mathematics.
250 of 1538 NOTE documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| NOTE-009 v2 | Spectral Sets | A survey of when a set X ⊃ σ(A) satisfies ‖f(A)‖ ≤ K sup_X|f| for all rational f. Its headline entries are that the closure of the numerical range is completely K-spectral with a universal K_Crouzeix ∈ [2, 11.08] (Fact 107.7.2a; completely 2-spectral for 2×2, Fact 107.7.2b) and that σ(A) is K-spectral iff A is diagonalisable, with K the eigenvector condition number (Example 107.2.5). §107.8 turns these into Krylov error bounds such as Arnoldi ε_m ≤ 2K_Crouzeix ρ_{m−1}(f,E) for convex E ⊃ W(A). | Read |
| NOTE-016 v2 | The sheaf-theoretic structure of non-locality and contextuality | For finite measurement scenarios, an empirical model has a factorizable (local / non-contextual) hidden-variable realization iff its compatible family of distributions extends to a global section (Thm 8.1). Over the reals every no-signalling model has such a section (Thm 5.9: “negative-probability LHV” ⇔ no-signalling, because both span the same subspace, of dimension D = Σ_{U∈𝒰}(l−1)^{|U|}, Thm 5.4). Contextuality comes in three strictly nested strengths (probabilistic ⊋ possibilistic ⊋ strong), with the Bell, Hardy and GHZ(n≥3) models as witnesses of each level. | Read |
| NOTE-022 v2 | Spectral Networks and Betti Lagrangians | For any rank n, and for Lagrangians with non-algebraic (positive-braid) asymptotics, spectral networks exist and are characterised by pseudo-holomorphic strips in the adiabatic limit (Theorems 1–2). For exact Betti Lagrangians, a Family Floer functor equals Gaiotto–Moore–Neitzke non-abelianization in the adiabatic limit (Theorem 3, i.e. 5.24) and is realised by A∞-operations in wrapped and partially wrapped Fukaya categories of T*S (Theorem 4). The analysis in §§4–5 leans on Ekholm 2007 and on the second author’s 2024 preprint, and Theorem 5 (spectral networks from Demazure weaves) is recorded without a written proof. | Read |
| NOTE-026 v2 | Forman-Ricci curvature of networks | The paper specializes Forman’s (2003) Bochner–Weitzenböck curvature on weighted CW complexes to graphs, giving the closed-form edge curvature Ric_F(e) = ω(e)[ω(v1)/ω(e) + ω(v2)/ω(e) − Σ_{e_v1∼e, e_v2∼e}(ω(v1)/√(ω(e)ω(e_v1)) + ω(v2)/√(ω(e)ω(e_v2)))] (Eq. 11). It then proposes three uses: an EMD distance between curvature distributions, which puts Facebook, Google and BioGrid graphs at 0.050–0.060 from a Barabási–Albert model against 0.17–0.36 from Erdős–Rényi and Watts–Strogatz (Table 1); a discrete “2-D-form” Ricci flow γ̃(e) − γ(e) = −Ric_F(γ(e))·γ(e) (Eq. 36) for change detection; and a Laplacian flow ∂G/∂t = (□₁ − F)G (Eq. 45) for denoising. The authors say the flow applications are “only theoretically proposed and illustrated solely on a small example” (§5). | Read |
| NOTE-035 v2 | Plávala, General probabilistic theories: An introduction | A self-contained textbook on finite-dimensional GPTs whose central results are that state spaces and effects are dual (K = {φ ∈ A(K)*+ : ⟨φ, 1_K⟩ = 1}, Thm 3.19) and that optimal two-state discrimination succeeds with probability ½(1 + ‖λx₀ − (1−λ)x₁‖) in base norm (Thm 3.43). Every composite lies between the minimal and maximal tensor products (Prop. 5.6), and four statements are equivalent: id_K is measure-and-prepare, id_K is self-compatible, K admits a universal broadcasting channel, and K is a simplex (Thm 7.7); minimal = maximal iff one factor is a simplex is cited to [134], not proved. | Read |
| NOTE-037 v2 | Debreu 1952, a social equilibrium existence theorem | Debreu proves existence of an equilibrium for an “abstract economy” (generalized game), in which each agent’s feasible set A_ι(ā_ι) depends on the others’ actions. The conditions: each action set 𝔄_ι is a contractible polyhedron; A_ι has a closed graph with non-empty compact values; f_ι is continuous on the graph into the extended reals; the value function φ_ι(ā_ι) = max f_ι is continuous; and each best-response set M_ā_ι is contractible. The proof applies the Eilenberg–Montgomery/Begle fixed-point theorem to the product best-response map. A Remark replaces the joint continuity of φ_ι with continuity (lower hemicontinuity) of A_ι plus a compact graph. A Corollary gives a saddle-point theorem that contains Kakutani’s and von Neumann’s. | Read |
| NOTE-046 v2 | Chu & Raginsky, expected soft maxima of Gaussian processes | Take a centered Gaussian process on a finite set T with max variance σ² and minimum separation a. The expected softmax-weighted average g(β) = E[Σ_t X_t e^{βX_t}/Z] satisfies g(β) ≤ √(2σ²·E D(ν_β‖ν₀)) for every β (Thm 1, D = KL from uniform). For β ≥ β*, g(β) ≥ c·a·√(E D(ν_β‖ν₀)), with c the Sudakov constant (1/17 via Marcus–Rosen) (Thm 2). For i.i.d. N(0,σ²) logits there is an exact identity, g(β) = βσ²(1 − E‖ν_β‖²₂) (Lemma 2). The quenched free energy is bounded above by √(2σ²·E D_{1/2}(ν_β‖ν₀)), with an i.i.d. lower bound of (cσ/2)√(E D_{1/2}) (Thms 4–5). | Read |
| NOTE-047 v2 | nLab, two-level type theory | The nLab page is an accurate but thin gloss on two-level type theory. It says nothing about the result that makes 2LTT usable: basic 2LTT is conservative over HoTT (ACKS Prop. 2.18: an inner type provably inhabited in 2LTT is inhabited in HoTT), which answers much of the page’s own “open question” about the principle of equivalence. It also omits that a fibrant-replacement type former would force UIP on the inner level (ACKS Thm 2.20). The paper it cites, ACKS arXiv:1705.03307, is the proper source: definitions via models, conservativity, the axiom menu (A1)–(A6)/(T1)–(T3), and internal Reedy-fibrant diagrams over inverse categories with a definition of univalent (∞,1)-category. | Read |
| NOTE-050 v2 | The relativity of causal knowledge | The paper defines SCMs as functors I → Prob and interventions as natural transformations (which requires invertible mixing functions). It encodes each non-intervened SCM as the convex space Δ(V, Ω) of all probability measures on its endogenous space, and puts these spaces on a graph as stalks of a network sheaf and cosheaf. The restriction maps are interventionally consistent α-abstractions and the extension maps are unspecified affine maps. “Relative causal knowledge” is the image of alternating restriction and extension along a path (Def. 11). The two substantive-looking theorems are either unsound or vacuous. Theorem 1 (every convex combination of a model’s interventional measures is a soft-interventional measure of it) rests on a proof step that is false. Theorems 2 and 3 hold for any pushforward map, so interventional consistency does no work in them. There are no results about the sheaf. | Read |
| NOTE-051 v2 | Cairo, counterexample to the Mizohata-Takeuchi conjecture | For every C² hypersurface Σ ⊂ ℝᵈ not lying in a hyperplane (implicitly: one with a point of non-zero curvature; see corrections), and every large R, there are f ∈ L²(Σ) and a weight w = |ĥ|² with supp ĥ ⊂ B_{CR} such that ∫|Ef|²w ≳ log R · ‖f‖²₂ · sup_ℓ ∫_ℓ w (Thm 1.2 via Prop. 3.1). So the Mizohata–Takeuchi conjecture and Stein’s conjecture (Conj. 1.4) are false as stated, and the CHV route to loss-free endpoint multilinear restriction is closed. The mechanism is a lattice Q of middle-layer subset sums of N ≍ log R points on Σ, chosen so that no hyperplane meets more than 2^{d−1} of the R⁻¹-balls around Q (Lemma 3.5). | Read |
| NOTE-062 v2 | Başar, Meyn & Perkins, control system theory notes | A conventional first-year graduate course in linear state-space control. It covers modelling and linearisation, LTI/LTV solutions, Lyapunov stability, controllability and observability with Kalman/Hautus tests and minimal realisations, then pole placement and observers, integral control, sensitivity, and finally HJB/LQR and a heuristic minimum principle. Its most useful non-standard content is the Pendubot thread (Chs 1, 9, 10): an observer-based design from one sensor is fragile (the compensator has an unstable pole at s ≈ 46.4), while full-state LQR has |S(jω)| ≤ 1, infinite gain margin and ±60° phase margin. Every worked example I recomputed is correct. | Read |
| NOTE-066 v2 | Backens & Duman, graphical calculus for Spekkens’ toy bit | Take the maximal-knowledge fragment of Spekkens’ toy bit theory with post-selected measurements, viewed as relations over {1,2,3,4}ⁿ. It has a ZX-style red–green graphical calculus that is universal (§4.3), sound (§4.4) and complete (Thm 17). Completeness goes through a proof that the toy theory’s binary “check-matrix” formalism coincides with the qubit stabilizer one (valid states SᵀJS = 0, valid transformations QᵀJQ = J, Lemma 3). With it come graph-state/local-complementation results for toy states (Thms 4–5), map–state duality (§5.2) and an rGS-LO normal form (Thms 10–11). The rewrite rules differ from the stabilizer ZX-calculus only through the phase group: Z₂×Z₂ against Z₄. That difference appears as an “11-commutation” rule in place of π-commutation (§4.5). | Read |
| NOTE-068 v2 | Tembine et al., mean-field-type games with Rosenblatt noise | The paper’s control and game results use linear state feedback, and they depend on the Rosenblatt noise only through its covariance, which is exactly fBM’s with the same H. They are therefore identical to the fBM results. The ergodic LQ gain (Thm 4), K̂ = −[b₁ + √(b₁² + 4H(1−H)b₂²q/r)]/(2b₂(1−H)), is correct and reduces to the Brownian LQR gain at H = ½. But the paper’s headline “suboptimality of noise approximation” (§VI-A, repeated for the variance-aware control, zero-sum, non-zero-sum and MFTG settings) holds only against surrogates with a different covariance; against fBM with the same H it is false. The ML claim that Rosenblatt “super-diffusion transformers” “significantly enhance the performance of generative machine intelligence models” (§XIV-C) has no model, training, dataset or experiment behind it anywhere in the paper. | Read |
| NOTE-069 v2 | Riehl & Shulman, type theory for synthetic ∞-categories | A three-layer type theory adds a directed interval 2 to HoTT. Its layers are cubes, topes (the coherent theory of a strict interval) and types, and simplices become shapes that can be mapped out of by extension types. One contractibility condition, “Segal” (unique fillers for the 2-1-horn), then yields associativity (Prop. 5.9) and 3-horn fillers (Prop. 5.21). It also makes every function a functor (Prop. 6.1), gives a dependent Yoneda lemma for covariant families (Thm. 9.5), and makes having a left adjoint a proposition when the adjoint’s codomain is Rezk (Thm. 11.23). The bisimplicial-sets semantics is sketched. Segal and Rezk types are shown to match Segal and complete Segal spaces (Props. A.21, A.25), and soundness is not proved in full. | Read |
| NOTE-074 v2 | The epic story of maximum likelihood | Stigler gives a history of maximum likelihood from Lagrange (1769) to Le Cam (1953). Fisher gave three optimality arguments. The 1922 one rests on the false belief that ML estimates are always sufficient. The 1925 “ANOVA” one derives the information inequality V ≥ 1/I(θ) from Var X = E Var(X|T) + Var E(X|T), treating approximate normality as exact. The third, sent privately to Hotelling in 1930, holds for multinomial models and smooth degree-0-homogeneous estimators, and was published only in disguise in 1935. Stigler’s verdict is that the Neyman–Scott growing-parameter example (Wald 1938, published 1948) is the practically important failure. He calls Hodges’s 1951 superefficiency example “an ingenious but minor technical achievement”, because the superefficiency set has Lebesgue measure zero (Le Cam 1953). | Read |
| NOTE-078 v2 | Filip, Notes on the Multiplicative Ergodic Theorem | Expository notes that prove the Oseledets theorem in four ways. §2 uses projective invariant measures and a splitting lemma, without Kingman. §3.5 checks Kaimanovich’s regularity criterion in the symmetric space GLₙ/Oₙ, using Kingman. §4 proves the Karlsson–Ledrappier horofunction theorem (for µ-a.e. ω there is a horofunction h_ω with −(1/N)h_ω(g_ω⋯g_{T^{N−1}ω}x₀) → l) for isometries of proper metric spaces. §5 gives an L² “mean” version via Monod’s direct integrals of CAT(0) spaces and Karlsson–Margulis tracking, with an L² mean Kingman theorem. Two of the notes’ own derivations do not work as written: Proposition 3.4.8(iii) is mis-stated, and the Lemma 5.1.5 step in the tracking proof does not apply as claimed. The main theorems are standard and not in doubt. | Read |
| NOTE-082 v2 | Barnum, Gaebler & Wilce, ensemble steering and self-duality | A bipartite state ω steers its B-marginal iff ω̂ : [0,u_A] → [0,ω_B] is a strong quotient map of ordered sets (Thm 5.4). When ω̂ is injective and the marginal is interior, a steering state is an order-isomorphism (Cor. 5.5). With Thm 4.1/Cor. 4.2, it follows that universal self-steering makes every irreducible state space homogeneous and weakly self-dual (Prop. 5.8). The step from there to C*-algebraic quantum theory needs strong self-duality, which the paper does not derive, and a Hanche-Olsen correspondence it defers (“as we will establish elsewhere”, §6). | Read |
| NOTE-085 v2 | Local separators and community structure | This is a comparison of the authors’ local 1- and 2-separator decompositions against Infomap, Label Propagation, Leiden and Louvain (igraph) on five sparse graphs (62 to 9,133 nodes). Local separators score lower overlapping modularity than the modularity optimisers everywhere, and lower than Infomap on 3 of 5 graphs (Table 2). Their edge is confined to high density thresholds on the two road networks, where the best radius gives a handful more qualifying clusters: about 12 against about 5 at β ≥ 1.25 on Euroroads, and about 50 against about 40 for Infomap at β ≥ 1.5 on NRW (Figs. 3–4). On Netscience all methods are similar. At δ = 1 on both road networks, and at δ = 1.25 on NRW, Infomap yields more qualifying clusters than any local-separator setting. | Read |
| NOTE-088 | The Homological Nature of Entropy | On a finite probability space, take an information structure S (a set of partitions closed under joins that have a common refinement in S and containing the trivial partition) and a probability family Q complete for it. The information cohomology H*(S;Q) is computed by a Hochschild-type bar complex whose action is Shannon’s averaged conditioning. If S is connected and “sufficiently rich” and cochains are measurable, then H¹ is one-dimensional and spanned by Shannon entropy (Thm 1). This makes entropy the unique measurable 1-cocycle, up to one multiplicative constant per connected component. In the unitary-invariant quantum case H¹_U = 0: Shannon entropy is the coboundary of minus the von Neumann entropy, which is the unique invariant 0-cochain with that property up to an additive constant (Thm 3). The tree (“arborescent”) version again gives H¹ spanned by entropy (Thm 4). | Read |
| NOTE-101 v2 | Luque — Philosophical analysis of the Price equation | Luque argues that the Price equation, Δz̄ = Cov(w, z) + E(wΔz) (eq. 1, p. 7), is a Kuhnian “generalization-sketch”, like f = ma. It is not a model, has no predictive power on its own, and becomes empirical only when a specific model interprets it. On that reading the critics’ charges (van Veelen 2005; van Veelen et al. 2012; Nowak & Highfield 2011) that it is “a tautology” and “lacks predictive power” are either false or ask too much, and he recommends that evolutionary biologists adopt it as their unifying framework. | Read |
| NOTE-118 v2 | Belot — Time and change in mechanics | For theories with ideal existence, uniqueness and a dynamical time-translation group, time is three intertwined ℝ-actions (on spacetime, on the symplectic space of solutions, and on the isomorphic space of initial data), and change is a slicing-indexed family of functions on those spaces. Singular dynamics, gauge freedom and time-dependence each damage this picture in a repairable way. Spatially compact vacuum GR breaks it, because the reduced Hamiltonian vanishes (h ≡ 0) and there is only one canonical isomorphism between the reduced spaces. That is the problem of time. It is not an artefact of the 3+1 split, it does not show change to be illusory, and it is absent from asymptotically flat GR and from artificially generally covariant theories. A parameterized “geometric time”, such as CMC, restores change and, in the CMC case, a time-dependent Hamiltonian flow. | Read |
| NOTE-133 v2 | Leitgeb & Ladyman — Criteria of identity and structuralism | Graph theory counts unlabelled graphs whose nodes, or whose isomorphic connected components, cannot be told apart even by an irreflexive relation. Examples are the edgeless two-node graph G′ and any graph with two or more isolated nodes (11 of the 156 unlabelled 6-node graphs). Mathematicians treat these as determinately distinct objects, so the authors conclude that the identity or difference of places in a structure is given by the structure itself, with identity counted among its relations, and not by any criterion of identity. The argument appeals to mathematical practice; it is not a proof. | Read |
| NOTE-135 v2 | Kleiner & Ludwig — Mathematical structure of experience | The paper proposes a definition, (MSC). A mathematical structure S = ((A_i), (S_j)) is a structure of conscious experience iff (S1) its domains are sets of experiential aspects, and (S2) every relation or function S_j has an “S_j-aspect”. An S_j-aspect is an aspect that a variation of experience changes exactly when the variation fails to preserve S_j. The older condition (MDC), that a structure is “of” experience if its domains are aspects and its axioms hold, survives only as the necessary condition for describing experience, because as a sufficient condition it admits incompatible structures (discrete and non-discrete topologies on the same set), arbitrary re-definitions (every C·d for C > 0) and consciousness-indifferent structures. Worked examples: experienced relative similarity is a strict partial order that satisfies (MSC), given transitivity (assumed). Visual regions form a topology that satisfies (MSC), given Bayne and Chalmers’s subsumptive unity thesis. | Read |
| NOTE-141 v2 | Corfield — Duality as a category-theoretic concept | A short programmatic paper. Its thesis is that category theory, not set theory, is the framework in which the scattered instances of mathematical duality can be drawn together. The constructions in §3 (dual adjunctions restricting to dual equivalences on fixed points, truth-value-enriched Galois connections, Isbell conjugation, dualising objects “keeping summer and winter homes”) all “revolve around” adjunction, though Corfield expects no “single monolithic account” (p. 59–60). On his usage, ‘duality’ is reserved for “involution with some form of structural reversal” (p. 57), so several physicists’ dualities (homological mirror symmetry as an A∞-equivalence, mirror symmetry’s Z2 on the Hodge diamond, S-duality as a vestige of SL(2,Z)) are better called equivalences or symmetries. He proposes, without argument beyond the examples, that philosophers of physics will need homotopy type theory rather than predicate logic for the sameness questions dualities raise. | Read |
| NOTE-163 v2 | SEP — Mereology | The entry argues that no mereological principle, not even the partial-order core M (Reflexivity, Transitivity, Antisymmetry), is merely a fixing of the meaning of “part”. It lays out the ladder M ⊂ MM (+Supplementation) ⊂ EM (+Strong Supplementation) ⊂ GEM (+Unrestricted Sum). GEM is “virtually” a complete Boolean algebra with zero removed, and it is reached from M by exactly three adequate pairings: Supplementation + Unrestricted Sum2, Strong Supplementation + Unrestricted Sum3, and Complementation + Unrestricted Sum1. The influential Simons (1987) pairing, Supplementation + Unrestricted Sum3, which the entry’s own 2003 edition also used, does not yield GEM. | Read |
| NOTE-177 v2 | Turner — Ultrafilters as Propositional Theories | An expository piece that redefines a filter over a set W as a “propositional theory” (a set of W-propositions closed under pairwise conjunction and implication, not containing ∅), proves maximal ⇔ prime ⇔ ultra (Thm 2), the ultrafilter extension theorem as a propositional Lindenbaum’s lemma (Thm 3, assuming choice), Łoś’s theorem M_T,[A] ⊨ φ iff ⟦φ⟧_A ∈ T for ultraproducts of “intensional spaces” (Thm 9), and first-order compactness from it (Thm 12); “None of the results are novel” (p. 2). | Read |
| NOTE-182 v2 | Arsiwalla et al. — Pregeometry and formal language | A programmatic essay with no construction, theorem or model of its own. It argues that inferential justification in a materialist ontology regresses without end (§3), that a “coherentist” constructivism escapes the regress, and that Wheeler’s pregeometry is therefore best read as “structureless structure”: the types and compositional rules of a computational language such as homotopy type theory (§§4, 6). Every construction of space from such rules that it points to is in the authors’ own earlier papers ([13, 17, 18]) or is promised as “a forthcoming publication” (fn. 1, p. 3). | Read |
