publications
publications by categories in reversed chronological order. generated by jekyll-scholar.
2026
- PreprintProbing neural audio codecs for distinctions among English nuclear tunesJuan Pablo Vigneaux, and Jennifer ColearXiv preprint arXiv:2603.14035, 2026
State-of-the-art spoken dialogue models (Défossez et al. 2024; Schalkwyk et al. 2025) use neural audio codecs to “tokenize” audio signals into a lower-frequency stream of vectorial latent representations, each quantized using a hierarchy of vector codebooks. A transformer layer allows these representations to reflect some time- and context-dependent patterns. We train probes on labeled audio data from Cole et al. (2023) to test whether the pitch trajectories that characterize English phrase-final (nuclear) intonational tunes are among these patterns. Results: Linear probes trained on the unquantized latents or some of the associated codewords yield above-chance accuracy in distinguishing eight phonologically specified nuclear tunes with monotonal pitch accents (top average test accuracy (TATA): 0.31) and the five clusters of these tunes that are robust in human speech production and perception (TATA: 0.45). Greater accuracy (TATAs: 0.74-0.89) is attained for binary distinctions between classes of rising vs. falling tunes, respectively used for questions and assertions. Information about tunes is spread among all codebooks, which calls into question a distinction between ‘semantic’ and ‘acoustic’ codebooks found in the literature. Accuracies improve with nonlinear probes, but discrimination among the five clusters remains far from human performance, suggesting a fundamental limitation of current codecs.
2025
- JournalThe Magnitude of Categories of Texts Enriched by Language ModelsTai-Danae Bradley, and Juan Pablo VigneauxTheory and Applications of Categories, Nov 2025
The purpose of this article is twofold. Firstly, we use the next-token probabilities given by a language model to explicitly define a category of texts in natural language enriched over the unit interval, in the sense of Bradley, Terilla, and Vlassopoulos. We consider explicitly the terminating conditions for text generation and determine when the enrichment itself can be interpreted as a probability over texts. Secondly, we compute the Möbius function and the magnitude of an associated generalized metric space of texts. The magnitude function of that space is a sum over texts (prompts) of the t-logarithmic (Tsallis) entropies of the next-token probability distributions associated with each prompt, plus the cardinality of the model’s possible outputs. A suitable evaluation of the magnitude function’s derivative recovers a sum of Shannon entropies, which justifies seeing magnitude as a partition function. Following Leinster and Shulman, we also express the magnitude function of the generalized metric space as an Euler characteristic of magnitude homology and provide an explicit description of the zeroeth and first magnitude homology groups.
- PreprintEntropies associated with orbits of finite groupsRyan Leal, Jingtong Sun, and Juan Pablo VigneauxarXiv preprint arXiv:2512.02257, Nov 2025
For certain groups, parabolic subgroups appear as stabilizers of flags of sets or vector spaces. Quotients by these parabolic subgroups represent orbits of flags, and their cardinalities asymptotically reveal entropies (as rates of exponential or superexponential growth). The multiplicative "chain rules" that involve these cardinalities induce, asymptotically, additive analogues for entropies. Many traditional formulas in information theory correspond to quotients of symmetric groups, which are a particular kind of reflection group; in this case, the cardinalities of orbits are given by multinomial coefficients and are asymptotically related to Shannon entropy. One can treat similarly quotients of the general linear groups over a finite field; in this case, the cardinalities of orbits are given by q-multinomials and are asymptotically related to the Tsallis 2-entropy. In this contribution, we consider other finite reflection groups as well as the symplectic group as an example of a classical group over a finite field (groups of Lie type). In both cases, the groups are classified by Dynkin diagrams into infinite series of similar groups A_n, B_n, C_n, D_n and a finite number of exceptional ones. The A_n series consists of the symmetric groups (reflection case) and general linear groups (Lie case). Some of the other series, studied here from an information-theoretic perspective for the first time, are linked to new entropic functionals.
