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

Abstract

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.

Lifting Dual Connections with the Riemann Extension

“…The set of solutions of the above equation will be denoted as J ∇ * ∇ , following the convention of [15]. When both X, Y are in J ∇ * ∇ , the tensor W X,Y simplifies to:…”

Topological Information Data Analysis: Poincare-Shannon Machine and Statistical Physic of Finite Heterogeneous Systems

“…Baez, Fritz and Leinster defining the category of finite probability and using Fadeev axiomatization, could show that the only family of functions that has the functorial property is Shannon information loss [31,32]. Boyom, basing his approach on information and Koszul geometry, developed a more geometrical view of statistical models that notably considers foliations in place of the random variables [33]. Introducing a deformation theoretic framework, and chain complex of random variables, Drumond-Cole, Park and Terilla [34][35][36] could construct a homotopy probability theory for which the cumulants coincide with the morphisms of the homotopy algebras.…”

Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature

“…The geometry of statistical manifolds also provides interesting issues for differential geometry, statistics, machine learning, etc. (see, e.g., [27][28][29][30][31][32]). In particular, the differential geometry field is focused on topics such as submanifold theory of statistical manifolds [33], Hessian geometry [34], statistical submersions [35], complex manifold theory of statistical manifolds ( [29,36,37]), contact theory on statistical manifolds [38], and quaternionic theory on statistical manifolds [39].…”

“…Moreover, the equality cases of Equations (27) and (28) hold identically at all points if and only if the imbedding curvature tensors h and h * of the submanifold associated with the dual connections∇ and∇ * satisfy Equation (17), i.e., M is a totally geodesic submanifold with respect to the Levi-Civita connection. Example 1.…”