| NOTE-195 | Yoneda lemma | The page states that for a locally small category C, every presheaf X and every object c, there is a canonical bijection Hom_{[C^op,Set]}(y(c), X) ≅ X(c), given by η ↦ η_c(id_c). It proves the bijection by chasing id_c around a naturality square. From it the page derives that the Yoneda embedding y: c ↦ Hom_C(−,c) is fully faithful and that y(c) ≅ y(d) ⇔ c ≅ d. It also shows by counterexample that the hypothesis doing the work is naturality: objects whose hom-sets are pointwise in bijection need not be isomorphic. | Read |
| NOTE-202 v2 | Kolmogorov’s structure functions and model selection | The structure function of a single data string determines all of its stochastic properties: within any complexity-constrained model class it picks out the best-fitting model with certainty. Every admissible shape of that function occurs for some data, and neither it nor the minimal sufficient statistic is computable. | Superseded — by NOTE-787 |
| NOTE-204 v2 | GNS construction (Wikipedia) | Any state (positive normalized linear functional) on a C*-algebra determines, essentially uniquely, a Hilbert space and a cyclic *-representation in which that state becomes a vector expectation ρ(a)=⟨π(a)ξ,ξ⟩, and the representation is irreducible exactly when the state is pure. | Superseded — by NOTE-857 |
| NOTE-206 v2 | Adler — Hilbert spaces and the Riesz representation theorem | Every continuous linear functional on a Hilbert space is the inner product with a unique vector of equal norm, and the representer can be written explicitly as the sum of φ(e_k)e_k over any orthonormal basis, which in turn yields the Radon–Nikodym theorem. | Superseded — by NOTE-854 |
| NOTE-211 v2 | Knowledge Sheaves (sheaf-theoretic KG embedding) | Knowledge-graph embedding can be recast as learning an approximate global section of a cellular sheaf on the schema graph, which subsumes Structured Embedding and TransE/TransR-style models and yields, via harmonic extension, a training-free way to answer composite multi-hop queries. | Superseded — by NOTE-843 |
| NOTE-212 v2 | Meaningful information (sophistication) | With total recursive functions as the model class, the algorithmic minimal sufficient statistic (“sophistication”) is a non-trivial measure of meaningful information. It matches the finite-set and probability-model versions up to additive terms, some objects are entirely meaningful with no residual randomness, and it is not computable. | Superseded — by NOTE-790 |
| NOTE-214 v2 | Riesz representation theorem (Wikipedia) | A Hilbert space is isometrically (anti-)isomorphic to its continuous dual — every continuous linear functional φ is x ↦ ⟨x, f_φ⟩ for a unique f_φ with ‖f_φ‖ = ‖φ‖ — and this identification is what defines adjoints and the bra–ket correspondence. | Superseded — by NOTE-861 |
| NOTE-215 v2 | Bra–ket notation (Wikipedia) | Dirac’s bra–ket notation writes vectors as kets and continuous linear functionals as bras, and on a Hilbert space the Riesz representation theorem is what licenses treating every bra as the conjugate of a ket, so that ⟨φ|ψ⟩ is at once a functional applied to a vector and an inner product. | Superseded — by NOTE-855 |
| NOTE-219 v2 | Duality of Bures and Shape Distances | 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. | Superseded — by NOTE-753 |
| NOTE-220 | Jorgensen & Tian, Noncommutative Analysis (GNS, RKHS) | The textbook puts the kernel construction and the GNS construction in one frame. It calls building a Hilbert space from a positive definite function “the GNS construction”, and proves (Cor. 1.35) that the feature map realising a kernel as inner products is unique up to a unitary among minimal realisations. It also states that in the abelian case states are exactly probability measures, so that “the GNS construction is non-commutative measure theory”. | Skimmed |
| NOTE-226 v2 | Kernel mean embedding of distributions (review) | 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. | Superseded — by NOTE-838 |
| NOTE-229 v2 | Information Theory with Kernel Methods | With a kernel normalised to k(x,x) = 1, the covariance operator Σ_p = E_p[ϕ(x)ϕ(x)*] is a density operator (PSD, unit trace), injective in p when k² is universal. Its von Neumann entropy and relative entropy behave like Shannon quantities, with D(Σ_p‖Σ_q) ≤ D(p‖q), and the empirical versions are computed from the normalised Gram matrix K/n. | Superseded — by NOTE-837 |
| NOTE-230 v2 | A Generalized Representer Theorem | For any strictly increasing regularizer g(‖f‖) and any (even non-convex, coupled) cost on the training outputs, every RKHS minimizer of c((x_i,y_i,f(x_i))_i) + g(‖f‖) is a finite kernel expansion f = Σ α_i k(·,x_i). The proof needs only the reproducing property and orthogonal decomposition onto span{k(·,x_i)}, not the Riesz theorem. | Superseded — by NOTE-836 |
| NOTE-231 v2 | Spectral Inference Networks (SpIN) | The top-N eigenfunctions of a symmetric kernel operator Kf = E_{x′}[k(x,x′)f(x′)] on L²(p) can be learned by a neural network that maximizes the generalized Rayleigh quotient Tr(Σ⁻¹Π), with Σ = E[u uᵀ] and Π = E[k(x,x′)u(x)u(x′)ᵀ]. Slow Feature Analysis is the special case where k is the graph Laplacian of adjacent video frames. | Superseded — by NOTE-824 |
| NOTE-236 v2 | Abramsky, Barbosa & Mansfield, contextual fraction | For any no-signalling empirical model, the contextual fraction CF (one minus the largest weight of a noncontextual sub-model) is a linear programme. Its dual gives a Bell inequality whose normalised violation equals CF, and no Bell inequality is violated more (Theorem 1). CF is invariant under relabelling, non-increasing under translation of measurements, coarse-graining and mixing, and NCF is multiplicative under products (Theorem 2). It lower-bounds the failure probability of Z2-linear MBQC (Theorem 3) and of k-consistent games (Theorem 4). | Read |
| NOTE-239 | The Geometry of Information Retrieval | The book recasts IR’s vector-space, probabilistic and logical models in one Hilbert-space language. Documents are state vectors. Relevance and aboutness are Hermitian observables that need not commute. Subspaces carry a non-distributive logic whose conditional is shown to be a Stalnaker conditional equal to the Sasaki hook (ch. 5). Gleason’s theorem makes tr(ρP) the probability of a subspace, which ch. 6 uses to rewrite cosine matching, cluster representatives, relevance and pseudo-relevance feedback, dynamic clustering and ostensive retrieval. It is a proposed language, not a tested model: no experiment is reported, and its “content hypothesis” that the probability of a concept is cos²θ is argued, not tested. | Read |
| NOTE-240 | The Lattice Representation Hypothesis of Large Language Models | Thresholding linear attribute directions (Fisher-LDA probes) gives a crisp object–attribute incidence, and the paper’s Theorem 1 is the standard fact that any such incidence has a complete concept lattice. The geometric meet is half-space intersection; the join is defined as the set union of two cones and “approximated by the conic hull” of their directions, which as written is not a lattice join. On five WordNet domains with GPT-4o-annotated attributes, the probes recover the incidence at F1 69.7–83.2 on three 7–8B models, and profile-based subsumption reaches F1 57.1–77.1. No lattice law is ever tested. | Read |
| NOTE-242 v2 | Attention Approximates Sparse Distributed Memory | Kanerva’s SDM read weights each stored pointer by the number of neurons in the intersection of two Hamming balls of radius d. The paper shows that this weight is roughly log-linear in Hamming distance for close patterns. As a result, with L²-normalised vectors and a regression-fitted β, softmax attention reproduces SDM’s retrieval behaviour on random and learnt-projection data (App. B.7). Two things are weaker than they look. The analytic derivation (App. B.2, Eqs. 16–21) gets the exponent wrong: by my recomputation its β is ≈2–2.5× too small. The biological case is Kanerva’s 1988 cerebellar mapping, restated rather than re-argued. It also needs r = 2⁶⁴ ≈ 1.8×10¹⁹ neurons in the Attention setting, which the paper itself calls “biologically implausible” (App. B.5). | Read |
| NOTE-244 | Neural Tangent Kernel of Matrix Product States: Convergence and Applications | In the sequential infinite-bond-dimension limit, with tensor variances σ_i²/√(|α_i||α_{i+1}|) and per-tensor learning rates (|α_i||α_{i+1}|)^{-1/2}, the NTK of a periodic MPS tends to K(x,x′)=Σ_k φ(x_k)·φ(x′k) Π{l≠k} σ_l² φ(x_l)·φ(x′_l). By my reduction this is (Σ_k σ_k⁻²) times the MPS’s own GP covariance, so gradient flow is kernel regression with a fixed product kernel. The Born-machine solution P_x(t)=1/m−(1/m−P_x(0))e^{−4mKt/Z} is correct (I checked it; Z is in fact conserved). The lazy-training lemma is heuristic, the positive-definiteness “proof” shows at most semi-definiteness, several constants are off by factors of n or 2^{−n}, and there are no experiments. | Read |
| NOTE-245 | Mathematical Foundations for a Compositional Distributional Model of Meaning | 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. | Read |
| NOTE-246 v2 | Reasoning about Meaning in Natural Language with Compact Closed Categories and Frobenius Algebras | 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. | Read |
| NOTE-247 | A new description of orthogonal bases | Theorem 5.1 proves that on a finite-dimensional complex Hilbert space, commutative †-Frobenius monoids in FdHilb and orthogonal bases are in bijection: the basis is the monoid’s copyable elements, and the monoid is the linear extension of copying and uniformly deleting that basis. §6 adds that the monoid is special (m∘δ = id) exactly when the basis is orthonormal. Corollaries 7.1 and 7.2 lift this to categories: with fully structure-preserving maps the category is equivalent to the groupoid of finite sets labelled by positive reals (the basis norms), and with comonoid homomorphisms it is equivalent to FinSet. | Read |
| NOTE-248 | H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics | In the category Hilb of complex Hilbert spaces of arbitrary dimension, a Frobenius algebra with a unit exists only in finite dimension (Lemma 3). So the paper drops the unit, and Theorem 22 proves that a nonunital commutative special †-Frobenius algebra corresponds to an orthonormal basis (the basis being the algebra’s copyable vectors) if and only if it is semisimple, if and only if it satisfies Ambrose’s H*-axiom (H), and if and only if it is a directed colimit of finite-dimensional unital ones. Whether (H) is automatic, i.e. whether a nonzero “radical” Frobenius algebra exists (Proposition 23), is left open. In categories of relations and of nonnegative ℓ²-matrices, Frobenius algebras decompose as disjoint unions of abelian groups and do satisfy (H) (Theorems 30–38). | Read |
| NOTE-249 v2 | Measurement as Sheafification: Context, Logic, and Truth after Quantum Mechanics | A programmatic essay, with no theorem, model or computation of its own. It restates Abramsky & Brandenburger’s criterion (contextuality and non-locality as the absence of a global section of a presheaf over measurement contexts) and adds three things. It declares measurement to be “sheafification” of that presheaf, without specifying a site, a topology or any sheafified object. It asserts that the Ghose–Patra seven-valued logic is a finite Heyting algebra, without giving an order or operations. It proposes that a quantum potential scaled by 0 ≤ λ ≤ 1 (λ = 1 quantum, λ = 0 classical Hamilton–Jacobi) carries the presheaf continuously to a sheaf, which is asserted, not modelled. Its one worked formula, v2’s Čech “obstruction class” [δs], is a coboundary and so is zero by construction. | Read |
| NOTE-250 v2 | The Cohomology of Non-Locality and Contextuality | For an empirical model e on a measurement cover 𝒰, a section s in the support of context C₁ gets a class γ(s) ∈ Ȟ¹(𝒰, F_{C̄₁}), relative Čech cohomology of the presheaf F = F_ℤ S_e of formal ℤ-linear combinations of support sections. γ(s) = 0 iff s belongs to a compatible family of such ℤ-combinations (Prop 4.2), so a non-vanishing γ(s) is a sufficient but not necessary witness of contextuality (Prop 4.3). γ is non-zero for every section of the PR box, GHZ, the triangle, the 18-vector Kochen–Specker set and the Peres–Mermin square, but it vanishes on the Hardy model and on one strongly contextual Kochen–Specker cover (§8). | Read |
| NOTE-251 | Contextuality, Cohomology and Paradox | For a model whose outcomes live in a commutative ring R, “All-vs-Nothing” (AvN_R) means the R-linear equations satisfied by every support section of every context have no global solution. Theorem 21 proves AvN_R(S) ⇒ SC(Aff S) ⇒ CSC_R(S) ⇒ CSC_ℤ(S) ⇒ SC(S), so every AvN model is strongly contextual with a non-vanishing Čech obstruction on every section. Theorem 4 shows that every state stabilised by an “AvN triple” of n-qubit Pauli operators yields such a model. The “paradox” half proves much less: an informal identification of Liar cycles with strongly contextual support tables, one exact case (the length-4 Liar cycle is the PR box), and the cohomology of paradox deferred to future work. | Read |
| NOTE-252 | On the Cohomology of Contextuality | Carù gives a possibilistic model on the two-party, two-setting, four-outcome Bell cover that is strongly contextual, yet whose Čech obstruction vanishes on all 22 support sections (Fig 4, Table 1). That refutes Abramsky–Mansfield–Barbosa’s Conjecture 8.1 for any symmetry or connectedness condition the Bell cover meets. I confirmed the refutation over ℤ and ℤ/2. The rest is weaker. Odd-degree “higher obstructions” form a hierarchy (Prop 5.5), but every level above the first is empty for every no-signalling model (Prop 5.6). Prop 6.3 restates the known torsor description of Ȟ¹. Prop 6.1 is false in the “only if” direction; the PR box refutes it. | Read |
| NOTE-253 | Towards a complete cohomology invariant for non-locality and contextuality | For cyclic scenarios, where the contexts’ intersection graph is a single chordless N-cycle, Carù proves that ℤ/2 Čech cohomology of the (N−1)-th iterated “joint model” is a complete invariant for logical and strong contextuality (Thm 7.7). I confirmed this on the Hardy model, on his own 2017 counterexample (detected on all 22 sections at level 3) and on 659 random contextual cycle models. The extension beyond cycles does not hold as stated. Prop 8.2 is false on the paper’s own Table 5 model. Worse, the one non-cyclic case the paper claims to repair, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa, is never detected at any level under the paper’s definitions (my proof and computation). That refutes the paper’s closing Conjecture 9.1. | Read |
| NOTE-254 | Logical and Topological Contextuality in Quantum Mechanics and Beyond | The thesis’s cohomology chapters republish the 2017 and 2018 papers with the “joint model” renamed the “line model”. The definitions, Thm IV.32 (complete on cyclic scenarios at level |M|−1), Prop IV.34, Thm IV.36 and Conjecture IV.41 carry over unchanged. So does every defect found in the preprint. Under the thesis’s own Def IV.26, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa is still detected on only 6 of 15 sections at levels 0–3 (my recomputation), and a lifted ℤ/2 family shows it is never fully detected at any level. The context-wise reading the worked examples use detects all 15 sections at level 2. The genuinely new material lies outside the cohomology programme: a probabilistic line-model construction (§IV.7), a valuation-algebra “theory of disagreement” with inference algorithms (Ch V, bound O(k^n·l^{k(n−1)+1}) on (n,k,l) scenarios), and the AvN-triple characterisation of stabiliser states (Ch VI). | Read |
| NOTE-255 | The Impossibility of a Paretian Liberal | Sen proves that no social decision function can satisfy three conditions at once: unrestricted domain (U), the weak Pareto principle (P), and “minimal liberalism” (L*). A social decision function only has to yield a best element in every subset, and need not be transitive or satisfy independence. L* says at least two individuals are each decisive, both ways, over at least one pair of alternatives. I checked the two-case proof and it is correct: it builds a strict social cycle over 3 or 4 alternatives. The Lady Chatterley’s Lover example is a correct 3-alternative instance. Sen’s moral is that “in a very basic sense liberal values conflict with the Pareto principle” (p. 157). | Read |
| NOTE-271 | A topos perspective on the Kochen–Specker theorem: I. Quantum states as generalized valuations | Isham & Butterfield show that, for bounded self-adjoint operators with discrete spectrum, a global section of the spectral presheaf Σ(Â) = σ(Â) on the category O_d, whose arrows are functional relations B̂ = f(Â), is literally a valuation obeying the value rule and FUNC. A global section of the dual presheaf D(W) = Hom(W, {0,1}) on the poset W of Boolean subalgebras of projectors is literally a Kochen–Specker 0/1 assignment. The Kochen–Specker theorem (cited, not reproved) is therefore equivalent to “no global sections” when dim H > 2 (§§2.2–2.3). In place of the missing global valuations, they construct contextual truth values in the Heyting algebra of sieves: ν^ρ(A ∈ Δ) = {f: B̂ →  | tr(ρ Ê[B ∈ f(Δ)]) = 1}. For every density matrix these satisfy functional composition, null, monotonicity, exclusivity and unit conditions (§4.4, §5.3, Theorem 5.1). | Read |
| NOTE-272 | The Theory of Representations for Boolean Algebras | The paper identifies Boolean algebras with Boolean rings with unit (xx = x), and generalized Boolean algebras with Boolean rings (Thms 1–4). Theorem 67 proves that every Boolean ring A is isomorphic to the algebra B(A) of classes 𝔈(a) = {prime ideals p : a ∉ p}, under symmetric difference and intersection. Its ideal lattice is isomorphic to the system of classes 𝔈(𝔞) under arbitrary unions and finite intersections. The existence of enough prime ideals (Thm 63) is proved by transfinite induction, and Theorem 70 shows that “every Boolean ring has an isomorphic algebra of classes” and “every ideal is the product of its prime divisors” are equivalent without choice. Theorems 68–69 classify all set-representations as restrictions of B(A) to subclasses of prime ideals. The topology, i.e. Stone spaces and duality, is explicitly deferred to “another paper”. | Read |
| NOTE-274 | Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges | 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). | Skimmed |
| NOTE-275 v2 | The logic of quantum mechanics | The paper argues heuristically that the experimental propositions of a quantum system correspond to closed linear subspaces of Hilbert space. Meet is intersection, join is closed linear span, and negation is orthogonal complement. The lattice satisfies L1–L4 and complementation L71–L73 but not the distributive law L6. The paper proposes the modular law L5 as its replacement. Under a finite-dimension assumption, it concludes that the propositional calculus of QM is an abstract projective geometry. The Appendix proves that orthocomplementations on P_{n−1}(F), n ≥ 4, are exactly the polarities of definite diagonal Hermitian forms with respect to an involutory anti-automorphism of F. | Read |
| NOTE-276 | A numerical algorithm for block-diagonal decomposition of matrix *-algebras with application to semidefinite programming | Given real symmetric A₁…A_m, the method finds an orthogonal P that block-diagonalises them all as simultaneously and finely as possible. That is the decomposition of the matrix *-algebra 𝒯 they generate into simple components (Algorithm 4.1) and then irreducible components (§4.2), with no group or algebra supplied in advance. The simple components are the connected components of the graph “Q_iᵀA_pQ_{i′} ≠ O” between the eigenspaces of one random combination A(r) = Σ r_pA_p. This is correct with probability 1 when span{I, A₁…A_m} = 𝒯 ∩ Sⁿ (Props. 3.2–3.4, 4.2). The structure theorem it relies on, Theorem 3.1(C) “if 𝒯 is irreducible then 𝒯 = Mₙ”, is false over ℝ as stated. The companion paper (Maehara & Murota 2010) handles the complex and quaternion types. | Read |
| NOTE-277 v2 | A topos foundation for theories of physics: I. Formal languages for physics | The programme paper. Its contention is that a theory of a physical system S is a representation of a typed “local language” L(S) (ground types Σ and R, function symbols A : Σ → R) in a topos. Physical quantities become arrows Σ_φ → R_φ, and propositions become sub-objects of Σ_φ, which form a Heyting algebra. Classical physics is the representation in Sets. The paper proves nothing. Its one argument is that the axiom ¬(A ε Δ) ⇔ A ε ℝ∖Δ is not validated by non-Boolean Heyting representations (§3.2.1). | Read |