2024
- PreprintA combinatorial approach to categorical Möbius inversion and pseudoinversionJuan Pablo VigneauxarXiv preprint 2407.14647, Nov 2024
We use Cramer’s formula for the inverse of a matrix and a combinatorial expression for the determinant in terms of paths of an associated digraph (which can be traced back to Coates) to give a combinatorial interpretation of Möbius inversion whenever it exists. Every Möbius coefficient is a quotient of two sums, each indexed by certain collections of paths in the digraph. Our result contains, as particular cases, previous theorems by Hall (for posets) and Leinster (for skeletal categories whose idempotents are identities). A byproduct is a novel expression for the magnitude of a metric space as sum over self-avoiding paths with finitely many terms. By means of Berg’s formula, our main constructions can be extended to Moore-Penrose pseudoinverses, yielding an analogous combinatorial interpretation of Möbius pseudoinversion and, consequently, of the magnitude of an arbitrary finite category.
2023
- ProceedingsOn the Entropy of Rectifiable and Stratified MeasuresJuan Pablo VigneauxIn Nielsen, Frank and Barbaresco, Frédéric, Geometric Science of Information: 6th International Conference, GSI 2023, Lecture Notes in Computer Science, 14071, 338–346, Nov 2023
We summarize some results of geometric measure theory concerning rectifiable sets and measures. Combined with the entropic chain rule for disintegrations (Vigneaux, 2021), they account for some properties of the entropy of rectifiable measures with respect to the Hausdorff measure first studied by (Koliander et al., 2016). Then we present some recent work on stratified measures, which are convex combinations of rectifiable measures. These generalize discrete-continuous mixtures and may have a singular continuous part. Their entropy obeys a chain rule, whose “conditional term” is an average of the entropies of the rectifiable measures involved. We state an asymptotic equipartition property (AEP) for stratified measures that shows concentration on strata of a few “typical dimensions” and that links the conditional term of the chain rule to the volume growth of typical sequences in each stratum.
- ProceedingsCategorical magnitude and entropyStephanie Chen, and Juan Pablo VigneauxIn Nielsen, Frank and Barbaresco, Frédéric, Geometric Science of Information: 6th International Conference, GSI 2023, Lecture Notes in Computer Science, 14071, 278-287, Nov 2023
Given any finite set equipped with a probability measure, one may compute its Shannon entropy or information content. The entropy becomes the logarithm of the cardinality of the set when the uniform probability is used. Leinster introduced a notion of Euler characteristic for certain finite categories, also known as magnitude, that can be seen as a categorical generalization of cardinality. This paper aims to connect the two ideas by considering the extension of Shannon entropy to finite categories endowed with probability, in such a way that the magnitude is recovered when a certain choice of "uniform" probability is made.
- JournalTypicality for stratified measuresJuan Pablo VigneauxIEEE Transactions on Information Theory, Nov 2023
We define \emphstratified measures on Euclidean space as convex combinations of rectifiable measures. They are possibly singular with respect to the Lebesgue measure and generalize discrete-continuous mixtures. A stratified measure ρcan thus be represented as \sum_i=1^k q_i \rho_i, where (q_1,..,q_k) is a probability vector and each \rho_i is absolutely continuous with respect to the m_i-Hausdorff measure \mu_i on a m_i-rectifiable set E_i (e.g. a smooth m_i-manifold if m_i>0 or a countable set if m_i=0). We introduce a set of strongly typical realizations of ρ^⊗n that occur with high probability; they are supported on a finite union of strata E_i_1\times ⋯\times E_i_n whose dimensions “per factor” concentrate around the \emphmean dimension \sum_i=1^k q_i m_i. For each n, an appropriate sum of Hausdorff measures on the different strata gives a notion of reference “volume”; the exponential growth rate of the typical set’s volume is quantified by Csiszar’s \emphgeneralized entropy of ρwith respect to μ=\sum_i=1^k \mu_i. This entropy satisfies a chain rule; the conditional term is related to the volume growth of the typical realizations in each stratum. Under suitable hypotheses, our notion of mean dimension coincides with Rényi’s information dimension when applied to stratified measures, but the generalized entropy used here differs from Rényi’s dimensional entropy.
