The Homological Nature of Entropy

Baudot, Pierre Yves; Bennequin, Daniel

doi:10.3390/e17053253

Cited by 56 publications

(159 citation statements)

References 36 publications

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“…Pierre Baudot and Daniel Bennequin recently discovered that the entropy function has a homological nature [31]. We recall that in 2002 Peter McCullagh raised a fundamental geometric-topological question in the theory of information: What Is a Statistical Model?…”

Section: Ige Stands For Information Geometrymentioning

confidence: 98%

“…This may be interpreted as a variant of the topology of the information. Another approach is to be found in Baudot-Bennequin [31].…”

Section: Theorem 1 There Exists a Functormentioning

confidence: 99%

“…For another viewpoint see [16,31]. Our purpose is to show the theory of statistical models has a homological nature in the category F B(Γ, Ξ).…”

Section: Then We Havementioning

confidence: 99%

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Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Boyom

2016

Entropy

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Let us begin by considering two book titles: A provocative title, What Is a Statistical Model? McCullagh (2002) and an alternative title, In a Search for Structure. The Fisher Information. Gromov (2012). It is the richness in open problems and the links with other research domains that make a research topic exciting. Information geometry has both properties. Differential information geometry is the differential geometry of statistical models. The topology of information is the topology of statistical models. This highlights the importance of both questions raised by Peter McCullagh and Misha Gromov. The title of this paper looks like a list of key words. However, the aim is to emphasize the links between those topics. The theory of homology of Koszul-Vinberg algebroids and their modules (KV homology in short) is a useful key for exploring those links. In Part A we overview three constructions of the KV homology. The first construction is based on the pioneering brute formula of the coboundary operator. The second construction is based on the theory of semi-simplicial objects. The third construction is based on the anomaly functions of abstract algebras and their abstract modules. We use the KV homology for investigating links between differential information geometry and differential topology. For instance, "dualistic relation of Amari" and "Riemannian or symplectic Foliations"; "Koszul geometry" and "linearization of webs"; "KV homology" and "complexity of models". Regarding the complexity of a model, the challenge is to measure how far from being an exponential family is a given model. In Part A we deal with the classical theory of models. Part B is devoted to answering both questions raised by McCullagh and B Gromov. A few criticisms and examples are used to support our criticisms and to motivate a new approach. In a given category an outstanding challenge is to find an invariant which encodes the points of a moduli space. In Part B we face four challenges. (1) The introduction of a new theory of statistical models. This re-establishment must answer both questions of McCullagh and Gromov; (2) The search for an characteristic invariant which encodes the points of the moduli space of isomorphism class of models; (3) The introduction of the theory of homological statistical models. This is a pioneering notion. We address its links with Hessian geometry; (4) We emphasize the links between the classical theory of models, the new theory and Vanishing Theorems in the theory of homological statistical models. Subsequently, the differential information geometry has a homological nature. That is another notable feature of our approach. This paper is dedicated to our friend and colleague Alexander Grothendieck.

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Section: Ige Stands For Information Geometrymentioning

confidence: 98%

“…This may be interpreted as a variant of the topology of the information. Another approach is to be found in Baudot-Bennequin [31].…”

Section: Theorem 1 There Exists a Functormentioning

confidence: 99%

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Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Boyom

2016

Entropy

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“…The structure of a dataset is mathematically represented as a simplicial complex, hence providing us the opportunity to apply the rich apparatus of algebraic topology [1]. Based on the Shannon information measure [2], we define the multi-dimensional entropies and depart from the hitherto research that relates the concepts of algebraic topology and information theory, such as the cohomological nature of information [3], persistent entropy [4,5], the graph's topological entropy [6] or higher-order spectral entropy [7]. The introduced vector-like entropies capture the (in)distinguishability of different layers of the rigorously partitioned structure of the dataset and, further, indicate the way that the changes of data affect the internal structural relationships of the dataset.…”

Section: Introductionmentioning

confidence: 99%

Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets

Maletić

Zhao

2017

Entropy

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Abstract:The emergence of complex datasets permeates versatile research disciplines leading to the necessity to develop methods for tackling complexity through finding the patterns inherent in datasets. The challenge lies in transforming the extracted patterns into pragmatic knowledge. In this paper, new information entropy measures for the characterization of the multidimensional structure extracted from complex datasets are proposed, complementing the conventionally-applied algebraic topology methods. Derived from topological relationships embedded in datasets, multilevel entropy measures are used to track transitions in building the high dimensional structure of datasets captured by the stratified partition of a simplicial complex. The proposed entropies are found suitable for defining and operationalizing the intuitive notions of structural relationships in a cumulative experience of a taxi driver's cognitive map formed by origins and destinations. The comparison of multilevel integration entropies calculated after each new added ride to the data structure indicates slowing the pace of change over time in the origin-destination structure. The repetitiveness in taxi driver rides, and the stability of origin-destination structure, exhibits the relative invariance of rides in space and time. These results shed light on taxi driver's ride habits, as well as on the commuting of persons whom he/she drove.

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“…The case x ∈[1,2] corresponds to Lemma 4.3. Suppose it is valid on [n − 1, n], for certain n ≥ 2; for x ∈ [n, n + 1],…”

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confidence: 99%

A functional equation related to generalized entropies and the modular group

Bennequin

Vigneaux

2020

Aequat. Math.

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We solve a functional equation connected to the algebraic characterization of generalized information functions. In order to prove the symmetry of the solution, we are led to the study of two homographies that generate the modular group; the group acts on a related functional equation. This suggest a more general relation between conditional probabilities and arithmetic.Mathematics Subject Classification (2010). Primary 97I70, 94A17 .

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The Homological Nature of Entropy

Cited by 56 publications

References 36 publications

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets

A functional equation related to generalized entropies and the modular group

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