| NOTE-280 | On hypercomplex numbers | For associative algebras with a finite basis over an arbitrary field F, the paper proves the structure theory. A semi-simple algebra (no nilpotent invariant sub-algebra) is uniquely the direct sum of simple algebras (Thms 10, 17). Every simple algebra is the direct product of a primitive (division) algebra and a simple matric algebra, i.e. a full n×n matrix algebra (Thms 21–23). The maximal nilpotent invariant sub-algebra N contains all others (Thm 13), and A/N is semi-simple. When every non-invertible element is nilpotent, A = B + N with B primitive (Thm 28, with a proof correction added 1 Feb 1908). Section 8 lists the results as (i)–(vi). | Read |
| NOTE-281 | A closed set of normal orthogonal functions | The paper defines a closed orthonormal set {φ} on (0, 1) of functions taking only the values ±1, recursively by (2) and explicitly by (3)–(4). In binary x = Σ a_i/2^i, the functions are signed products (−1)^{a_{i₁}+…+a_{i_r}}; the nth has exactly n sign changes. It proves Fourier-like properties: uniform convergence for continuous F with the terms grouped in dyadic blocks (Thm II); pointwise convergence to ½[F(a−0)+F(a+0)] for bounded variation (Thm IV); localisation (Thm V); uniform Cesàro summability for continuous F (Thm VII); a continuous function with a divergent φ-series at any chosen point (Thm VIII); and uniqueness of expansions (Thms IX–X). It never mentions groups, characters or (ℤ/2)ⁿ. | Read |
| NOTE-282 v2 | On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups | 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. | Read |
| NOTE-284 | The theory of topological commutative groups | For countable discrete abelian 𝔊, the character group X = Hom(𝔊, K), K = ℝ/ℤ with pointwise convergence, is compact and second countable (Thm 1). Characters separate points and subgroups, with annihilator correspondence and quotient/subgroup duality (Thms 2–4). An orthogonal pair (X, 𝔊) with trivial annihilators has each the character group of the other (Thm 5). By the First Fundamental Theorem, every compact second-countable abelian group is the character group of its discrete character group; the proof uses Peter–Weyl via Haar. Hence duality between countable discrete and compact second-countable abelian groups. Chapters III–IV give structure theorems for locally compact connected abelian groups (compact ⊕ ℝⁿ). Duality for general locally compact abelian groups is not proved here. | Read |
| NOTE-285 | A topos foundation for theories of physics: IV. Categories of systems | Proposes axioms for a category Sys of physical systems, symmetric-monoidal under composite ⋄ and under disjoint sum ⊔, whose arrows are translations of local languages. Each arrow j : S₁ → S is realised by a left-exact functor τ(j) between the systems’ topoi, together with arrows φ(j) : Σ_{S₁} → τ(j)Σ_S and β(j) : τ(j)R_S → R_{S₁}. Classical physics satisfies the scheme exactly. In quantum theory the disjoint sum H₁⊕H₂ satisfies it (the pulled-back arrow of Â₁⊕Â₂ is δ̆(A₁)), but the composite H₁⊗H₂ does not: the natural pull-back of δ̆(A₁) differs from δ̆(A₁⊗1̂) off the image of V(H₁), which the authors attribute to “operator entanglement”. | Read |
| NOTE-288 v2 | Group Equivariant Convolutional Networks | 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). | Read |
| NOTE-289 | Sur une espèce de géométrie analytique des systèmes de fonctions sommables | A priority note, not a paper of proofs. Riesz says his aim was an “analytic geometry” of square-summable functions, completed by his existence theorem (every square-summable coordinate sequence is the image of a square-integrable function). He states two consequences without proof. First, every linear operation U on square-summable functions that is continuous for convergence in mean is given by U(f) = ∫ f k for some square-summable k; uniqueness of k is not stated. Second, a completeness criterion for Schmidt’s problem. | Read |
| NOTE-290 | A topos foundation for theories of physics: III. The representation of physical quantities with arrows δ̆°(A): Σ̲ → ℝ̲≽ | Each bounded self-adjoint  becomes an arrow δ̆^o(A) : Σ → ℝ^⪰, where ℝ^⪰ is the presheaf of real-valued order-reversing functions on ↓V. At λ ∈ Σ_V it sends V′ ⊆ V to λ(δ^o(Â)_{V′}), and δ^o(Â)_V is the spectral-order-least operator in V above Â. The map  ↦ δ̆^o(A) is injective, but ℝ^⪰ is only an additive monoid, not the real-number object. Its Grothendieck completion k(ℝ^⪰) is an abelian-group object whose global elements are exactly the functions of bounded variation on V(H). | Read |
| NOTE-291 | Analysis of Boolean Functions | Every f: {−1,1}ⁿ → ℝ has a unique multilinear (Fourier–Walsh) expansion f = Σ_S f̂(S)χ_S over the parities χ_S(x) = ∏{i∈S} x_i, which are an orthonormal basis (Thm 1.1, Thm 1.5) and are the characters of the group F₂ⁿ, χ_S(x+y) = χ_S(x)χ_S(y) (eq. 1.5; §3.2; §8.5). The central operators are all diagonal in this basis with eigenvalues depending only on |S|: the noise operator T_ρχ_S = ρ^{|S|}χ_S (Prop. 2.47) and the Laplacian Lχ_S = |S|χ_S (Prop. 2.37). Influence, noise stability and Arrow’s Theorem (via Stab{−1/3}) become statements about the Fourier weight at each level (Thms 2.20, 2.38, 2.49, 2.56). | Skimmed |
| NOTE-292 v2 | A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory | A projector P̂ is mapped to the clopen sub-object δ(P̂) of the spectral presheaf Σ over the poset V(H) of abelian von Neumann subalgebras, by “daseinisation” (de Groote’s V-support): δ(P̂)V = the least projector in V above P̂. The map is injective and preserves ∨, but only δ(P̂∧Q̂) ⪯ δ(P̂)∧δ(Q̂). This represents the propositional language PL(S) in the Heyting algebra Sub_cl(Σ). A pure state gives the truth object T^ψ, and “A ε Δ” gets the sieve-valued truth value {V′ ⊆ V | ⟨ψ|δ(Ê[A∈Δ]){V′}|ψ⟩ = 1}. | Read |
| NOTE-301 | The approximation of one matrix by another of lower rank | For a real n×N matrix a, the rank-r matrix β nearest in the least-squares (Frobenius) distance is obtained by writing a = u′λU (u, U orthogonal, λ diagonal ≥ 0, “Theorem I”) and zeroing all but the r largest diagonal elements. The solution is unique unless λ_r = λ_{r+1}. The squared error is Σ_{i>r} λ_i², so the error is at most l(1−r/R)^{1/2}, with l the length of a and R its rank. The paper also argues that the resulting canonic components cannot straightforwardly be read as psychological factors. | Read |
| NOTE-302 | The Linear Representation Hypothesis and the Geometry of Large Language Models | Three readings of “concepts are directions” are made precise with counterfactual pairs, and proved to line up. The shared direction of word-pair differences in the unembedding space is a logit-linear probe (Thm 2.2). The embedding-space direction is a steering vector that leaves causally separable concepts unchanged (Thm 2.5). Under any inner product making separable concepts orthogonal, the Riesz map sends the first to the second (Thm 3.2). Softmax training fixes the representation only up to γ ↦ Aγ + β, λ ↦ A⁻ᵀλ, so no inner product is identified, and the paper chooses M = Cov(γ)⁻¹ by setting a free diagonal D to the identity (Thm 3.4). Evidence: 27 concepts on LLaMA-2 7B (26 show a common direction; thing⇒part does not), read from histograms and heatmaps with no test statistic, plus a Gemma-2B heatmap comparison. | Read |
| NOTE-303 | On the imbedding of normed rings into the ring of operators in Hilbert space | Theorem 1 is the first abstract characterisation of C*-algebras. Every normed -ring (a unital Banach -algebra with ‖xx‖ = ‖x‖·‖x‖, ‖x*‖ = ‖x‖, and e + x*x invertible) is isometrically *-isomorphic to a norm-closed -subalgebra of the bounded operators on some, not necessarily separable, Hilbert space. The proof builds, for each maximal left ideal M, a positive functional f vanishing on M. The inner product is (ξ, η) = f(yx) on R/M, and the construction takes the direct sum over all M. This is the GNS construction. Lemma 1 is the commutative case: such an algebra is isometrically *-isomorphic to C(𝔐) for its compact maximal-ideal space. §4 adds that the simple weakly closed rings on separable Hilbert space are exactly the factors of types I_n, II₁ and III (Theorem 3). | Read |
| NOTE-305 | Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe | For a “closed continuous group”, a compact Lie group with invariant volume, the Fundamentalsatz (p. 752) proves the Parseval equality over all inequivalent irreducible representations E = (e_ik) of order n. It is n Σ_{i,k} |α_ik|² + … = V ∫ |x(s)|² ds for every continuous x, with α_ik = ∫ x ē_ik ds. For continuous class functions it reads |α|² + … = V ∫ |x|² ds with α = ∫ x χ̄ ds. The proof applies Hilbert–Schmidt theory to the convolution kernel z(st⁻¹), z = x x̃. The consequences are that finite combinations of matrix coefficients uniformly approximate every continuous function (Approximationssatz), that representations separate points (Satz I), and that characters separate conjugacy classes (Satz II). | Read |
| NOTE-307 | Learning the Irreducible Representations of Commutative Lie Groups | Proposes “Weyl’s principle” (via Kanatani 1990) as a definition of disentangling: the elementary components of data are the irreducible subspaces of the representation of a symmetry group acting on it. It learns such a decomposition for one class of group fixed in advance, compact commutative subgroups of SO(D) (“toroidal”), from pairs (x, y) of raw data vectors related by an unobserved group element, y = W R(φ) Wᵀx + ε. The orthogonal basis W (2-D invariant subspaces) is fitted by SGD on a closed-form marginal likelihood, and the integer weights ω_j are then read off a batch rotated by a known 0.1°. The only experiment trains 100 filters on 250,000 white-noise 16×16 patches paired with random rotations of themselves (learned ω from −11 to 12), and uses the invariant √κ̂ for 1-NN on rotated MNIST, reported in a figure with no numbers. Spectral degeneracy plays no role. | Read |
| NOTE-311 | Applications of the theory of Boolean rings to general topology | Stone topologises the set 𝔈 of prime ideals of a Boolean ring by taking the sets 𝔈(a) as a basis. 𝔈 is then a totally disconnected, locally bicompact Hausdorff space, bicompact exactly when the ring has a unit (Thm 1), and every such space arises this way (Thm 2). Theorem 4 states that “the algebraic theory of Boolean rings is mathematically equivalent to the topological theory of Boolean spaces”: isomorphism ↔ homeomorphism, ideals ↔ open sets, quotients ↔ closed subsets, unit ↔ bicompactness, and subrings ↔ continuous images (Thm 7). The same paper also contains what are now called the Stone–Čech compactification (Thms 78–79, 88), the Stone–Weierstrass theorem (Thm 82), the Banach–Stone theorem (Thm 83) and closed ideals of C(X) ↔ closed sets (Thm 85). | Read |
| NOTE-312 | Studies on the Foundation of Quantum Mechanics. I | Husimi derives Birkhoff and von Neumann’s lattice axioms from statistical premises. Implication is P̄ ≤ Q̄ in every state (and requires compatibility), negation is Q̄ = 1 − P̄, and compatible propositions obey classical logic. Meet is obtained from “Ehrenfest’s principle” (additivity of mean values) via the top eigenvalue of aP + bQ (pp. 778–779), and the modular law, equivalently the parallelogram law |P| + |Q| − |P∩Q| = |P∪Q|, is proved under a finite-chain assumption. The orthomodular identity is used but never named or stated as an axiom. It appears as the step “since we admit traditional logic to any classical part … S + P∩Q = R” for R ⊃ P∩Q (p. 784), and as the case “(Q − P) + P = Q provided Q ⊃ P” of a classical identity (p. 780). | Read |
| NOTE-313 | An Introduction to Hyperplane Arrangements | The notes define the regions of a real arrangement (connected components of the complement) and the intersection poset L(A) with its Möbius function and characteristic polynomial χ_A(t) = Σ μ(x) t^{dim x}. They prove Zaslavsky’s theorem (Thm 2.5): r(A) = (−1)ⁿ χ_A(−1) and b(A) = (−1)^{rank A} χ_A(1), so the numbers of regions and bounded regions depend only on L(A), not on the face structure (Cor. 2.1, Fig. 2). The same machinery counts faces of every dimension (Thm 2.6: f_k = Σ_{x≤y, dim x=k} |μ(x,y)|). Later lectures add Whitney’s theorem, deletion–restriction, matroids and geometric lattices, broken circuits, supersolvability, the finite-field method, and the metric d(R, R′) = number of separating hyperplanes, whose zero set shows that a region is determined by its separating set (p. 89). | Read |
| NOTE-315 | The Theory of Signal Detectability. Part I: The General Theory; Part II: Applications with Gaussian Noise | Peterson and Birdsall show that the criterion approach (maximise P_SN(A) − β·P_N(A), or maximise detection at fixed false-alarm k) and the Woodward–Davies a-posteriori approach are all solved by one receiver that computes the likelihood ratio ℓ(x) = f_SN(x)/f_N(x) and thresholds it (Part I, Theorems 1–7). They introduce the receiver operating characteristic, whose slope at each point is the threshold β (Theorem 8, Eq. 2.51). For a signal known exactly in white Gaussian noise, ln ℓ is normal with equal variance 2E/N₀ under both hypotheses and means 2E/N₀ apart, so the ROC is a one-parameter family indexed by the “detection index” d = (M_SN − M_N)²/σ² = 2E/N₀ (Part II, Eqs. 4.1–4.8, Fig. 4.1). The receiver is a correlator or matched filter (Eq. 5.3, 4.10). | Skimmed |
| NOTE-321 v3 | The two dragons of cognition: recursive condensation for predictive processing | A Hypothesis-and-Theory essay, the cognitive statement of Li’s structural-learning programme. Intelligence pays once for exploratory search, condenses the result into stable discrete tokens, and then navigates cheaply. The cortex and the sleep rhythms are proposed as this cycle’s implementation (gamma as the expand/contract clock, the laminar circuit, STDP, SWRs nested in Slow-4). Charitably read, Savitch stands for memoisation and Urysohn for “collapsed, separated classes admit a simple readout”. On that reading the core is a coherent restatement of amortised inference and complementary learning systems, with untested and largely non-discriminating predictions. The headline scaling law, linear cortex giving exponential reach (Theorem 2), is stated without argument, and the claim that collapse defeats the curse of dimensionality leaves the cost of learning the collapse unaddressed. | Read |
| NOTE-323 | Local Urysohn Width: A Topological Complexity Measure for Classification | >- It defines local Urysohn width uw_{D₀}(P, γ): the least number of connected sets of diameter ≤ D₀, each carrying a continuous classifier correct on the margin-γ safe region it meets, needed to cover that safe region. It proves: - uw = w on a bouquet of w circles with safe balls at the antipodes, when 3γ/2 ≤ D₀ < L/2 − 3γ/4 (Thm 4.1); - uw ≥ w·m, with m = Θ(L/D₀) safe balls per loop, so uw = Ω(w·L/D₀) (Cor 5.1); - two-way non-determination with VC dimension (Thm 6.1); - an Ω(w log w) sample lower bound via label-permuted coupon collection (Thm 7.1); - uw ≥ 2β₁/Δ₀ for good, convex, bounded-adjacency covers (Thm 8.5). The proofs are correct. The obstruction is metric, as the paper concedes in Remark 8.11: patches cannot reach across the wedge point. Because Urysohn’s lemma makes local correctness automatic on any patch, the measure is a connected covering number of the safe region that does not depend on the labels. | Read |
| NOTE-324 | Structural Learning Theory: A Metric-Topology Factorization Approach | >- The formal core of Xin Li’s structural-learning programme. Width w(P; γ, δ) is the least number of open cells on each of which some predictor is γ-contractive with conditional risk ≤ δ. The paper proves that width is incomparable with the per-cell VC dimension (Thm 2.2). It shows that with K < w cells there is an n-independent error floor η(w, K), under a non-degeneracy assumption stated in the body (Thm 2.4, Assumption 2.3). Width is at least c·β₁·L/D₀ under a bounded-overlap assumption (Thm 2.5), and discovering all basins costs Ω(w log w) samples (Thm 2.6). The CS-Laplacian eigenvalue count converges uniformly to the predictor-relative count w_G(P) when n ≳ (H_G + d_X log(1/r_x) + log(1/δ′))/(r_x^{d_X} g_eff²) (Thm 3.5); no result shows that w_G equals width. There are no experiments. | Read |
| NOTE-326 | ’What is a Thing?’: Topos Theory in the Foundations of Physics | The Döring–Isham programme in one volume. A theory of a system is a representation of a typed language L(S) in a topos. In quantum theory, presheaves over the abelian von Neumann subalgebras V(H), daseinisation sends projectors injectively to clopen sub-objects of the spectral presheaf Σ, preserving ∨ but not ∧. A pure state gives sieve-valued truth values. Every bounded self-adjoint  becomes an arrow Σ → ℝ^↔ whose value at a point is a pair of monotone functions, an interval that widens as contexts coarsen. Most of it reprints papers I–IV. New are the inverse-image corollary linking PL(S) and L(S), values in a pseudo-state, and the algebra of the quantity-value candidates. The Heyting structure on Sub_cl(Σ) is still only sketched. | Read |
| NOTE-327 | A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category | Hamilton, Isham & Butterfield move the series’ base category from operators (O) and Boolean subalgebras (W) to the poset V of commutative von Neumann subalgebras of B(H). There they restate Kochen–Specker as “the spectral presheaf Σ(V) = Gelfand spectrum of V has no global elements” (dim H > 2, cited, §2.2). They identify the coarse-graining presheaf G(V) = L(V), G(i)(P̂) = inf{Q̂ ∈ L(V₂) | P̂ ≤ Q̂} (Def. 3.1), with the presheaf CloΣ of clopen subsets of the spectra; the isomorphism is asserted, and part IV proves it. A state ρ yields the sieve-valued valuation ν^ρ_V(P̂) = {V′ ⊆ V | ρ(G(P̂)) = 1} (Def. 3.4) and the support projectors inf{P̂ ∈ L(V) | ρ(P̂) = 1}. These form a global element of G and hence an “interval valuation”, a subobject of Σ (§§4.2–4.4). The probability-r valuations give subobjects of G but not global elements. | Read |
| NOTE-328 | A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations | Butterfield & Isham show, over the poset V of commutative von Neumann subalgebras, that sieve-valued valuations and “interval valuations” (assignments of support projectors, or of subsets of Gelfand spectra) determine each other under stated conditions. If supports s(α, V) = inf{P̂ | α_V(P̂) = true_V} form a global element of the coarse-graining presheaf G, then α_{V₁}(P̂) = {V₂ ⊆ V₁ | G(s(α, V₁)) ≤ G(P̂)} is a sieve-valued valuation obeying FUNC (Theorem 3.1). There is an analogue for “tight” subobjects of Σ (Theorem 3.2). Both are satisfied by every state’s valuation ν^ρ. The ordering ≤ (or ⊆) is shown to be the natural relation securing sievehood, null, monotonicity, exclusivity and unit (§4). The review §2.4 also supplies the proof, via state extension, that G ≅ CloΣ: restricting the clopen set of P̂ to V₂ gives the clopen set of its coarse-graining. Part III had only asserted this. | Read |
| NOTE-329 | A Topos Perspective on the Kochen-Specker Theorem: II. Conceptual Aspects, and Classical Analogues | Butterfield & Isham argue that the sieve-valued “generalised valuations” of part I are not a quantum peculiarity but the natural valuations for any presheaf of propositions on any small category. Three supports are offered. (1) Classical physics: the value presheaf Υ(Ā) = Ā(S) on the category M of measurable functions has global sections (one per microstate, eq. 3.7), yet a macrostate R ⊆ S yields a sieve-valued valuation ν^R(A ∈ Δ) = {f_M | B̄(R) ⊆ f(Δ)} (eq. 3.10) with the same FUNC property as the quantum ν^ρ. (2) A proof that a sieve-valued valuation on any presheaf G obeys generalised FUNC, ν(B, G(f)(d)) = f*(ν(A, d)), iff it is a natural transformation G → Ω, i.e. a subobject of G (Theorem 4.1). (3) An informal argument, explicitly “not a genuine deduction” (§5.2), that if partial truth is fixed by which weakenings of a proposition are totally true, the truth value is a sieve. | Read |
| NOTE-333 | Topos Theory and Consistent Histories: The Internal Logic of the Set of All Consistent Sets | Isham takes the poset B of all Boolean subalgebras (“windows”) of the history orthoalgebra UP, ordered by coarse-graining, and shows that the presheaf A_d(W) = {α | some d-consistent W′ ⊇ W contains α} is a subobject of the constant presheaf ΔUP, whereas the naive “realisable in W” assignment is not a presheaf at all (§4.1–4.2). The probability claim ⟨α, p⟩ then gets the sieve-valued semantic value {W ⊆ W₀ | ∃ d-consistent W′ ⊇ W with α ∈ W′} if d(α, α) = p and ∅ otherwise (4.16), a global element 1 → Ω in Set^B, so the logic of “all consistent sets at once” is the distributive but non-Boolean Heyting algebra of sieves on B. | Read |
| NOTE-334 | Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity | Isham & Butterfield argue, without proving anything new, that presheaf toposes over a base of “contexts” suit physics because the subobject classifier Ω is fixed by the base. They say this answers the objection that many-valued logics are arbitrary (§2.2), and that it keeps the logic distributive (§1.1). They then sketch seven applications. Five are speculative one-paragraph proposals (frames, observers, causal sets, TQFT, synthetic-differential-geometry time in continuous histories). One (§3.7) recaps their Kochen–Specker results: KS ⇔ the dual presheaf D(Â) = Hom(W_A, {0,1}) on the operator category O has no global sections (dim H > 2), and every state ψ gives a FUNC-obeying sieve-valued valuation ν^ψ(A ∈ Δ) = {f_O : Â → B̂ | Ê[B ∈ f(Δ)]ψ = ψ}. | Read |
