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Based on Koszul theory of sharp convex cone and its hessian geometry, Information Geometry metric is introduced by Koszul form as hessian of Koszul–Vinberg Characteristic function logarithm (KVCFL). The front of the Legendre mapping of this KVCFL is the graph of a convex function, the Legendre transform of this KVCFL. By analogy in thermodynamic with Dual Massieu–Duhem potentials (Free Energy and Entropy), the Legendre transform of KVCFL is interpreted as a "Koszul Entropy". This Legendre duality is considered in more general framework of Contact Geometry, the odd-dimensional twin of symplectic geometry, with Legendre fibration and mapping. Other analogies will be introduced with large deviation theory with Cumulant Generating and Rate functions (Legendre duality by Laplace Principle) and with Legendre duality in Mechanics between Hamiltonian and Lagrangian. In all these domains, we observe that the "Characteristic function" and its derivatives capture all information of random variable, system or physical model. We present two extensions of this theory with Souriau's Geometric Temperature deduced from covariant definition of thermodynamic equilibriums, and with Balian quantum Fisher metric defined and built as hessian of von Neumann entropy. Finally, we apply Koszul geometry for Symmetric/Hermitian Positive Definite Matrices cones, and more particularly for covariance matrices of stationary signal that are characterized by specific matrix structures: Toeplitz Hermitian Positive Definite Matrix structure (covariance matrix of a stationary time series) or Toeplitz-Block-Toeplitz Hermitian Positive Definite Matrix structure (covariance matrix of a stationary space–time series). By extension, we introduce a new geometry for non-stationary signal through Fréchet metric space of geodesic paths on structured matrix manifolds. We conclude with extensions towards two general concepts of "Generating Inner Product" and "Generating Function".

DOI: 10.1007/978-3-319-05317-2_7

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