- JournalA characterization of generalized multinomial coefficients related to the entropic chain ruleJuan Pablo VigneauxAequationes mathematicae, Nov 2023
There is an asymptotic correspondence between the multiplicative relations among multinomial coefficients and the (additive) recursive property of Shannon entropy known as the chain rule. We show that both types of identities are manifestations of a unique algebraic construction: a 1-cocycle condition in *information cohomology*, an algebraic invariant of presheaves of modules on certain categories of observables. Depending on the coefficients, the 1-cocycles can be information measures (Shannon entropy, Tsallis α-entropy) or generalized (Fontené-Ward) multinomial coefficients. In each case the 1-cocycle condition encodes a system of functional equations. We obtain in particular a combinatorial analogue of the “fundamental equation of information theory”: a simple functional equation that uniquely characterizes the generalized binomial coefficients. The asymptotic correspondence mentioned above extends to any α-entropy and certain multinomial coefficients with compatible asymptotic behavior, shedding new light on the meaning of the chain rule and its deformations.
- JournalA formula for the categorical magnitude in terms of the Moore-Penrose pseudoinverseStephanie Chen, and Juan Pablo VigneauxBull. Belg. Math. Soc. Simon Stevin, Nov 2023
The magnitude of finite categories is a generalization of the Euler characteristic. It is defined using the coarse incidence algebra of rational-valued functions on the given finite category, and a distinguished element in this algebra: the Dirichlet zeta function. The incidence algebra may be identified with the algebra of n×n matrices over the rational numbers, where n is the cardinality of the underlying object set. The Moore-Penrose pseudoinverse of a matrix is a generalization of the inverse; it exists and is unique for any given matrix over the complex numbers. In this article, we derive a new method for calculating the magnitude of a finite category, using the pseudoinverse of the matrix that corresponds to the zeta function. The magnitude equals the sum of the entries of this pseudoinverse.
2021
- ProceedingsEntropy under disintegrationsJuan Pablo VigneauxIn Nielsen, Frank and Barbaresco, Frédéric, Geometric Science of Information: 5th International Conference, GSI 2021, Lecture Notes in Computer Science, 12829, 340–349, Nov 2021
We consider the differential entropy of probability measures absolutely continuous with respect to a given σ-finite reference measure on an arbitrary measurable space. We state the asymptotic equipartition property in this general case; the result is part of the folklore but our presentation is to some extent novel. Then we study a general framework under which such entropies satisfy a chain rule: disintegrations of measures. We give an asymptotic interpretation for conditional entropies in this case. Finally, we apply our result to Haar measures in canonical relation.
- ProceedingsInformation cohomology of classical vector-valued observablesJuan Pablo VigneauxIn Nielsen, Frank and Barbaresco, Frédéric, Geometric Science of Information: 5th International Conference, GSI 2021, Lecture Notes in Computer Science, 12829, 537–546, Nov 2021
We provide here a novel algebraic characterization of two information measures associated with a vector-valued random variable, its differential entropy and the dimension of the underlying space, purely based on their recursive properties (the chain rule and the nullity-rank theorem, respectively). More precisely, we compute the information cohomology of Baudot and Bennequin with coefficients in a module of continuous probabilistic functionals over a category that mixes discrete observables and continuous vector-valued observables, characterizing completely the 1-cocycles; evaluated on continuous laws, these cocycles are linear combinations of the differential entropy and the dimension.
2020
- PreprintExtra-fine sheaves and interaction decompositionsarXiv preprint 2009.12646, Nov 2020
We introduce an original notion of extra-fine sheaf on a topological space, and a variant (hyper-extra-fine) for which Čech cohomology in strictly positive degree vanishes. We provide a characterization of such sheaves when the topological space is a partially ordered set (poset) equipped with the Alexandrov topology. Then we further specialize our results to some sheaves of vector spaces and injective maps, where extra-fineness is (essentially) equivalent to the decomposition of the sheaf into a direct sum of subfunctors, known as interaction decomposition, and can be expressed by a sum-intersection condition. We use these results to compute the dimension of the space of global sections when the presheaves are freely generated over a functor of sets, generalizing classical counting formulae for the number of solutions of the linearized marginal problem (Kellerer and Matúš). We finish with a comparison theorem between the Čech cohomology associated to a covering and the topos cohomology of the poset with coefficients in the presheaf, which is also the cohomology of a cosimplicial local system over the nerve of the poset. For that, we give a detailed treatment of cosimplicial local systems on simplicial sets. The appendixes present presheaves, sheaves and Čech cohomology, and their application to the marginal problem.