| NOTE-401 | The Genetical Evolution of Social Behaviour. I | A one-locus, non-overlapping-generation model in which genotypes affect relatives’ fitness. Splitting each effect into a part on genes identical by descent (weight r) and a “diluting” part on other genes shows that the former alone fixes the direction of selection, and defines inclusive fitness. Mean inclusive fitness provably increases when no genotype harms others; with harm it only points uphill. A costly act spreads iff −k > 1/r: twice the cost to a sib, eight times to a cousin. | Read |
| NOTE-404 | The Logic of Animal Conflict | Five strategies for contests with dangerous weapons, simulated 2,000 times per pairing with payoffs +60 win, −100 serious injury, −2 per scratch, up to +20 for brevity. Hawk is not an ESS; Retaliator is called one though Mouse ties it, so by the paper’s own second condition it is only neutrally stable. In a war of attrition no pure strategy is stable; the exponential mixture p(x) = (1/v)e^(−x/v) is. Individual selection can explain limited war. No Hawk–Dove game, no ownership. | Read |
| NOTE-409 | Five Rules for the Evolution of Cooperation | A four-page review. Five mechanisms, each with a b/c threshold derived from a 2×2 game: kin selection r > c/b, direct reciprocity w > c/b, indirect reciprocity q > c/b, network reciprocity b/c > k, group selection b/c > 1 + n/m. For kin, network and group mechanisms the same condition makes cooperators ESS, risk-dominant and advantageous; for the reciprocities RD and AD need more. Derivations are in the unread supplement; the review itself proves nothing. | Read |
| NOTE-411 | The Evolution of Cooperation | TIT FOR TAT won Axelrod’s two tournaments (14 entries plus random, 200 moves; 62 entries, expected 200 moves) and went to fixation in an ecological replay. Against it, only ALL D or alternation need be checked, so no strategy does better iff w ≥ (T−R)/(T−P) and w ≥ (T−R)/(R−S). ALL D is stable for all w; kinship or a cluster with enough in-cluster interactions lets reciprocity in, and a nice stable strategy resists clusters. ALL C ties with TIT FOR TAT, so stability is neutral, not strict. | Read |
| NOTE-418 | Pattern formation outside of equilibrium | Read for the record’s question only. Patterns away from equilibrium are classified by the linear instability of a uniform state (type I_s, III_o, I_o) and described near onset by universal amplitude equations. There is in general no free energy or Lyapunov functional selecting them. Bénard’s own 1900 hexagons were surface-tension-driven, not Rayleigh’s buoyancy convection. Closed chemical systems must reach equilibrium, so their patterns are transients or need pumping. No global minimization principle exists beyond a perturbative one near threshold. | Skimmed |
| NOTE-427 | The Chemical Basis of Morphogenesis | Turing’s reaction–diffusion theory of pattern. Linearize about a homogeneous steady state of M morphogens on a ring of N cells: each Fourier mode grows or decays independently, and the fastest-growing one sets the pattern. Six forms of onset are possible. For two morphogens, stationary waves of finite wave-length need bc < 0 and 4√(μ′ν′)/(μ′+ν′) < (d−a)/√(−bc) < (μ′+ν′)/√(μ′ν′) (9.4a), with instability if d√(μ′/ν′)/√(−bc) − a√(ν′/μ′)/√(−bc) > 2 (9.4b). These cannot both hold when μ′ = ν′. A computed 20-cell ring gives three lobes. | Read |
| NOTE-473 | A Widely Applicable Bayesian Information Criterion | The Bayes free energy of a singular model is nL_n(w0) + λ log n − (m − 1) log log n + O_p(1), with λ the RLCT. The posterior mean of nL_n(w) at inverse temperature 1/log n (WBIC) reproduces the first two terms without knowing λ, equals BIC for regular models, and two temperatures estimate λ. In reduced-rank regression WBIC chose the true rank 100/100 times, and estimated RLCTs within about 0.5 of theory. | Read |
| NOTE-475 | Deep Equals Shallow for ReLU Networks in Kernel Regimes | For κ(xᵀx′) on S^(d−1), µ_k = (ω_(d−2)/ω_(d−1)) ∫ κ(t)P_k(t)(1 − t²)^((d−3)/2) dt (Eq. 8), shared by N(d, k) ~ k^(d−2) harmonics. Theorem 1: an endpoint expansion κ(1 − t) = p(t) + c t^ν + o(t^ν) with ν non-integer gives µ_k ~ C k^(−d−2ν+1), parity-dependent. ReLU kernels have ν = 1/2 (NTK) and 3/2 (random features) at any depth, so the deep NTK decays as k^(−d) and the deep RF kernel as k^(−d−2) (Corollaries 2–3). MNIST and Fashion-MNIST accuracy is flat in depth from L = 2 to 5 (Table 1). | Read |
| NOTE-518 | Exact expressions for double descent and implicit regularization via surrogate random design | Surrogate design S^n_μ = Det(μ, K), density ∝ pdet(XXᵀ), K Poisson(γn) truncated at d. E[#rows] = n (Lemma 1); E[I − X̄⁺X̄] = (γnΣ + I)⁻¹ (Lemma 2); E[tr((X̄ᵀX̄)⁺)] = γn(1 − det((I/γn + Σ)⁻¹Σ)) (Lemma 3); λn = 1/γn. These give exact MSE in all regimes (Theorem 1) and E[X̄⁺ȳ] = (Σ + λnI)⁻¹v for n < d, with n = tr(Σ(Σ + λnI)⁻¹) (Theorem 2). For sub-Gaussian rows the i.i.d.-design MSE converges to the surrogate formula (Theorem 3), at an observed O(1/d) rate. | Read |
| NOTE-519 | A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent | Q̄ = ((N/n)(Kcos/(1+δcos) + Ksin/(1+δsin)) + λI)⁻¹ with δσ = (1/n) tr KσQ̄ is a deterministic equivalent for E_W[(ΣXᵀΣX/n + λI)⁻¹] (Theorem 1); Ē_train and Ē_test follow (Theorems 2–3). As λ → 0, δ ~ 1/λ for 2N < n and O(1) for 2N > n; det(Ω⁻¹) ~ λ, so Ē_test → ∞ at 2N = n unless the test set is within σ² ≲ λ of the training set. N/n → ∞ recovers Gaussian kernel regression. Theory matches MNIST, Fashion- and Kannada-MNIST. | Read |
| NOTE-528 | Understanding Mode Connectivity via Parameter Space Symmetry | Minimum of full-rank ‖Y − W_l…W₁X‖² ≅ GL_h(ℝ)^{l−1}: 2^{l−1} components, independent of width. 1-D three-layer: 4 components, 3 with a skip. h ≥ 2: permutations connect all components. Homogeneous last layers: rescaled minima in one component with unbounded midpoint loss, even under last-layer permutations (Props. 5.3–5.4). Symmetry curves γ(t) = exp(t log g)·w have constant loss; curvature ≤ κ bounds the chord’s distance from the level set by (1/κ)(1 − √(1 − (κ‖Δw‖/2)²)) ≈ κ‖Δw‖²/8. | Read |
| NOTE-543 | Proving Linear Mode Connectivity of Neural Networks via Optimal Transport | Layerwise neuron alignment equals a Wasserstein distance between empirical weight distributions (Birkhoff), so LMC modulo permutation follows when neuron weights are i.i.d. and layers are wide: for mean-field two-layer SGD (Theorem 3.1), and for deep Gaussian or sub-Gaussian nets with m̃_ℓ = Õ((T_ℓ/ε)^{m̃_{ℓ−1}}) (Theorem 5.2), tight by Theorem 5.3. Low-dimensional weights relax this (Theorem 5.4). | Read |
| NOTE-576 | The Method of Characteristics Revisited: A Viability Approach | Viability is a property of a constraint set under a dynamics: K is viable if from every state some evolution stays in K, invariant if all do. Nagumo’s theorem reduces viability to a tangency condition. When K is not viable, the viability kernel is the largest viable part of it, and everything else in K leaves in finite time. Stability in the Lyapunov sense is not discussed. | Read |
| NOTE-588 | Fritz — A synthetic approach to Markov kernels | Markov categories (symmetric monoidal, a commutative copy/discard comonoid on every object, discard natural) carry enough structure to define determinism, conditionals, four kinds of conditional independence with the semigraphoid rules, almost-sure equality, sufficiency, completeness and ancillarity. Basu’s theorem then holds in every Markov category; Fisher–Neyman (Theorem 14.5) and Bahadur (Theorem 16.3) need strict positivity, which existence of conditionals implies but which Stoch has without them. The abstract Fisher–Neyman is checked against the classical one only in FinStoch. | Read |
| NOTE-589 | Conditional Simulation Using Diffusion Schrödinger Bridges | Posterior sampling p(x|y) when only joint samples (X, Y) can be drawn, recast as a Schrödinger bridge on the extended space between the joint p(x, y) and a reference p_ref(x|y)p(y) in which y is held fixed (Proposition 1), and solved by iterative proportional fitting with learned forward and backward drifts (CDSB). Its first iteration is the ordinary conditional score-based model, so the rest refine it; the reference can be an informed guess (upsampled image, EnKF, SRFlow) rather than noise. Better than the conditional score model at equal, small numbers of steps on most but not all image metrics, and it filters Lorenz-63 with 20 steps where the score model diverges. | Read |
| NOTE-595 | Equivalent Comparisons of Experiments | Gives a new minimax proof of the Sherman–Stein theorem (for finite outcome spaces, an experiment at least as informative as another in every decision problem can reproduce it by a Markov matrix) and, by a martingale argument, removes the finiteness restriction, so for a finite parameter set “more informative” and “sufficient for” coincide in general. Adds k-decision problems, with three equivalent forms, and shows that for dichotomies two-action problems, and so the type I/type II error trade-off, already decide the order. | Read |
| NOTE-598 | Baez & Fong — A Noether theorem for Markov processes | For a Markov semigroup exp(tH) on a finite set, an observable O commutes with H iff the expectations of O and O² are both constant for every initial distribution, iff the expectations of every polynomial in O are, iff O is constant on each connected component of H’s transition graph. Constancy of the mean alone is not enough; a 3-state example shows it. On a σ-finite measure space, [O, exp(tH)] = 0 for all t iff the means of O and O² are constant, by way of a one-step Markov-chain version proved with Chebyshev’s inequality. | Read |
| NOTE-600 | Contextuality-by-Default theory | A system is a set of stochastically unrelated bunches (one per context) linked by connections (one per content). It is noncontextual if some coupling of the bunches makes every connection a maximal coupling, which separates context-dependent marginals (“direct influences”) from contextuality. The test is linear-programming feasibility (Theorem 4.1). Signed quasi-couplings always exist (Theorem 6.1), and the minimal total variation minus 1 measures the degree of contextuality. | Read |
| NOTE-601 | Seven Sketches in Compositionality | From Chapter 1 and Section 3.4: an observation is a monotone map, and a generative effect is its failure to preserve joins, Φ(a) ∨ Φ(b) < Φ(a ∨ b); left adjoints preserve joins, and out of a preorder with all joins a monotone map preserves joins iff it is a left adjoint. Database schemas are categories, instances functors to Set, and a schema map induces migrations Σ ⊣ Δ ⊣ Π. The other five chapters were seen only at their openings and summaries. | Skimmed |
| NOTE-612 | The Topology of Causality | Builds the sheaf-theoretic framework for causality over spaces of input histories: causal functions are free assignments of outputs to tip events, causality is continuity in the lowerset topology, and the presheaf of causal functions is a sheaf exactly when the space has no “solipsistic contextuality” (always when tight). Empirical models are compatible families on any open cover; standard empirical models on causal switch spaces, including total orders, are always local. | Read |
| NOTE-618 | Structuralism | From a one-page-in-four preview of chs. I–IV: a structure is defined by wholeness, transformation and self-regulation. Its elements are subordinated to laws of composition; “all known structures… are, without exception, systems of transformation”; and in logic and mathematics regulation is perfect because every operation has an inverse. Structures whose transformations unfold in time, linguistic or social, are regulated but not fully reversible. Structures are real only if governed from within, by a subject defined as “the center of functional activity”. | Skimmed |
| NOTE-623 | Probabilistic Foundations of Contextuality | In Kolmogorovian probability, joint distributedness is transitive, so treating a measurement as the same random variable in overlapping contexts already forces an overall joint distribution. A Kochen–Specker or Bell contradiction therefore refutes that identification, not the joint distribution. Contextuality is restated as the impossibility of a coupling of context-indexed variables whose same-content copies satisfy a property C. For inconsistently connected binary systems, C is multimaximality, characterized by adjacent pairs (Theorem II.3). | Read |
| NOTE-624 v2 | The Geometry of Causality | Empirical models on any cover of a space of input histories are exactly the points of a causaltope, a product of simplices sliced by linear causality equations, which makes supported and causally separable fractions linear programs. Defines causal separability relative to an ambient space and shows, with entangled and contextually controlled quantum switches, that the causally separable fraction is bounded below by the separable local fraction and can equal it. | Skimmed |
| NOTE-625 | Backprop as Functor | Learners (P, I, U, r), with a request function r : P × A × B → A, compose and form a symmetric monoidal category Learn. For step size ε and an error e with ∂e/∂x(x₀, −) invertible, sending a differentiable parametrised function to (P, I, p − ε∇ₚE, f_a(∇ₐE)) is a faithful strong symmetric monoidal functor Para → Learn: gradient descent on a composite is the composite of local updates exchanging requests, which is backpropagation, and the proof is the chain rule. | Read |
| NOTE-629 | Sheaf-theoretic structure of definite causality | Extends the Abramsky–Brandenburger sheaf framework from non-locality (discrete order) to any finite causal order of events. Sections are causal functions over families of inputs on lower sets. Empirical models are causal conditional distributions forming a polytope. Locality is a global section, equivalently a mixture of deterministic causal functions. The authors withdrew the paper in 2024 because its “locale of inputs” is not fit for purpose. This reading finds that Proposition 5’s meet can leave the poset and that distributivity fails. | Read |
| NOTE-630 | Getting to the Bottom of Noether’s Theorem | Noether’s theorem in Hamiltonian form, “a generates symmetries of b iff b generates symmetries of a”, follows in Poisson algebras, complex *-algebras and Banach–Lie algebras from the antisymmetry of the bracket plus uniqueness of solutions; for a bilinear bracket antisymmetry is self-conservation, {a, a} = 0. What carries content is the map from observables (a Jordan algebra) to generators (a Lie algebra): i in complex quantum mechanics, none in the real or quaternionic case, and Alfsen–Shultz’s dynamical correspondence for JB-algebras. | Read |
| NOTE-635 | The Combinatorics of Causality | Replaces causal orders by spaces of input histories (join-prime sets of partial input assignments), which can express causal constraints that depend on inputs. Defines free choice, tip events, causal completeness and tightness, three kinds of composition, and a lattice of spaces; proves that the maximal causally complete spaces are exactly the causal switch spaces; and finds 2644 complete spaces on 3 binary-input events, against 19 from definite orders. | Read |
| NOTE-640 | Contextuality in canonical systems | Contextuality is to be judged on a canonical representation in which every variable is replaced by binary splits and connections are coupled multimaximally. For two content-sharing k-valued variables with all splits kept, the canonical system is noncontextual if and only if one nominally dominates the other: its probabilities fall below the other’s for at most one value (Theorem 4.6). Only the 1- and 2-splits matter (Theorems 4.1, 4.3). For continuous densities, all splits make any difference in distribution contextual. | Read |
| NOTE-643 | Contextuality-by-Default 2.0 | Replaces CbD 1.0’s maximal coupling of each connection by a multimaximal one (every subset maximally coupled). For binary variables this coupling exists, is unique and has an explicit staircase form (Theorem 1), so a noncontextual system’s subsystems are noncontextual (Theorem 3) and partial and complete (non)contextuality coincide. Cyclic, consistently connected and two-copy systems are judged as before. For non-binary variables multimaximal couplings can fail to exist, can be non-unique, and are not stable under coarse-graining (Examples 1–3). | Read |
| NOTE-644 | Semantic Unification | Puts basic DRT into sheaf form: a DRS is a section of a presheaf of consistent literals, an anaphoric resolution is a cover by variable maps, and the discourse’s meaning is the gluing, unique if it exists (Proposition 1). All the interpretive work is in choosing the cover; the only obstruction to gluing shown is inconsistency, so the paper models context dependence and ambiguity, not contextuality in the Abramsky–Brandenburger sense. Its probabilistic section ranks covers by summed corpus counts, not by gluing distributions. | Read |
| NOTE-646 | Combining contextuality and causality | Extends sheaf-theoretic contextuality to causal scenarios by an enabling relation on measurements and a game reading: Nature’s deterministic strategies, which may depend on causal history, replace the event sheaf, and the Abramsky–Brandenburger definitions then go through. Recovers flat contextuality and causal Bell scenarios, and expresses adaptive MBQC by Experimenter strategies. The strategy presheaf is not a sheaf: compatible deterministic local strategies can have no gluing or several. | Read |
| NOTE-647 | A comonadic view of simulation and quantum resources | Makes simulation between empirical models adaptive: measurement protocols form a comonoidal comonad MP, and a simulation of e by d is a deterministic map MP(d ⊗ c) → e with c non-contextual. Proves that such simulations are exactly the conversions by terms in the free operations of the resource theory of contextuality, which now include conditional measurement, with an equational theory and normal forms; the non-contextual fraction is monotone and contextual models cannot be cloned. | Read |
| NOTE-650 | Closing Bell | Answers which maps between the empirical models of two scenarios are induced by classical, non-adaptive procedures: exactly those given by a non-contextual model of a hom scenario [S, T] whose outcomes are procedures, so deciding it is a linear program. A family of compatible local procedures glues into one global procedure exactly when that model is non-contextual; the rest are “contextual simulations”. The hom construction makes scenarios with predicates a closed category. | Read |
| NOTE-651 | Categories of Empirical Models | Gives empirical models on different measurement scenarios a category: a simulation answers each target measurement with a jointly measurable set of source measurements and a stochastic outcome map natural in the context. Non-contextuality becomes simulability from nothing, the non-contextual fraction becomes a functor (contextuality cannot grow along a simulation), strong contextuality is reflected, Graham reductions are simulations, and contextual models cannot be cloned. | Read |
| NOTE-659 | Small Singular Values Matter | Uses the Marchenko–Pastur law as a null model for trained transformer weights and finds departures at both ends of the spectrum, the lower end only in rectangular matrices, whose null has a positive lower edge. In the MLP up and gate projections the below-bulk directions align with the input activations’ principal directions, and in rectangular matrices the smallest decile is never the least important to remove. A Gaussian teacher–student ensemble shows one way it can happen: the loss suppresses noise along a learned direction, a negative spike that holds a weak signal below the bulk. Overlap and ablation importance disagree for several matrix types, so “departs from the null” and “carries information” are not shown to coincide in general. | Read |
| NOTE-661 | Single-Head Attention in High Dimensions | In a solvable model, one tied softmax attention layer trained by ridge-penalized ERM on Gaussian sequences with n ∝ d² samples, derives (non-rigorously, by AMP and random-matrix theory) the test error, interpolation and recovery thresholds, and the spectrum of the learned query–key map, η·ReLU(S0 + δZ − ε): outliers are recovered target directions, the bulk is finite-sample noise, and weight decay acts as a nuclear norm. Power-law targets are recovered mode by mode, giving heavy-tailed spectra and LASSO-rate scaling laws. | Read |
| NOTE-663 | Topological Invariance and Breakdown in Learning | Proves that a permutation-equivariant update with a K-Lipschitz update map changes the distance between any two neurons in one step by a factor in [1 − ηK, 1 + ηK], so coincident neurons never split, and for ηK < 1 the induced map on the set of neuron vectors is a bi-Lipschitz homeomorphism (a C¹ diffeomorphism under smoothness), and so no two neurons merge in finitely many steps. The proofs hold. The two-phase “topological simplification” reading and the Betti-number experiments go beyond them. The measured changes are fragmentation, not merging, at ηλ_max far below 1. | Read |
