whose page offered no file. It stays Unavailable until a copy is supplied. |
| 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-1810 v2 | Structure-Mapping: A Theoretical Framework for Analogy | Gentner (1983). An analogy maps the relations of a base domain onto a target and drops the attributes of its objects; of the relations, those bound into a system by higher-order relations are more likely to be carried over than isolated ones (systematicity), so in “the atom is like the solar system” MORE MASSIVE THAN maps and HOTTER THAN does not. The rules look only at the form of the representation, the arity and order of predicates, which sorts comparisons into literal similarity, analogy, abstraction and mere appearance. The predicate structure is taken as given, and the author concedes it is a modelling choice; the evidence is one study summarized without statistics. | Active — read in full 2026-10-29 (NOTE-1349) from a scan of the published article on the author’s Northwestern lab pages; worth reading as the founding statement that analogy maps relations rather than attributes and that higher-order relations select which relations go across, which is what a single-relation alignment such as Gromov–Wasserstein leaves out. |
| 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. |