- JournalA functional equation related to generalized entropies and the modular groupDaniel Bennequin, and Juan Pablo VigneauxAequationes mathematicae, Nov 2020
We solve a functional equation connected to the algebraic characterization of generalized information functions. To prove the symmetry of the solution, we study a related system of functional equations, which involves two homographies. These transformations generate the modular group, and this fact plays a crucial role in solving the system. The method suggests a more general relation between conditional probabilities and arithmetic.
- JournalInformation structures and their cohomologyJuan Pablo VigneauxTheory and Applications of Categories, Nov 2020
We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings. Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement. We extend Baudot and Bennequin’s definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature. Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions. The only 1-cocycles are Shannon entropy or Tsallis α-entropy, depending on the value of the parameter.
2019
- ThesisTopology of statistical systems: a cohomological approach to information theoryJuan Pablo VigneauxUniversité Sorbonne Paris Cité, Nov 2019
This thesis extends in several directions the cohomological study of information theory pioneered by Baudot and Bennequin. We introduce a topos-theoretical notion of statistical space and then study several cohomological invariants. Information functions and related objects appear as distinguished cohomology classes; the corresponding cocycle equations encode recursive properties of these functions. Information has thus topological meaning and topology serves as a unifying framework. Part I discusses the geometrical foundations of the theory. Information structures are introduced as categories that encode the relations of refinement between different statistical observables. We study products and coproducts of information structures, as well as their representation by measurable functions or hermitian operators. Every information structure gives rise to a ringed site; we discuss in detail the definition of \emphinformation cohomology using the homological tools developed by Artin, Grothendieck, Verdier and their collaborators. Part II studies the cohomology of discrete random variables. Information functions—Shannon entropy, Tsallis α-entropy, Kullback-Leibler divergence—appear as 1-cocycles for appropriate modules of probabilistic coefficients (functions of probability laws). In the combinatorial case (functions of histograms), the only 0-cocycle is the exponential function, and the 1-cocycles are generalized multinomial coefficients (Fontené-Ward). There is an asymptotic relation between the combinatorial and probabilistic cocycles. Part III studies in detail the q-multinomial coefficients, showing that their growth rate is connected to Tsallis 2-entropy (quadratic entropy). When q is a prime power, these q-multinomial coefficients count flags of finite vector spaces with prescribed length and dimensions. We obtain a combinatorial explanation for the nonadditivity of the quadratic entropy and a frequentist justification for the maximum entropy principle with Tsallis statistics. We introduce a discrete-time stochastic process associated to the q-binomial probability distribution that generates finite vector spaces (flags of length 2). The concentration of measure on certain \emphtypical subspaces allows us to extend Shannon’s theory to this setting. Part IV discusses the generalization of information cohomology to continuous random variables. We study the functoriality properties of conditioning (seen as disintegration) and its compatibility with marginalization. The cohomological computations are restricted to the real valued, gaussian case. When coordinates are fixed, the 1-cocycles are the differential entropy as well as generalized moments. When computations are done in a coordinate-free manner, with the so-called \emphgrassmannian categories, we recover as the only degree-one cohomology classes the entropy and the dimension. This constitutes a novel algebraic characterization of differential entropy.
- JournalInformation theory with finite vector spacesJuan Pablo VigneauxIEEE Transactions on Information Theory, Nov 2019
Whereas Shannon entropy is related to the growth rate of multinomial coefficients, we show that the quadratic entropy (Tsallis 2-entropy) is connected to their q-deformation; when q is a prime power, these q-multinomial coefficients count flags of finite vector spaces with prescribed length and dimensions. In particular, the q-binomial coefficients count vector subspaces of given dimension. We obtain in this way a combinatorial explanation for the nonadditivity of the quadratic entropy, which arises from a recursive counting of flags. We show that statistical systems whose configurations are described by flags provide a frequentist justification for the maximum entropy principle with Tsallis statistics. We introduce then a discrete-time stochastic process associated to the q-binomial probability distribution, that generates at time n a vector subspace of \mathbb F_q^n (here \mathbb F_q is the finite field of order q). The concentration of measure on certain "typical subspaces" allows us to extend the asymptotic equipartition property to this setting. The size of the typical set is quantified by the quadratic entropy. We discuss the applications to Shannon theory, particularly to source coding, when messages correspond to vector spaces.