| NOTE-664 | Symmetry in Language Statistics and Representation Geometry | Proves that 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, with amplitudes from the kernel’s Fourier transform, and solves the exponential kernel exactly for periodic and open boundaries. The symmetry is measured in Wikipedia, the geometry seen in Gemma 2 by PCA, and its robustness to deleting the words’ mutual co-occurrences explained by a latent variable shared by many words. | Read |
| NOTE-668 | Hippocampal spatial representations exhibit a hyperbolic geometry | In rat dorsal CA1, the rank order of pairwise spike correlations matches points drawn uniformly from a 3D hyperbolic ball, not from a low-dimensional Euclidean cube, and the ball’s radius grows with the logarithm of exploration time. Place-field sizes are near exponential, which a Poisson model shows is more informative than uniform or log-normal sizes. The evidence separates “hyperbolic ball, uniform” from “Euclidean cube, uniform”; it does not separate a hyperbolic metric from an exponential distribution of field sizes, and the hierarchy it describes is nesting of field scales, not a recovered tree. | Read |
| NOTE-670 | An argument for hyperbolic geometry in neural circuits | A perspective, not a result: hyperbolic geometry should suit neural circuits because it approximates trees, makes networks navigable, and shares with Zipf’s law an exponential growth of states. The Zipf link is an analogy that any power-law rank distribution, and any system with extensive entropy, would satisfy; the case for three dimensions leans on Mostow rigidity, which does not bear on noisy embeddings; and the only data shown are the olfactory embedding of Zhou, Smith & Sharpee (2018). | Read |
| NOTE-671 | Hyperbolic Entailment Cones for Learning Hierarchical Embeddings | Nested, axially symmetric, rotation-invariant cones in the Poincaré ball define a partial order, but nesting forces apertures of at most π/2 and r sin ψ(r)/(1 − r²) non-increasing. So no cone exists near the origin, and the widest family is sin ψ = K(1 − r²)/r, with a closed-form membership angle. The transitivity of that family is asserted, not proved. Learned on WordNet noun hypernymy, the cones predict held-out closure edges better than Poincaré and order embeddings once some closure edges are in training. Trained on the transitive reduction alone, every method fails. | Read |
| NOTE-672 | Hyperbolic geometry of the olfactory space | In four natural odour sources, rank-ordered concentration correlations match points near the surface of a 3D hyperbolic ball and not uniform Euclidean cubes; human odour-descriptor profiles match a full 3D hyperbolic ball. The evidence separates “hyperbolic shell” from “uniform Euclidean cube” only. On a thin shell, hyperbolic distance is nearly a monotone function of angle, so a sphere in flat space, which was not tested, would be a close rival, and the hierarchy that motivates the model is not recovered from the data. | Read |
| NOTE-675 | Hyperbolic Diffusion Embedding and Distance | Embeds each point’s diffusion densities at dyadic time scales in a product of Poincaré half-spaces and proves that the ℓ1 product distance, a scale-weighted Hellinger sum, is equivalent up to constants to a thresholded snowflake min{1, d_T^(2α)} of a latent tree metric when the diffusion operator behaves like the heat kernel on that tree. The tree heat-kernel bounds are cited rather than proved and one Hölder step holds only for d_T ≥ 1. Empirically it keeps tree neighbourhoods (best MAP) and not global distances (more distortion than hyperbolic MDS). | Read |
| NOTE-678 | Emergent Hyperbolic Network Geometry | Shows that simplicial complexes grown by gluing d-simplices to (d − 1)-faces, by a rule with no geometry in it, have exponential or power-law degrees (γ = 2 + 1/(d + s − 1)) by a master equation, high clustering and modularity, and a tree as their dual; and draws them, with all links of equal length, as ideal simplices tiling the Poincaré ball. The hyperbolicity is that drawing, which the authors say depends on the equal-length assumption; it is not measured or proved. A fitness on faces gives a simulated transition to growth in one direction. | Read |
| NOTE-679 v2 | Representation Tradeoffs for Hyperbolic Embeddings | Trees embed in the hyperbolic plane with worst-case distortion 1 + ε, but because a hyperbolic distance d takes about d bits to write, the coordinates need Θ((ℓ/ε) log deg_max) bits in the plane: cheap in branching, expensive in depth. More dimensions divide the upper bound by r for bushy trees. For general metrics, an exact hyperbolic MDS by PCA on cosh of the distances after centring at a new mean, with a perturbation bound growing as sinh² of the diameter; hyperbolic PGA has spurious local minima and only a local-convexity guarantee. | Read |
| NOTE-686 | Hyperbolic Neural Networks | Möbius gyrovector operations give the Poincaré ball closed-form exponential and logarithmic maps, parallel transport from the origin as a scaling, and geodesic hyperplanes with a closed-form distance; from them come hyperbolic logistic regression, feed-forward layers and GRUs that reduce to the Euclidean ones as c → 0. Because linear maps and nonlinearities are lifted through the origin’s tangent space, the curvature acts only through biases and output distances. Gains over Euclidean models are confined to a synthetic tree-like task and to separating WordNet subtrees in low-dimensional Poincaré embeddings. | Read |
| NOTE-693 | The power of deeper networks for expressing natural functions | Proves that a single hidden layer of smooth, bias-free units needs exactly ∏(rᵢ+1) neurons to Taylor-approximate x₁^r₁⋯xₙ^rₙ while a deep network needs O(Σ log rᵢ), extends the gap to sparse and univariate polynomials, and bounds a product of n inputs at depth k by about 2^(n^(1/k)). Its uniform-approximation lower bounds rest on an unproved step, the depth-k tightness is a conjecture, and the gap is set by degree, so bounded-degree polynomials show none. | Read |
| NOTE-753 v2 | Harvey, Larsen & Williams — Duality of Bures and Shape Distances | For centred linear kernels K_X = CXXᵀC, the Bures distance equals the Procrustes distance, B(K_X, K_Y) = P(X, Y), and normalized Bures similarity equals the cosine of the Riemannian shape distance. Both follow from F(K_X, K_Y) = ‖XᵀCY‖_*. By Uhlmann’s theorem, the fidelity is the largest overlap between representations consistent with the two kernels. CKA is only loosely tied to NBS, inside a rank-dependent envelope. The result is finite-dimensional and linear; the infinite limits are heuristic. | Read |
| NOTE-757 | Calo et al. — Bisimulation metrics are optimal transport distances | For two finite Markov chains with real-valued state labels, the discounted bisimulation metric of Ferns et al. and van Breugel and Worrell solves the same fixed-point equation as the optimal-transport distance over bicausal couplings with ground cost (1−γ)|r(x)−r(y)|, so the two are one object. That distance equals the value of a finite linear program over discounted occupancy couplings, and an entropy-regularised alternating-projection method (Sinkhorn Value Iteration) reaches ε accuracy in O(‖c‖²∞ log(|X||Y|)/((1−γ)⁵ε²)) iterations under exact inner projections. | Read |
| NOTE-763 v2 | Calo et al. — Distances for Markov chains from sample streams | The bicausal transport distance between two finite Markov chains, which the authors’ earlier paper identified with the bisimulation metric, has a second linear program whose constraints use only the chains’ discounted state-transition occupancy measures, not their kernels. A stochastic primal-dual method on it gets an ε-accurate estimate with probability 1−δ from O((|X||Y|(|X|+|Y|) + (|X|+|Y|) log(1/δ)) / ((1−γ)²ε²)) independent transition samples per chain. The guarantee assumes independent draws from the occupancy measures, which needs restarts; samples from one running trajectory are outside it. | Read |
| NOTE-784 v2 | Coifman et al. — Diffusion maps | From a symmetric, positivity-preserving, positive semi-definite kernel, normalized into a Markov operator conjugate to a symmetric one, the top eigenfunctions scaled by powers of their eigenvalues embed the data so that Euclidean distance equals diffusion distance. A one-parameter normalization separates geometry from sampling density, recovering the Laplace–Beltrami operator at α = 1. The symmetry of the kernel is what yields the spectral decomposition; non-symmetric kernels are not treated. | Superseded — by NOTE-1441 |
| NOTE-787 v2 | Vereshchagin & Vitányi — Kolmogorov’s structure functions and model selection | For a string x of length n and every complexity bound α, the minimal two-part code length λ_x(α), the structure function h_x(α) + α and the best achievable fit β_x(α) + K(x) are one function to within O(log n), and any set witnessing λ_x or h_x also witnesses β_x. Every non-increasing shape occurs for some x. h_x and λ_x are upper semicomputable but not computable; β_x is not even semicomputable. Finite sets only; the extension to computable distributions costs O(log n). | Read |
| NOTE-790 | Vitányi — Meaningful Information | With total recursive functions as models, the shortest model that completes an optimal two-part code for a string (its sophistication) is a non-trivial split of its information into structure and noise. With partial recursive models the split collapses to zero for every string. The total-function, computable-distribution and finite-set classes agree to within logarithmic terms. Some strings of every length are almost all structure, and sophistication is not computable. | Read |
| NOTE-814 | Salter-Duke — Tangled webs | An exact but exponential algorithm finds edge-connectivity tangles in graphs of up to about 200 vertices. On eleven small real networks, tangle communities resemble those of standard methods only where community structure is strongly delineated; three networks have no distinguishable tangles at the orders reached, and where tangles do distinguish regions, some hold only one to three vertices. On images, lines one pixel wide cannot be distinguished. The author concludes the tangle notion of cohesion “is not as compelling as I had hoped”. | Read |
| NOTE-817 | Sayed Ahmed — Back and forth between algebraic geometry, algebraic logic, sheaves and forcing | Distributive lattices with join-preserving idempotent operators are represented, after Comer, as continuous sections of a sheaf of quotient algebras over the prime spectrum of their fixed-point part. A small site is said to give a notion of forcing, with Kripke–Joyal clauses, and a Łukasiewicz conjunction is forced through a tensor product. Most theorems are stated without proof, and negation is never written out. | Read |
| NOTE-821 | Kupiainen — Rigorous renormalization group | A three-page orientation by a principal of rigorous RG. It covers Wilson’s block-spin map as an iteration on Hamiltonians, made rigorous by cluster (polymer) expansions that control large fields, and what this proved. The massless 4D lattice φ⁴ model is Gaussian at long distances. The continuum 4D φ⁴ limit is always Gaussian (Aizenman and Duminil-Copin). The Gross–Neveu model and Hall-conductivity universality have been constructed or proved. The method stays near Gaussian, and a non-perturbative formalism is missing. The piece makes no new claim of its own. | Read |
| NOTE-822 | Ovsjanikov et al. — Functional maps | A bijection T: M → N is fully encoded by the linear map f ↦ f∘T⁻¹ it induces on functions, a matrix C in chosen bases. If T preserves volume (or the bases are discretely orthonormal), C is orthogonal, so its inverse is its transpose; if T is an isometry, it commutes with the Laplace–Beltrami operator, so C maps each eigenspace to the matching one. Descriptor, landmark, segment and commutativity constraints are linear in C, so matching reduces to a least-squares solve with a 100-eigenfunction basis, which beat Blended Intrinsic Maps on SCAPE and TOSCA. | Read |
| NOTE-824 | Pfau et al. — Spectral Inference Networks | For a symmetric kernel k on a space with a sampling density p, the span of the top N eigenfunctions of Kf = E_x′[k(x,x′)f(x′)] maximizes Tr(Σ⁻¹Π) over functions, with Σ = E[uuᵀ] and Π = E[k(x,x′)u(x)u(x′)ᵀ]; a Cholesky-masked gradient orders them and moving averages of Σ and its Jacobian remove the minibatch bias. Slow Feature Analysis is the graph-Laplacian case. The ordering argument is informal by the authors’ own statement, and the paper never considers contrastive kernels or positive pairs. | Read |
| NOTE-829 | Jorgensen & Tian — Noncommutative Analysis (GNS, RKHS) | The book calls the kernel-to-Hilbert-space construction “the GNS construction” (Remark 1.34) and states that minimal realizations ϕ(x,y) = ⟨Φ(x), Φ(y)⟩ are unique up to a unitary (Cor. 1.35). It does not prove the corollary; the proof is “left as an exercise”. The parallel uniqueness for states and cyclic representations is proved (Thm 4.35), and the argument transfers. The commutative GNS space L²(μ) is written out explicitly (Ex. 4.86, Lemma 3.28). | Skimmed |
| NOTE-831 | Farrell & Ioannou — Generalized Stability Theory I | For a linear system du/dt = Au with A non-normal, the eigenvalues fix only the t → ∞ behaviour. Growth over a finite time t is the largest singular value of e^{At}, bounded below by the fastest mode and at t → 0 set by the numerical abscissa, the top eigenvalue of (A + A†)/2. A system with every mode damped can amplify a perturbation a hundredfold, and its forced and stochastic responses exceed those of the normal system with the same spectrum. | Read |
| NOTE-836 | Schölkopf, Herbrich & Smola — A Generalized Representer Theorem | For any nonempty set X, real positive-definite kernel k, strictly increasing g and any cost c on the training triples, even coupled, non-convex or infinite-valued, every minimizer of c + g(‖f‖) over the class F of countable kernel expansions with finite RKHS norm is a finite expansion Σ α_i k(·, x_i) at the training points. The proof uses only the reproducing property and Pythagoras. Existence of a minimizer, the extension from F to the completed RKHS, and the semiparametric Theorem 2 are not proved here. | Read |
| NOTE-837 | Bach — Information Theory with Kernel Methods | With a continuous kernel normalised to k(x,x) = 1 on a compact set, the uncentred covariance operator Σ_p is a density operator. Its quantum relative entropy is sandwiched between the KL of two kernel-smoothed distributions and the KL itself, D(p̃‖q̃) ≤ D(Σ_p‖Σ_q) ≤ D(p‖q), with equality on the right only for an orthonormal, hence discrete, embedding. On the torus the gap is O(σ²) in the bandwidth. Kernel mutual information characterizes independence; conditional independence only one way. | Read |
| NOTE-838 | Muandet et al. — Kernel Mean Embedding of Distributions | A survey whose load-bearing chain is short and explicit. An RKHS is a Hilbert space with bounded evaluation, so Riesz gives k_x (Thm 2.4, Prop. 2.1). If E_P√k(X,X) < ∞, the expectation functional is bounded and Riesz gives μ_P with E_P f = ⟨f, μ_P⟩ (Lemma 3.1, proved). MMD is ‖μ_P − μ_Q‖ (eq. 3.29), a metric on distributions exactly when k is characteristic. HSIC is the squared Hilbert–Schmidt norm of the centred cross-covariance. Every result is for a fixed kernel. | Read |
| NOTE-839 | Trefethen et al. — Hydrodynamic Stability Without Eigenvalues | The linearized operators of plane Couette and Poiseuille flow are far from normal. With every eigenmode decaying, perturbations can grow transiently by O(R) and resonate by O(R²), and an operator perturbation of norm O(R⁻²) makes the flow eigenvalue-unstable, all read off the pseudospectra. That transition to turbulence then follows from linear growth plus energy-neutral nonlinear mixing is a conjecture, from a 2 × 2 model. | Read |
| NOTE-842 | Farrell & Ioannou — Generalized Stability Theory II | For du/dt = A(t)u there are no modes, but the propagator’s singular vectors and the Lyapunov exponents and vectors carry generalized stability theory over. A time-dependent operator that is stable, or neutral, at every instant can be asymptotically unstable, and only if it is non-normal. Random variation of the operator destabilizes generically, with less variation needed the more non-normal and the less damped the mean operator. | Read |
| NOTE-843 | Gebhart, Hansen and Schrater — Knowledge Sheaves | A knowledge-graph embedding is a knowledge sheaf on the schema (one pair of restriction maps per relation type) with a 0-cochain of its pullback to the graph. The summed Structured Embedding score is the sheaf Laplacian form xᵀLx; adding a relation vector gives TransR, and identity maps TransE. Since the zero cochain is always a section, training needs negative triples and a margin. A multi-hop query is scored by harmonic extension, a Schur complement of the query graph’s Laplacian, which beats summing TransE vectors on ip and pi queries. | Read |
| NOTE-854 | Adler — Hilbert spaces and the Riesz representation theorem | An undergraduate expository proof, twice over, that every continuous linear functional on a Hilbert space is v ↦ ⟨u, v⟩ for a unique u with ‖u‖ = ‖φ‖: once from the projection theorem, once as u = Σ_k conj φ(e_k) e_k over any orthonormal basis, with ‖φ‖² = Σ_k |φ(e_k)|². It ends with von Neumann’s proof of the Radon–Nikodym theorem for finite positive measures by applying Riesz to f ↦ ∫ f dν on L²(ν + μ). | Read |
| NOTE-855 | Bra–ket notation (Wikipedia) | Dirac’s notation writes vectors as kets and linear functionals as bras. On a Hilbert space the inner product, antilinear in its first slot, turns each ket into a bra, and the article cites the Riesz theorem for the reverse direction; on a Banach or untopologized space the bracket is only a functional applied to a vector, because Riesz does not apply. The article is a reference with internal inconsistencies, including an unsourced edit of 2026-09-30 to the date of the notation’s origin. | Read |
| NOTE-857 | Gelfand–Naimark–Segal construction (Wikipedia) | For a state ρ on a C*-algebra A, the semi-inner product ⟨a, b⟩ = ρ(b*a) on A, quotiented by its null left ideal and completed, carries A acting by left multiplication, with the class of the identity as a unit cyclic vector and ρ(a) = ⟨π(a)ξ, ξ⟩. Any two cyclic representations with cyclic vectors reproducing the same state are unitarily equivalent, by the unitary π(a)ξ ↦ π′(a)ξ′. The representation is irreducible exactly when the state is pure. The article says nothing about kernels. | Read |
| NOTE-861 | Riesz representation theorem (Wikipedia) | On a real or complex Hilbert space every continuous linear functional φ is x ↦ ⟨x, f_φ⟩ for a unique f_φ, with ‖f_φ‖ = ‖φ‖; the map y ↦ ⟨·, y⟩ is an antilinear isometric bijection H → H*. Completeness is used only to split H = ker φ ⊕ (ker φ)^⊥. The article then defines the adjoint of a bounded operator between Hilbert spaces through this bijection, and shows it is the transpose read through the two Riesz maps. | Read |
| NOTE-863 | Goldreich — Foundations of Cryptography, Vol. 1 (1995 fragments): computational indistinguishability and pseudorandom generators | In the 1995 draft of Volume 1, computational indistinguishability is a strictly coarser equivalence than statistical closeness, and the book proves this two ways. Proposition 3.3 does it unconditionally: a counting argument shows there are ensembles supported on at most 2^(n/2) strings that no circuit of size 2^(n/8) tells from uniform with gap 2^(−n/8) or more, but such an ensemble cannot be sampled efficiently. §3.3.1 shows that the output of any pseudorandom generator is statistically far from uniform yet, by definition, indistinguishable from it. That witness is efficiently samplable, and it needs pseudorandom generators, which exist if and only if one-way functions do (Theorem 3.29, stated without proof). That efficient post-processing preserves indistinguishability is set as Exercise 1 of §3.8.4, not proved in the text. All of this is asymptotic, about ensembles indexed by n, not about single pairs of distributions. | Skimmed |
| NOTE-880 | Håstad, Impagliazzo, Levin & Luby — A pseudorandom generator from any one-way function | There is a pseudorandom generator if and only if there is a one-way function (Theorem 6.2.2), in the uniform setting: adversaries are algorithms, and every reduction is an oracle Turing machine that turns an adversary for the constructed generator into an inverter for the given function. The route runs one-way function → false entropy generator → pseudoentropy generator → pseudorandom generator, where “computational entropy” is the Shannon entropy of some P-samplable distribution that no adversary of the given strength tells apart from the output. The general reduction is only weak-preserving: composed literally it needs seeds of length n^34, and the paper’s direct construction n^10. The paper attributes to Goldreich [G : 89] the equivalence of pseudorandom generators with P-samplable pairs that are computationally indistinguishable but statistically very different; with Theorem 6.2.2 that gives the one-way function form, but the paper never writes that combined sentence. | Read |
| NOTE-891 | Yao — Should tables be sorted? | For storing a set of n keys from a universe {1,…,m} in a table so that the membership query (is x in S?) is answered with the fewest worst-case adaptive probes, Yao proves that when the table cells may hold only keys of S or a bounded number of pointer symbols, sorted order with binary search (⌈lg(n+1)⌉ probes) is optimal once m exceeds a Ramsey-number threshold N(n, p, q), whatever the arrangement rule, table size or hashing scheme. The threshold is astronomically large, and for small universes sorted order is beaten (one probe suffices for m = 3 with n = 2, and for m ≤ 2n − 2 with n > 2). The optimality also fails as soon as one cell may hold an arbitrary value from the universe: with a single extra “directory” cell, two probes suffice for m ≥ 2^{16n²}. Only query cost is counted; building the table is free and unconstrained, so the paper says nothing about the cost of arranging or about trading arrangement cost against lookups. | Read |
| NOTE-893 | Karlin, Manasse, Rudolph & Sleator — Competitive snoopy caching | For a shared-bus multiprocessor whose caches can either keep a replicated block (and pay one bus cycle for every write to it) or drop it (and pay p to fetch it again if it is read), the paper gives counter-based on-line algorithms whose total bus cost on every request sequence is at most twice that of the best off-line algorithm, and proves that no deterministic on-line algorithm can guarantee better than 2. The rule is: drop a copy once the writes it has caused since it was last read have cost as much as refetching it. With a third option (reading or writing through without fetching) the lower bound rises to 3, and the algorithm given is 3-competitive ignoring writebacks, 4 counting them; with associative caches the bound becomes the paging ratio. The paper never names rent-or-buy or ski rental; its rent is upkeep paid on an idle copy, and its purchase is the refetch after dropping, the reverse orientation of building a bin. Costs are fixed constants, the guarantees are worst case only, and the authors call the practical significance “unclear”. | Read |
| NOTE-902 | Piatetsky-Shapiro — The optimal selection of secondary indices is NP-complete | For one relation, a known mix of equality queries and updates, single-attribute secondary indexes, and a fixed upkeep charge per index on each update, choosing the index set that minimizes average transaction cost is NP-hard: minimum set cover reduces to it, each subset becoming an attribute and each element a query that any one of its attributes’ indexes can answer. The paper calls the problem NP-complete but states only the optimization version, and the proof is a sketch of about half a page. Storage is not in the model; the only price of an index is upkeep on updates. A variant with off-line updates under an update-cost budget is asserted NP-complete without proof. The paper does not cite or discuss Comer 1978. | Read |
| NOTE-903 | Sleator & Tarjan — Amortized efficiency of list update and paging rules | For a list searched sequentially from the front, where reaching the i-th item costs i, moving each accessed item to the front costs at most twice what any other rule costs on the same sequence, including rules that know the whole sequence in advance (four times, counting move-to-front’s own moves), with no probabilistic assumption. A fixed list sorted by access frequency, and the transpose and frequency-count rules, are not within any constant factor of that. Rearranging ahead of need never pays in this model (Theorem 3). For paging, where only a small fast memory is cheap, no rule that cannot see the future can do better in the worst case than n/(n − n_MIN + 1) times the offline optimum MIN, and LRU and FIFO attain that bound up to an additive n_MIN; with equal memories the ratio is the memory size. The model is a single linear list (or a two-level memory) with uniform items and access cost depending only on position; it says nothing about partitions, bins or search that is not a scan. | Read |
| NOTE-905 | Indyk & Motwani — Approximate nearest neighbors: towards removing the curse of dimensionality | Approximate nearest-neighbour search in d dimensions reduces, with small overhead, to a decision problem (is there a point within r, or none within (1+ε)r), and that decision problem is solved by locality-sensitive hashing: if hash functions make near pairs collide with probability at least p1 and far pairs with at most p2, then n^ρ random hash tables with ρ = ln(1/p1)/ln(1/p2) answer queries with O(n^ρ) hash evaluations in O(dn + n^{1+ρ}) space, with constant success probability. For Hamming space by bit sampling, and for ℓ_p with p in [1, 2] by embedding, ρ ≤ 1/(1+ε) in the 1999 text (1/ε, for ε > 1, in STOC). The paper also gives the other end of the trade: Õ(d) query time for n·O(1/ε)^d or (nd)^{O(1)} preprocessing. It is an upper bound only: it proves no lower bound, does not claim its exponents are best, solves only the approximate problem with constant probability, and counts preprocessing as space. | Read |
| NOTE-913 | Green, Karvounarakis & Tannen — Provenance semirings | Annotating each input tuple with an element of a commutative semiring, and computing union and projection with + and join with ·, gives one positive relational algebra (RA+) that is set semantics over the Booleans, bag semantics over the naturals, c-tables over positive Boolean expressions, event tables over sets of outcomes, and lineage over sets of tuple ids. The polynomials N[X] in the tuple ids are the most general such annotation: for any RA+ query and any commutative semiring K, the K-annotated answer is the provenance polynomial evaluated at the input’s K-annotations (Theorem 4.3). For datalog the paper needs ω-continuous semirings and formal power series N∞⟦X⟧, with the analogous factorization (Theorem 6.4) and a criterion for infinite coefficients (Theorem 6.5). Everything is positive: no negation, difference or aggregation, and revaluing inputs can only remove output tuples, never add them; trust and deletion propagation are not treated. | Read |
| NOTE-914 | Panigrahy, Talwar & Wieder — Lower bounds on near neighbor search via metric expansion | Cell-probe lower bounds for near-neighbour search follow from the expansion of the graph joining points within distance r: a deterministic t-probe structure with m cells of w bits must have (mwt/n)^t at least the vertex expansion, and a randomized one must have m^t w/n at least the robust expansion at measure 1/m^t, on random data sets with weakly overlapping neighbourhoods. For the hypercube this recovers known bounds and gives, for example, space n^{1+Ω(1/(ct))} for approximation c and n^{Ω(1/(tε²))} for 1 + ε; for symmetric (Cayley) graphs a one-probe structure matches the bound, and for dynamic low-contention structures the update time must be at least the robust expansion over poly(t). The bounds are average case, for approximate decision search, and weak in t: they fall off exponentially in the number of probes, so they are tight only for a constant number of probes and say little about structures that probe n^ρ cells. | Read |
| NOTE-917 | Beyer, Goldstein, Ramakrishnan & Shaft — When is “nearest neighbor” meaningful? | For a fixed number n of data points and a query drawn independently of them, if the variance of the p-th power of the query-to-point distance, divided by its mean, goes to zero as dimension m grows, then the farthest point is, with probability tending to one, within any factor 1 + ε of the nearest (Theorem 1). The condition holds for IID dimensions and for several correlated or non-identical workloads, and fails when data and queries lie on a low-dimensional subspace or when queries fall inside well-separated clusters. Simulations put the collapse of contrast within 10 to 20 dimensions for IID data, and two real image-feature sets showed weak contrast. The condition is sufficient, not shown necessary; the theorem is about the ratio of distances with n held fixed, and says nothing about real data of low intrinsic dimension except that such data can escape it. | Read |
| NOTE-918 | Andoni, Laarhoven, Razenshteyn & Waingarten — Optimal hashing-based time–space trade-offs for approximate near neighbors | For c-approximate near-neighbour search (find a point within cr when one lies within r) over n points in Euclidean space, the paper gives a data structure for every pair of exponents with c²√ρq + (c²−1)√ρu ≥ √(2c²−1), using space n^{1+ρu+o(1)} + O(dn) and query time n^{ρq+o(1)} + dn^{o(1)}, on worst-case data (for Hamming and ℓ_1, c√ρq + (c−1)√ρu ≥ √(2c−1)). It proves the curve cannot be beaten by list-of-points structures, the model of all known data-independent hashing schemes, on random instances, and unconditionally in the cell-probe model only at one end (query n^{o(1)}) for one or two probes. So the exact trade between space spent organizing and per-query time is known within the hashing model; outside that model, for more than two probes, it is not, and data-dependent schemes are covered only by conjecture. | Read |
| NOTE-923 | Kleinberg — The small-world phenomenon: an algorithmic perspective | On an n × n lattice where each node also gets q long-range links drawn with probability proportional to d^(−r), short paths exist for r = 0, but no algorithm that sees only the links of nodes the message has visited finds them faster than about n^(2/3) steps in expectation. At r = 2 the greedy rule (pass to the contact nearest the target in lattice distance) delivers in O((log n)²) expected steps (proved for p = q = 1); for 0 ≤ r < 2 every such algorithm needs Ω(n^((2−r)/3)) and for r > 2 Ω(n^((r−2)/(r−1))). So efficient decentralized search depends on how the long-range links are distributed relative to the lattice, not on the links’ existence or on the graph’s diameter. The k-dimensional version (r = k) is stated without proof, and the model is a lattice with a known distance that every node can compute to the target. | Read |
| NOTE-925 | Larsen & Nelson — optimality of the Johnson–Lindenstrauss lemma | For n, d ≥ 2 and lg^0.5001 n / √min{n, d} < ε < 1 there is an n-point set in R^d such that every map into R^m, linear or not, that keeps all squared distances within 1 ± ε needs m = Ω(ε⁻² lg(ε²n)), which is Ω(ε⁻² lg n) once ε > 1/min{n, d}^0.4999. So the Johnson–Lindenstrauss bound is optimal up to a constant, and the distributional lemma plus a union bound is the best route. The proof is an encoding argument over k-sparse point sets with k = ε⁻²/256, and the count it ends on is m = Ω(k lg(d/k)). | Read |
| NOTE-926 | Achlioptas — database-friendly random projections | A random projection keeps all pairwise distances of n points within 1 ± ε with probability at least 1 − n^−β when k ≥ (4 + 2β)/(ε²/2 − ε³/3) · log n and the matrix entries are independent ±1, or √3 times {+1, 0, −1} with probabilities 1/6, 2/3, 1/6. This is the same k₀ as the best bound for Gaussian projections. The proof shows that every even moment of a unit vector’s projection is at most that of the all-equal vector (1, …, 1)/√d, and that this vector’s moments are at most the Gaussian ones. | Read |
| NOTE-927 | Lee & Naor — extending Lipschitz functions via random metric partitions | A random partition of a metric space whose pieces are small and whose points are, on average, far from piece boundaries yields a “gentle” partition of unity, and averaging a Lipschitz map over it extends the map into any Banach space. This gives ae(X) ≤ C log n/log log n for every n-point metric space X (Theorem 1.10), improving Johnson, Lindenstrauss and Schechtman’s C log n; ae(X) ≤ C log λ(X) for doubling X; ae ≤ C r² on K_r-minor-free graph metrics; e(ℓ₂ᵈ, Z) ≤ C√d; and, with the Johnson–Lindenstrauss lemma, eₙ(L₂, Z) ≤ C√(log n). For maps into Hilbert space it does not improve Johnson and Lindenstrauss’s K√(log n), and it leaves √(log n) open. | Read |
| NOTE-928 | Linial, London & Rabinovich — the geometry of graphs | Every n-point metric embeds in ℓ₂ with distortion O(log n) in O(log n) dimensions (Theorem 3.2(1), attributed to Bourgain). The paper reproves the distortion bound by a randomized Bourgain-style construction into ℓ_p^{O(log² n)}, and it gets the dimension from Johnson–Lindenstrauss. Every embedding of a constant-degree expander into ℓ_p, 1 ≤ p ≤ 2, of any dimension, has distortion Ω(log n) (Proposition 4.2), so the bound is tight. The lower bound comes from Leighton and Rao’s flow–cut gap, and the same embeddings give an O(log k) max-flow/min-cut gap for k commodities. | Read |
| NOTE-929 | Alon & Klartag — optimal compression of approximate inner products | Storing every inner product and squared distance of n points of norm at most 1 in R^k to additive error ε takes Θ(n log n/ε²) bits once k ≥ log n/ε², Θ(nk log(2 + log n/(ε²k))) for log n ≤ k ≤ log n/ε², and Θ(nk log(1/ε)) below, for 1/n^0.49 ≤ ε ≤ 0.1. Comparing these counts across dimensions shows that some n points in R^k cannot be mapped into fewer than Ω(min{k, log(2 + ε²n)/ε²}) dimensions with additive error ε, for every ε ≥ 2/√n. A bipartite version of the Johnson–Lindenstrauss lemma reaches O(log(2 + ε²n)/ε²) dimensions; the one-family version at that size is left open. | Read |
| NOTE-930 v2 | Johnson & Lindenstrauss — extensions of Lipschitz mappings into a Hilbert space | Lemma 1: any n points in ℓ₂ⁿ map into ℓ₂ᵏ, k = [K(τ) log n], with ‖f‖_lip‖f⁻¹‖_lip ≤ (1+τ)/(1−τ), by a Haar-random rank-k orthogonal projection, with positive probability. The proof concentrates the norm of a projected unit vector about its median by Lévy’s isoperimetric inequality and takes a union bound over the normalized differences of the points; K ≥ (10/τ)² suffices. The lemma is a tool for Theorem 1: Lipschitz maps from n-point subsets of any metric space into ℓ₂ extend to the whole space with Lipschitz constant at most C(log n)^{1/2} times the original. | Read |
| NOTE-931 | Baraniuk, Davenport, DeVore & Wakin — a simple proof of the restricted isometry property | If an n × N random matrix satisfies the concentration inequality used in JL proofs, Pr(|‖Φx‖² − ‖x‖²| ≥ ε‖x‖²) ≤ 2e^{−n c₀(ε)}, then a union bound over a (12/δ)^k-point net on each of the C(N, k) ≤ (eN/k)^k coordinate k-subspaces, with a short bootstrapping step from net to subspace, gives the restricted isometry property for all k ≤ c₁n/log(N/k) with probability at least 1 − 2e^{−c₂n} (Theorem 5.2). Gaussian, ±1 and Achlioptas’s sparse ensembles qualify with c₀(ε) = ε²/4 − ε³/6, and the same argument gives the property in any fixed orthonormal basis. | Read |
| NOTE-932 | Dasgupta & Gupta — an elementary proof of the Johnson–Lindenstrauss lemma | For n points in ℝᵈ and 0 < ε < 1, a projection onto a random k-dimensional subspace, scaled by √(d/k), keeps every squared pairwise distance within a factor 1 ± ε with probability at least 1/n once k ≥ 4(ε²/2 − ε³/3)⁻¹ ln n. The proof bounds each tail of the squared length of a random unit vector’s first k coordinates by Markov’s inequality on an exponential of Gaussian squares, and then takes a union bound over the n(n−1)/2 pairs. | Read |
| NOTE-933 | Donoho & Tanner — observed universality of phase transitions | For Gaussian matrices, ℓ1 minimization (and its nonnegative LP form) recovers a k-sparse vector from n random measurements in dimension N with probability tending to 1 below a curve ρ(δ; Q) in the plane δ = n/N, ρ = k/n, and to 0 above it; this was proved earlier and is restated here as Theorem 2.1. In 2,948,000 linear programs over eight non-Gaussian ensembles at N = 200, 400 and 1600, the 50% success points fall on the same curves, but exact finite-N agreement is rejected: the differences from the Gaussian decay as N^(-1/2), so the paper supports only an asymptotic (“weak”) universality, as an observation, not a theorem. | Read |
| NOTE-965 | Rauh, Bertschinger, Olbrich and Jost — Reconsidering unique information | Theorem 2: no redundancy function on the Williams–Beer lattice satisfies symmetry, self-redundancy, monotonicity, non-negative atoms and the identity axiom. In XORCOPY (two independent bits, their XOR, and the target their joint copy) identity forces three 1-bit atoms {i}{jk} under a 2-bit total, so {12}{13}{23} ≤ −1. Bertschinger et al.’s bivariate measures cannot be extended to that lattice. Their unique information is monotone in each argument as expected; their shared and synergistic information violate left monotonicity. | Read |
| NOTE-969 | Eyink and Aluie — Localness of energy cascade I | Builds an exact band-pass decomposition of kinetic energy from smooth filters, a multiscale Germano identity, in which each band’s energy changes by the flux into it minus the flux out. It then restates, with heuristic sketches, Eyink’s (2005) bounds: under a scaling hypothesis on velocity increments with exponents strictly between 0 and 1, the mean energy flux across ℓ takes at most O((ℓ/Δ)^(2/3)) from scales above Δ and O((δ/ℓ)^(2/3)) from scales below δ. It argues, without proof, and checks on one 512³ simulation, that sign cancellations give Kraichnan’s faster 4/3 decay. Locality here is in scale, for one field. Nothing in it bears on units. | Read |
| NOTE-973 | Parish and Duraisamy — Non-Markovian closure for LES via Mori–Zwanzig | Derives large-eddy-simulation closures from the Mori–Zwanzig identity by Stinis’s finite-memory approximation, which turns the memory integral into auxiliary relaxation equations, a finite Markovian augmentation of the resolved modes. Its error is not bounded and its memory length is fitted (Burgers: τ₀ ≈ 0.2 over the spectral radius of the resolved Jacobian) or preset. Against filtered DNS it matches dynamic Smagorinsky on isotropic turbulence and channel flow, beats Smagorinsky and the t-model on the Taylor–Green vortex, and puts the subgrid drain at the cutoff, as DNS does. | Read |
| NOTE-974 v2 | Isett — A proof of Onsager’s conjecture | The negative half of Onsager’s conjecture. For every α < 1/3 there is a nonzero weak solution of the 3D incompressible Euler equations on the periodic box, in C^α in space and time, that vanishes outside a finite time interval and so does not conserve energy. It is built by iterating a Main Lemma that first glues exact local Euler solutions so that the error lives on separated short time intervals, then cancels each piece with a single transported Mikado-flow wave. For the record, it shows that a coarse-grained energy flux can survive the limit of vanishing scale for rough fields; it does not show a one-way dissipative cascade, since these solutions gain energy and then lose it. | Skimmed |
| NOTE-975 | Chorin, Hald and Kupferman — Optimal prediction with memory | Writes the resolved components of an ODE with random unresolved initial data as an exact generalized Langevin equation (Markovian term, memory, noise) under three projections, and approximates the memory two ways: by solving a Volterra equation for its expansion coefficients from Monte Carlo correlation functions, and by the t-damping (t-model) equation dΦ/dt = R(Φ) + tS(Φ), which needs no prior data. On one 4-variable Hamiltonian example both restore the decay of the conditional mean that the Markovian truncation misses, the Volterra scheme to about t ≈ 8. No error bound for any approximation, and no Markovian augmentation. | Read |
| NOTE-977 | Shpak, Stadler, Wagner and Altenberg — Simon–Ando decomposability and fitness landscapes | States Simon and Ando’s (1961) theorem for x(t+1) = (A* + εC)x, with A* block-diagonal, as existence statements: as ε → 0 the non-leading eigenvectors approach the blocks’, and times T0 < T1 < T2 < T3 exist that separate within-block relaxation, quasi-equilibrium, aggregate cross-block dynamics and global equilibrium. No rate in ε and no bound on aggregation error is given. It then applies an entrywise proxy (minimal within-class rate ≫ maximal cross-class rate) to mutation–selection on hypercubes, which low mutation rates meet trivially, and to recombination, which rarely meets it. It concludes that Simon–Ando aggregation is one source of dynamical sufficiency among several, distinct from symmetry (equitable partitions) and from linkage-equilibrium character aggregation. | Read |
| NOTE-979 | Shpak, Stadler, Wagner and Hermisson — Aggregation of variables and system decomposition | Separates aggregation of variables, a lower-dimensional map y = Qx whose dynamics close, from decomposition into non-interacting subsets: either can hold without the other. For a linear system the aggregate operator must satisfy ĀQ = QA. For a partition, the block sums close for every state exactly when the partition is column-equitable. Equitable partitions of mutation–selection matrices reduce single- and two-peak fitness landscapes exactly, and fail for three or more peaks unless the classes are refined to overlap classes. Simon–Ando aggregation is treated as the approximate limiting case, without error bounds. | Read |
| NOTE-980 | No, Ingber and Weissman — Strong successive refinability | Extends successive refinement to second order. For finite alphabets, a two-stage code can reach both decoders’ rates with their dispersion terms if the source is successively refinable throughout a neighbourhood of its law (Corollary 6), which holds for any discrete source under Hamming distortion; the Gaussian under squared error is shown directly. Layered codes then use L_n refinement stages to cut encoding complexity, at a rate penalty of order L_n log n / n, for binary and Gaussian sources. It does not treat logarithmic loss, which the reading that named it had recalled; the log-loss refinability result is No and Weissman’s, concerns direct log-loss with soft reconstructions of the source itself, and does not reach the information bottleneck, whose loss is remote. | Read |
| NOTE-982 | Constantin, E and Titi — Onsager’s conjecture on energy conservation | A one-page proof that a weak solution of the 3D incompressible Euler equations on the periodic box conserves energy if it lies in L³(0,T; B^α_{3,∞}) ∩ C([0,T]; L²) with α > 1/3. The flux of energy into scales below a mollification scale ε is written exactly as a commutator, (u⊗u)^ε − u^ε⊗u^ε = r_ε − (u−u^ε)⊗(u−u^ε), of size ε^(2α), against a gradient of size ε^(α−1), so the flux is O(ε^(3α−1)) and vanishes. This is the positive half of Onsager’s conjecture only. It does not show that rougher solutions dissipate. Two printed formulas in the manuscript are wrong as typeset, without effect on the result. | Read |
| NOTE-984 | Coifman and Maggioni — Diffusion wavelets | A self-adjoint diffusion operator T with spectrum in [0, 1] defines a multiresolution analysis by thresholding its dyadic powers at a precision ε: V_j is spanned by the eigenvectors with λ^{t_j} ≥ ε, so the V_j are exactly nested spectral subspaces and the wavelet spaces W_j are spectral bands, about one octave of the generator each. The paper’s contribution is an algorithm that builds localized orthonormal bases for approximations of these spaces without computing eigenvectors, by squaring the operator on each compressed basis. As written, that algorithm thresholds T^{2^{j−1}}, not T^{2^j−1} as the paper states, and the paper’s own Fig. 9 follows the former. Localization, stability and the near-linear complexity are argued or observed, not proved for the compressed algorithm. | Read |
| NOTE-985 | Coifman and Lafon — Diffusion maps | A symmetric non-negative kernel normalized into a reversible Markov chain gives a family of diffusion distances D_t, one per number of steps t, and an embedding by eigenvectors weighted by λ_l^t in which Euclidean distance is D_t. An α-normalization by a kernel density estimate makes the small-bandwidth limit the graph Laplacian with a density potential (α = 0), the Langevin generator whose invariant density is the sampling density (α = 1/2) or the Laplace–Beltrami operator (α = 1). The truncation is claimed accurate to relative precision δ, which is wrong: the bound is absolute. Several displays carry errors, and the heat-kernel proof has a gap. | Read |
| NOTE-986 | Nadler, Lafon, Coifman and Kevrekidis — Diffusion maps and reaction coordinates | Ties diffusion maps to the dynamics that generated the data. For samples from p = e^{−U}, the small-ε generator of the α-normalized walk is Δψ − 2(1 − α)∇ψ·∇U: the graph Laplacian (α = 0) gives Fokker–Planck with potential 2U, α = 1/2 gives the Langevin dynamics with potential U itself, and α = 1 gives Laplace–Beltrami. So the slow eigenfunctions of the α = 1/2 map approximate the slow modes of the sampled stochastic system, and the paper proposes them as reaction coordinates. The expansions are formal, and with the paper’s kernel every printed generator is too large by a factor of two. Its double-well example is separable. From equilibrium samples the slow eigenfunction is recovered well and the slow eigenvalue is not. That slow eigenfunctions are closed in mean, and in law only under a condition, is the record’s addition. | Read |
| NOTE-989 | Chorin, Hald and Kupferman — Optimal prediction and the Mori–Zwanzig representation | Writes Zwanzig’s nonlinear Mori–Zwanzig identity with the projection taken as conditional expectation on the resolved variables under an invariant measure: ∂ₜSᵗA = R(SᵗA) + ∫₀ᵗ K(s, S^{t−s}A) ds + S⊥ᵗF, with R(A) = E[LA|A], F = LA − E[LA|A] and K(s, A) = E[LS⊥ˢF|A]. First-order optimal prediction, dv/dt = R(v), is that identity with the memory and noise dropped, and for a canonical Hamiltonian system it is Hamiltonian in a renormalized 𝓗 = −log∫e^{−H}. In a perturbative nonlinear Schrödinger example with one resolved mode, a memory kernel cut off at a hand-chosen time recovers the decay of the conditional mean that the Markovian scheme misses. The paper states no fluctuation–dissipation relation and no criterion for when the Markovian term suffices. | Read |
| NOTE-1010 | Random Hyperbolic Graphs: Degree Sequence and Clustering | Proves for the threshold hyperbolic random graph (curvature −1, density exponent α > 1/2, R = 2 log n + C) that the average local clustering coefficient is Θ(1) with high probability and that the degree sequence is a power law with exponent 2α + 1 up to the maximum degree n^(1/(2α)+o(1)), with exact asymptotics for the fraction of each degree. Skimmed: the statements and the method were read, the proofs only in outline. | Skimmed |
| NOTE-1011 | Clustering and the hyperbolic geometry of complex networks | For Krioukov et al.’s hyperbolic random graph with 0 < ζ/α < 2, proves that below temperature 1 (β > 1) the global clustering coefficient converges to an explicit positive constant when ζ/α < 1 and to zero when 1 ≤ ζ/α < 2, and that at or above temperature 1 it converges to zero; clustering among vertices of bounded type stays positive for all ζ/α < 2 when β > 1. Skimmed: §1 and the theorem statements read, the proofs not. | Skimmed |
| NOTE-1013 v2 | Buckmaster, De Lellis, Székelyhidi and Vicol — Onsager’s conjecture for admissible weak solutions | The admissible version of the negative half of Onsager’s conjecture. For every β < 1/3 and every smooth strictly positive e(t) on [0, T] there is a C^β weak solution of 3D Euler on the torus with ∫|v|² = e(t) exactly, so with e decreasing the energy only falls and the coarse-grained flux runs into vanishing scales on every time interval (THEORY-776). That is the one-way transfer Isett’s solutions lacked, but one-way only for the box as a whole (no local energy inequality), and chosen by the prescribed profile, since an increasing e works just as well. | Read |
| NOTE-1020 | Del Vecchio and Slotine — Contraction and singular perturbation | Contraction gives global, explicit, uniform-in-time bounds for singular perturbation: the fast state reaches an εχ_z(d + K)/λ_z neighbourhood of the slow manifold at rate λ_z/ε, and the reduced (quasi-steady-state) trajectory is O(ε)-accurate for all time. Theorem 3 as printed is false: it needs the reduced system, not f alone, to be contracting. The robustness lemma is proved for constant metrics only; the reading re-derives it for state- and time-dependent ones, and with it the two steps THEORY-771 lacked. | Read |
| NOTE-1022 | Giri, Kwon and Novack — The L³-based strong Onsager theorem | The L³-based strong Onsager theorem: for every β < 1/3 there are weak Euler solutions in C⁰_t(B^β_{3,∞} ∩ L^{1/(1−3β)}) satisfying the local energy inequality and strictly dissipating total energy. In the proof the Duchon–Robert measure is exactly the smooth density −e e′ρ₀², positive in a periodic array of pipes and zero outside, fixed at the first step; so the coarse-grained flux into vanishing scales is one-signed locally in space and time, in the limit (THEORY-772), with its place and rate chosen by the construction. | Skimmed |
| NOTE-1025 | Bousquet and Slotine — Contraction and near-decomposability | Under contraction of the fast modules and of the reduced slow system, and coupling terms bounded affinely in the off-manifold state, a nearly decomposable system stays within O(ε) of its singular-perturbation reduction, uniformly in time after a transient of order (1/β_f + 1/(εβ_g)) log(1/ε), for ε below an explicit critical separation ε_c. Deterministic trajectory error only. Remark 8 prints ε_c inverted, and the slow-error bound needs an unstated d_g = 0. | Read |
| NOTE-1027 | Isett — Continuous, globally dissipative Euler flows | The first continuous weak Euler solutions satisfying the local energy inequality with nonzero dissipation: C^α_{t,x}, α < 1/15, on T^d, d ≥ 3, with a Cantor family sharing one initial datum and one dissipation measure. In the construction that measure is a function of time alone, so the coarse-grained flux into vanishing scales is one-way at every place in the limit and uniform in space, switched on only while the solution is drained to zero (THEORY-821). | Skimmed |
| NOTE-1028 | Chorin and Lu — Discrete stochastic parametrization by NARMAX | Model reduction done wholly in discrete time: advance the resolved variables by a fixed integrator and fit the exact one-step discrepancy as a NARMAX series, so memory lives in a few lags of the discrepancy, the resolved variable and the innovations. On Lorenz 96 with ε = 0.5 one or two lags reproduce x₁’s statistics and double forecast lead time at δ = 0.05. The link to Mori–Zwanzig is analogy, not derivation; nothing is proved and no error is bounded. The record’s check: a partially observed linear Gaussian VAR(1) is exactly ARMA, with order 1 + the rank of its MZ memory kernels, so a memory non-zero at every lag is carried exactly by finitely many history-measurable coordinates. | Read |
| NOTE-1029 | Lin and Lu — Wiener projections and the Koopman–Mori–Zwanzig formalism | An exact link between Mori–Zwanzig and NARMAX, with the approximation named. For an invertible map with an invariant measure, projecting onto the whole past of chosen functions of the resolved variables removes the memory term: the reduced equation is the causal Wiener filter on the coarse variable’s own history, with a residual orthogonal to all of it. NARMAX follows only by an “uncontrolled” rational approximation of the filter; no error is bounded. The record’s checks: Mori’s kernels and the Wiener filter are both fixed by the coarse variable’s autocovariance; for a scalar coarse variable of a linear Gaussian VAR(1) both are rational with the same numerator, the rational form is exact, and the memory Hankel rank is d − 1 for any rational autocovariance of rank d. | Read |
| NOTE-1030 | Rosas et al. — Software in the natural world | Three exact closures of a coarse-graining Z of a stationary discrete process X, built from computational mechanics. Informational closure (the micro past adds nothing to predicting Z, which is Barnett–Seth dynamical independence) is proved equivalent to causal closure (Z’s ε-machine equals its υ-machine). For spatial maps it implies computational closure (Z’s ε-machine is a coarse-graining of X’s), and the converse fails. For Markov chains closure is tied to strong lumpability. The proofs of the main equivalences hold, re-derived in THEORY-820 and THEORY-822. The lattice claims are asserted and in part false. Several examples contain slips. The modular random walk closes by symmetry, not modularity. Nothing is said about approximate closure. | Read |
| NOTE-1059 | Lee-Youngzie et al. — Towards a mathematical structure of global phenomenal consciousness | A hypothesis article with no data. It maps the parts of an experience, ordered by subsumption into a finite topology, to similarity-judgement matrices of their qualia by a presheaf, and reads compatibility of narrow qualia on overlaps as the condition for phenomenal binding, and the presheaf’s being a sheaf as the broad quale’s being determined by its parts. It reads a sheaf failure as disunity in binocular rivalry and as context dependence in the Ebbinghaus illusion, and gives no rule for telling the two apart. Its worked topology example is malformed, and by its own definitions disagreement on an overlap is not a sheaf failure. | Read |
| NOTE-1099 | Statistical ranking and combinatorial Hodge theory | Rank aggregation as combinatorial Hodge theory. Aggregated pairwise comparisons form an edge flow, and its orthogonal decomposition gives a least-squares global ranking (the gradient), local inconsistency (curl) and global inconsistency (harmonic). The size of the cyclic residual certifies whether any global ranking is meaningful. The framework is cardinal, generalizes Borda, and is not Kemeny. | Read |
| NOTE-1112 | Theory of Value: An Axiomatic Analysis of Economic Equilibrium | Read for §4.6 only. Not every complete preorder has a utility function: the lexicographic ordering of R² has none. Under a continuity assumption (closed upper and lower contour sets), a complete preorder on a connected subset of Rˡ, and more generally on a connected separable space, has a continuous utility function, and conversely. Debreu credits the separable form to Eilenberg (1941), and his own 1954 paper for dropping connectedness on second-countable spaces. Utility is defined only up to increasing transformation. | Skimmed |
| NOTE-1116 | Flows and Decompositions of Games: Harmonic and Potential Games | A finite game is read as a flow, the unilateral payoff differences on the graph of strategy profiles, and the Helmholtz decomposition of that flow splits the game uniquely into potential, harmonic and nonstrategic parts. Potential games are those with no harmonic part. Harmonic games, the pure-conflict part, generically have no pure equilibrium. The nearest potential game is a closed-form l2 projection. | Read |
| NOTE-1155 | Balduzzi et al. — Re-evaluating evaluation | A methods paper with two short applications. It reads evaluation data as antisymmetric matrices. Elo is then the row mean of the log-odds, exact when they are curl-free and movable by copies, and the Schur decomposition gives a multidimensional Elo. Nash averaging scores agents and tasks against the maximum-entropy Nash equilibrium of a zero-sum meta-game. Its proved invariance covers exact copies, and only when that equilibrium is unique. A checked counterexample shows what happens otherwise. | Read |
| NOTE-1215 | Ashwin, Wieczorek, Vitolo & Cox — Tipping points in open systems | A classification and a criterion, not an empirical study. Tipping in an open system ẋ = f(x, λ(t)) is split into B-tipping (a bifurcation of the quasi-static attractor), N-tipping (noise carries the system out of its neighbourhood) and R-tipping, where “the system fails to track a continuously changing quasi-static attractor”. In a linear model with a fixed stable M and a tipping radius R, the attractor never changes stability, yet steady drift r tips the system when ‖M‖⁻¹|r| > R and cannot when ‖M⁻¹‖|r| < R. Three examples, one with a unique globally stable equilibrium, and a global energy balance model show each type alone. Neither N- nor R-tipping needs any change of stability, so there is no reason to expect critical-slowing warnings for them. | Read |
| NOTE-1217 | Stewart, Golubitsky & Pivato — Symmetry groupoids and patterns of synchrony | In a coupled cell network, an equivalence relation on the cells gives a synchrony subspace invariant under every vector field the network admits if and only if it is balanced: every two equivalent cells have an input isomorphism that maps each input to an equivalent one (Theorem 6.5). Symmetry is one source of balance, not the only one, so robust synchrony occurs in networks with no symmetry. The dynamics on such a subspace are admissible for a smaller quotient network (Theorem 9.2), which can have a symmetry the original lacks. The paper treats existence only, not stability, and says so. | Read |
| NOTE-1218 | Torres-Hugas et al. — Cycle holonomy and remote synchronization | For a phase model whose edges carry antisymmetric phase lags, the twisted Laplacian Lα = δα*δα has a zero mode exactly when every cycle holonomy is 0 mod 2π, and otherwise its smallest eigenvalue measures the residual incompatibility; for an isolated ℓ-cycle that eigenvalue is largest at holonomy π. The symmetric Sakaguchi–Kuramoto model is a different model, but on cactus graphs its lag gives each ℓ-cycle a path mismatch (ℓ − 2)α, which an associated connection is built to carry as holonomy. In the triangle–pentagon remote-synchronization motif, the pentagon’s mismatch reaches π at α = π/3, where the locked branch is exactly a two-group state; but the branch has already lost transverse stability at αc ≈ 1.0164, so the holonomy marks geometry, not stability. | Read |
| NOTE-1221 | Robinson — Sheaves are the canonical data structure for sensor integration | An expository paper. It posits six axioms for an integrated sensor system and defines a sheaf as whatever satisfies them, so “canonical” is argued by that axiomatization, not proved; the paper itself calls sheaves one of “many co-universal information representations”. Its tools need less than a sheaf: on a presheaf of pseudometric spaces, the consistency radius of an assignment (the largest disagreement between an observation and the restriction of another) bounds its distance to every global section below by ε/(1 + K) when the restrictions are K-Lipschitz, and data fusion is the search for the nearest global section. In one simulated search-and-rescue scenario fusion cut the crash-location error 2–8×, and the per-restriction discrepancies singled out a biased sensor by inspection. Cohomology, for linear sheaves, flags configurations that can be locally but not globally consistent. | Read |
| NOTE-1222 | Hansen & Ghrist — Opinion dynamics on discourse sheaves | A discourse sheaf gives each agent a private opinion space and each edge a discourse space, with a linear expression map from each agent into each of its edges. Opinion diffusion by the sheaf Laplacian converges to the projection onto the global sections: agreement in what neighbours express, not in what they hold. Letting the expression maps diffuse instead converges to the nearest sheaf in which the fixed opinions agree in expression, which the authors call “learning to lie”. Nothing in the dynamics checks expression against what is held, so the agreement reached is coherence, not accuracy. | Read |
| NOTE-1223 | Baez & Fong — A compositional framework for passive linear networks | Open passive linear circuits (resistors, inductors and capacitors, the latter two through the Laplace transform over the field R(s)) are the morphisms of a hypergraph category Circ of decorated cospans, and black-boxing, which keeps only the Lagrangian relation a circuit imposes between its terminals’ potentials and currents, is a hypergraph functor Circ → LagRel (Thm 1.1, Thm 7.7). The relation is the graph of the differential of the power functional, the Dirichlet form left when the interior nodes are eliminated by the principle of minimum power; series resistors are the worked example. Dirichlet forms alone compose by that elimination but have no identities, so the target is Lagrangian relations. | Read |
| NOTE-1224 | Singer — Angular synchronization | Angles θ₁, …, θₙ are estimated, up to one global phase, from noisy offsets θᵢ − θⱼ of which an unknown fraction are pure outliers, by the top eigenvector of the Hermitian matrix Hᵢⱼ = e^{iδᵢⱼ} or by an SDP like Goemans–Williamson’s. On the complete graph a random-matrix argument puts the eigenvector’s threshold at p > 1/√n good offsets, with the expected squared correlation near 1/(1 + 1/(np²)). Fano’s inequality shows no algorithm can do much better: the information bound is of order √(n/m), and the polynomial method’s threshold is within a factor √((L − 1)/(2 log₂ L)) of it, where 2π/L is the angular resolution, while exhaustive maximum likelihood costs O(mLⁿ). | Read |
| NOTE-1225 | Spivak — Compositional dynamics in learning and mechanics | A category-theoretic syntax, the 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 functorial semantics into polynomial coalgebras, i.e. discrete-time input-output machines. They differ only in the integrator: Φconf stores position and takes a descent step, Φphase stores position and momentum and takes a symplectic-Euler step. Φconf turns a parameterized map plus a loss into gradient descent with backpropagation, and a closed system with inverse-Hessian response into 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. The main technical result is lens internalization, a lax symmetric monoidal functor Lens_C → C. | Read |
| NOTE-1231 | Clarke, Grumberg, Jha, Lu & Veith — Counterexample-guided abstraction refinement | A model checker runs on an existential abstraction that merges concrete states, so a property true of the abstraction is true of the program (Theorem 1). An abstract counterexample is checked against the concrete system by image computation; if it has no concrete counterpart, the first abstract state where it breaks is split to separate the concrete states reachable along it from those with the transition onward, and checking runs again. The coarsest such split is stated to be NP-hard (proof omitted), so a polynomial, non-optimal split is used. For finite-state programs and ACTL* properties whose counterexamples are paths or loops, the loop ends in a real counterexample or a proof. The property is fixed throughout. | Read |
| NOTE-1233 | Hansen & Ghrist — Toward a spectral theory of cellular sheaves | An exploratory programme, not a single result: the Hodge Laplacian of a weighted cellular sheaf, whose kernel in degree k is the cohomology H^k (in degree 0, the global sections), with sheaf versions of spectral graph theory’s basics: harmonic extension, an eigenvalue bound for normalized sheaves, interlacing, pushforward and covering spectra, product spectra, effective resistance and a sparsification theorem. The heat flow ẋ = −Δ^k x converges exponentially to the orthogonal projection onto the harmonic cochains (Prop. 8.1). Kron reduction fails for general sheaves, because an interior vertex can impose constraints on the boundary that no pairwise sheaf expresses. A Cheeger inequality is not proved; the applications in §8 are sketches. | Read |
| NOTE-1323 | Alvarez-Melis et al. — Hyperbolic OT hierarchy matching | Two Poincaré embeddings of the same WordNet hierarchy, from different random seeds, differ by more than a hyperbolic isometry: branches move relative to one another and pairwise distances change. Euclidean aligners and hyperbolic OT with an orthogonal map therefore fail (P@10 at most 11% across languages). Registering one space with a hyperbolic neural network trained through a Sinkhorn divergence recovers P@10 of 38–60% across languages and P@1 of 22% when matching a hierarchy to itself. The matching uses the symmetric hyperbolic distance, and nothing in it preserves the direction of entailment. | Read |
| NOTE-1324 | Alvarez-Melis et al. — Structured optimal transport | Replaces the linear cost of discrete optimal transport by the Lovász extension of a submodular set function on assignments, so that the cost of a coupling depends on which assignments are made together. The problem stays convex and is a zero-sum game against cost matrices in the base polytope, solved by saddle-point mirror prox with Sinkhorn and base-polytope projections; it reduces to ordinary OT when the set function is modular. Its own example cost, eq. (12), is not concave, so the cost it defines is not submodular. | Read |
| NOTE-1325 | Flamary et al. — POT | A software paper for POT, an MIT-licensed Python library of optimal transport solvers: exact transport by a C++ solver, entropic transport by Sinkhorn and stabilized variants, Wasserstein barycenters, partial and unbalanced transport, regularized transport, Gromov–Wasserstein, sliced Wasserstein and domain adaptation, with tests covering 92% of lines. It reports no experiments. Fused and partial Gromov–Wasserstein and co-optimal transport, which the record uses POT for, are not in this paper; they came in later releases. | Read |
| NOTE-1326 | Sarkar, Mukherjee and Sato — Relational composition as parallel transport | The companion paper’s premise, stated by its authors: per-entity frame freedom is “a chosen inductive bias, not a consequence of fresh names”. Under it, scores go through ⟨xᵢ, Tᵢⱼxⱼ⟩ and transports must be supplied. Its parsed-benchmark controls show the premise’s orthogonality helping on StepGame and costing 64.6 points on CLUTRR. | Skimmed |
| NOTE-1327 | Yu et al. — AvAtar active OT alignment | For alignment by entropic optimal transport, AvAtar picks which source objects to send to an oracle, who names each one’s true match. It scores a candidate by the derivative of a proxy utility of the transport plan (how near a permutation, or how consistent with the graphs, the plan is) with respect to supervising each possible match, averaged over matches by the current plan. The derivative is exact, from one singular positive semidefinite linear system solved by conjugate gradients. At a fixed budget of 20% extra labels on top of 20% given, it beats uncertainty baselines by up to 2.8 MRR points and random choice by up to 16, without variances reported. | Read |
| NOTE-1328 | Vayer et al. — Fused Gromov–Wasserstein | FGW 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. For α in (0, 1), q = 1 and distance-matrix structures it is a metric that vanishes exactly when a bijection preserves weights, features and structure. As α tends to 0 or 1 it tends to W or GW. On benchmark graph classification it is competitive with graph kernels, but the supplement shows Wasserstein on features alone matching or beating it on several datasets. | Read |
| NOTE-1329 | Hashimoto et al. — Word embeddings as metric recovery | If text were generated by a random walk whose steps depend on distance in a latent Euclidean or Riemannian space, negative log co-occurrence over a window that grows with the vocabulary, less a term for each word, would approach the squared latent distance. GloVe, word2vec and the PMI-SVD method then have the true embedding among their optima, up to isometry and unigram terms, and a negative-binomial regression on the same relation embeds words and manifolds about as well. On real corpora the evidence for a Euclidean semantic space is a correlation of 0.75 between PMI and human similarity ratings and two nearest-neighbour statistics; the authors say the random walk is not a model of language. | Read |
| NOTE-1330 | Chowdhury & Mémoli — Network Gromov–Wasserstein | The Gromov–Wasserstein distance, defined by minimizing the distortion of a coupling, is a pseudometric on all measure networks: any Polish node set with a probability measure and any bounded, possibly asymmetric, edge-weight function. It is zero exactly when the two networks are weakly isomorphic, that is, when both are images of one network that preserves weights and measures. Cheap invariants give a hierarchy of lower bounds, and the third lower bound, a transport problem whose costs are themselves transport problems, reduces to one-dimensional transport in closed form. The bounds are one-sided: networks can agree on every invariant and still be apart. | Read |
| NOTE-1331 | Sarkar, Mukherjee and Sato — Attention as lattice gauge theory | Under per-entity O(d) frame freedom, transports carry invariant information only around cycles, a triangle’s invariants are class functions of its holonomy, and an equivariant edge update is Ψ(H)Tᵢⱼ with Ψ conjugation-equivariant: a sound conditional theorem, standard in lattice gauge theory and synchronization, whose “unique form” is a family. The untrained S4 experiments show composition doing the work and Wilson-loop routing adding 0.6–7.8 points; the 0.99 AUC is a ceiling the data’s construction sets, as the paper itself computes. | Read |
| NOTE-1332 | Bodnar et al. — Neural sheaf diffusion | Bodnar et al. read sheaf diffusion on a graph as a projection onto the harmonic space of a sheaf Laplacian and ask which classes of restriction maps can make that projection linearly separate a node labelling. A symmetric one-dimensional sheaf, which is a weighted graph Laplacian, cannot separate the two sides of a balanced bipartite graph; signed maps separate any two classes; no one-dimensional sheaf separates three; diagonal maps with d ≥ C stalks, and O(d) bundles for C ≤ 2d at d = 2 and 4, separate any number. Every one of these sheaves is built from the labels. Learned from features, such sheaves give competitive node classification on six small heterophilic benchmarks. The claim that the O(d) construction exists only at d = 2 and 4 is wrong: it works at d = 8 too. | Read |
| NOTE-1333 | Alvarez-Melis et al. — OT with global invariances | Optimal transport between two unregistered spaces is posed as a joint minimisation over couplings and one global linear map from a Schatten ℓp ball. Under angle preservation or whitening it reduces to maximising a dual Schatten norm of the transported cross-covariance XΓYᵀ: the nuclear norm for orthogonal maps (p = ∞) and the Frobenius norm for p = 2, which is exactly GW with cosine similarity and squared loss. Alternating SVD and Sinkhorn steps with annealed entropy solve it without adversarial initialisation. On word translation it is level with GW and behind self-learning on nine of ten directions. | Read |
| NOTE-1334 v2 | Zeng et al. — HOT multi-network alignment | Aligns K networks at once by one multi-marginal coupling tensor. The networks are first co-clustered through a fused Gromov–Wasserstein barycenter, then each cluster’s nodes are aligned by multi-marginal fused GW with a cost from anchor-based random-walk embeddings, solved by a KL proximal point method. The joint objective at any tensor is the sum of the pairwise objectives at its marginals, so it bounds the sum of pairwise optima from above. On three-network benchmarks whose true correspondences are consistent by construction, it beats pairwise and multi-network baselines in pairwise and high-order hits; no experiment isolates the joint coupling from the new cost and the clustering. | Read |
| NOTE-1335 | Redko et al. — CO-Optimal Transport | COOT matches two data matrices of different shapes by optimizing two couplings at once, one between rows (samples) and one between columns (features), under a bilinear cost. With equal sizes and uniform weights it is zero exactly when the matrices are equal up to a permutation of rows and a separate permutation of columns, so its plans are determined at best up to the matrix’s joint row–column symmetries; the paper proves no uniqueness. For symmetric matrices it is a lower bound on GW, equal to it for squared Euclidean distance matrices. | Read |
| NOTE-1336 | Alvarez-Melis & Jaakkola — Gromov-Wasserstein word alignment | Unsupervised word translation is cast as a Gromov–Wasserstein problem between the two languages’ intra-language cosine-similarity matrices, solved by entropic projected gradient (each step a Sinkhorn problem), with no anchors. It matches adversarial-plus-refinement methods on related languages trained on comparable corpora at a fraction of the cost, and does markedly worse on English–Russian and English–Finnish. The objective sees only within-space similarities, so it cannot distinguish a correspondence from one composed with a symmetry of either space; the paper does not say so. | Read |
| NOTE-1342 | Felekis et al. — Causal optimal transport of abstractions | COTA learns a stochastic map from a low-level to a high-level causal model’s state space from samples of paired interventional distributions, given both DAGs and the map between interventions, by solving one entropic optimal-transport problem per intervention with a cost that rewards compatibility with the paired interventions and a term meant to impose do-calculus relations between plans. As written, that term depends only on the plans’ marginals, which the transport constraints fix, so it is constant on the feasible set and its weight acts only by rescaling the other terms. The reported gains come from the compatibility cost when interventions are many; with few, a Hamming cost does better. | Read |
| NOTE-1343 | Peyré et al. — Gromov–Wasserstein averaging | Defines an entropic Gromov–Wasserstein discrepancy between similarity matrices of any size and loss, and their barycenter as a Fréchet mean of user-chosen size and weights. It is computed by alternating Sinkhorn- based coupling updates with a closed-form matrix update that, for square loss, averages the couplings’ realignments of the inputs with the barycenter weights. The algorithm is proved to keep PSD and infinitely divisible kernels in their classes; its convergence at the step size used is observed, not proved, and only stationary points are sought. | Read |
| NOTE-1348 | Goguen & Burstall — Institutions | An institution is a category of signatures with a sentence functor, a contravariant model functor and a satisfaction relation per signature, subject to one axiom, the Satisfaction Condition: a translated sentence holds in a model exactly when the original holds in the model’s reduct. From that axiom alone theories and model classes form a Galois connection, theories can be glued by colimits whenever signatures can (Theorem 11), constraints that force standard or free interpretations can be added as sentences without losing this, and an institution morphism with surjective model translation lets a prover for one logic decide entailment for another (Proposition 35). The Satisfaction Condition is a criterion of correct translation relative to a given model translation; it says nothing about where that model translation comes from. | Read |
| NOTE-1350 | Sengupta, Tozzi, Cooray, Douglas and Friston — Towards a neuronal gauge theory | An essay proposing that variational free-energy minimization be cast as a gauge theory. The only connection it names is the Levi-Civita connection of the Fisher information metric on the manifold of a belief’s sufficient statistics, so its gauge freedom is reparameterization of the variational family; its claimed payoff is that attention is precision weighting forced by that manifold’s curvature. No gauge group, invariant Lagrangian or compensating field is written down. | Read |
| NOTE-1379 | Hofbauer and Schreiber — Permanence via invasion graphs | For ecological ODEs whose invasion growth rates have well-defined, nonzero signs on each subcommunity, it defines the invasion graph (edges S → T when every species in T but not S can invade S and every species lost cannot invade T) and proves that when the graph is acyclic, robust permanence holds if, and essentially only if, each species i has a positive invasion growth rate at every −i community. With cycles, signs are not enough. | Read |
| NOTE-1395 | Mehta, Cui, Wang and Marsland — Constrained optimization as ecological dynamics | Any minimization of f(R) over R ≥ 0 under constraints g_i(R) ≤ 0 has the KKT points of a consumer–resource dynamics as its steady states: variables are resources, each constraint is a species whose abundance is its Lagrange multiplier, and binding constraints are the surviving species. For quadratic programs the dynamics is MacArthur’s model, and the Lagrangian dual is a generalized Lotka–Volterra system. A cavity solution then gives the statistics of random quadratic programs, in agreement with numerics at M = 100. | Read |
| NOTE-1401 | Désidéri — Multiple-gradient descent algorithm (MGDA) | Defines Pareto-stationarity (some convex combination of the criteria’s gradients is zero), proves it necessary for Pareto-optimality, and shows that when it fails the minimum-norm element ω of the gradients’ convex hull gives a direction −ω along which every criterion decreases, with (u, ω) ≥ ‖ω‖² for every u in the hull. Steepest descent along −ω, the MGDA, has a subsequence converging to a Pareto-stationary point when the criteria are positive and coercive. The direction depends on how the gradients are scaled; the author calls the right scaling open. | Read |
| NOTE-1422 | Serván and Allesina — Tractable models of ecological assembly | For competitive Lotka–Volterra communities with A = D(v)BD(w), B symmetric with positive entries, assembled by rare, low-density, well-separated invasions, MacArthur’s Lyapunov function V orders the feasible sub-communities, so the assembly graph is acyclic and every feasible sub-community can be assembled without extinctions; with the paper’s Theorem 1 this makes sequential and all-at-once assembly reach the same end-points. One end-point when B is positive definite, several when it is not. | Read |
| NOTE-1427 | Apt — Potential Games (slides) | Apt’s slides define exact, ordinal and generalized ordinal potentials and the finite improvement property, and state, crediting Monderer and Shapley 1996, that a finite game has a generalized ordinal potential if and only if it has the FIP; one direction is proved and the converse sketched by counting improvement sequences. A two-by-two game with the FIP and no ordinal potential shows the equivalence cannot be with ordinal potentials. | Read |
| NOTE-1441 | Coifman et al. — Diffusion maps (full reading) | With the equations visible, the paper’s diffusion distance is the distance induced by the symmetric kernel ã^(m), D_m² = ã^(m)(x, x) + ã^(m)(y, y) − 2ã^(m)(x, y), with ã’s eigenvalues written λ_i², so that the map x ↦ (λ_i^m φ_i(x)) is an exact isometry for it. The truncation is claimed with a relative error 1 + O(e^{−αm}). Theorem 2.1, the ε → 0 heat-kernel limit of the α = 1 normalization, and the α-family (Eq. 14) are stated without proof; the sampling condition is that each ball of radius √ε hold enough points. Eq. 5 and Eq. 14 give the α = 0 potential opposite signs. | Read |
| NOTE-1465 | Santolin et al. — Freidlin-Wentzell and stochastic thermodynamics | For overdamped diffusions with weak noise and a nonconservative force, the Freidlin–Wentzell quasi-potential is expanded about the detailed-balance free energy, each order integrated along the detailed-balanced relaxation, and the first order saturates the emergent second law. The weak-noise escape rate from an attractor is bracketed below by minus the entropy produced in relaxing into it and above by the entropy produced along the most likely escape path. The bracket closes at detailed balance and widens with the nonconservative force; in a sheared double well it stays usefully narrow for weak shear. | Read |
| NOTE-1466 | Saligrama — information before commitment in safe adaptive control | A safe adaptive controller must stay safe under every model it has not ruled out. Its first action that leaves no safe continuation under an alternative is a commitment, and the evidence for it must be in hand before the action: what that action reveals comes too late. With the information gathered before commitment measured as a KL divergence between stopped records, a data-processing argument bounds any uniformly safe policy’s regret below by the cost of not committing with probability ½e^{−I_pre} − α. So bounded pre-commitment information leaves a fixed fraction of the oracle gap, and a linear gap gives linear regret. Safe recovery at O(√(T log T)) is proved when a safe experiment is repeatable, informative and returns to a common state. | Read |
| NOTE-1476 | Dal Cengio, Lecomte & Polettini — Geometry of nonequilibrium reaction networks | Extends the Hill–Schnakenberg cycle analysis from linear networks (graphs) to interacting ones (hypergraphs) by row reduction of the stoichiometric matrix: cycles span its kernel and carry steady currents and non-conservative affinities, cocycles span the orthogonal complement and carry transient currents and conservative forces, with an integrator that recovers chemical potentials. One chemostat cannot drive a network; two can. Suppressing the reactions of a cocycle creates a new conserved quantity. Near equilibrium the relaxation and driven-response Onsager matrices share their spectrum, and the cycle decomposition gives a linear-regime reconstruction of metabolic fluxes that respects Kirchhoff’s laws. | Read |
| NOTE-1477 | Kolathaya & Ames — input-to-state safe control barrier functions | Defines input-to-state safety of a set C = {h ≥ 0}: under a bounded input disturbance d, the larger set C_d = {h + γ(‖d‖∞) ≥ 0} is forward invariant, for a class-K γ. A barrier condition with a disturbance term, ḣ ≥ −α(h) − ι(‖d‖), implies it (Theorem 1). Given an ordinary control barrier function, adding L_g hᵀ to a safeguarding controller, or the constraint term −εL_g h L_g hᵀ in a CLF–CBF quadratic program, yields such a function with ι(‖d‖) = ‖d‖²/(4ε) (Theorem 2). The enlargement is a class-K function of the disturbance bound in barrier units, quadratic in the universal construction; it is not proportional to the bound. | Read |
| NOTE-1503 | Ben-Tal, Goryashko, Guslitzer & Nemirovski — Adjustable robust solutions | For an uncertain LP whose variables split into here-and-now ones and adjustable ones chosen after the data are revealed, the Adjustable Robust Counterpart (∀ data ∃ adjustment) can be much less conservative than the usual Robust Counterpart (∃ one solution ∀ data); they coincide under constraint-wise uncertainty. The ARC is usually NP-hard; restricting adjustments to affine functions of the data (AARC) is tractable with fixed recourse. Each variable may have an information basis, a prescribed portion of the data it can depend on; in an inventory example, poorer information bases raise cost and the non-adjustable RC is infeasible. | Read |
| NOTE-1504 | Heymann & Vanden-Eijnden — The geometric minimum action method | The Freidlin–Wentzell quasi-potential, a double minimization over paths and their duration, is rewritten as a single minimization over curves of a parametrization-free action, which has minimizers when the end points are equilibria and is computed by a string-method-like relaxation (gMAM). For a diffusion it is V = 2 inf ∫ |b| sin²(θ/2) ds. In gradient systems with identity noise the most likely transition paths are parallel to the drift; in the nongradient Maier–Stein model they depart from the heteroclinic orbit once the drift is far enough from gradient. | Skimmed |