Séance Séminaire

Colloquium de Mathématiques

Géométrie de l'Information de Koszul et Thermodynamique des groupes de Lie de Souriau: application à la médiane de matrices structurées de type Toeplitz hermitiennes définies positives

The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from "Characteristic Functions", was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of "Information Geometry" theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean "Moment map" by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. We will synthetize the analogies between both Koszul and Souriau models, and will reduce their definitions to the exclusive Cartan "Inner Product". Interpreting Legendre transform as Fourier transform in (Min,+) algebra, we conclude with a definition of Entropy given by a relation mixing Fourier/Laplace transforms: Entropy = (minus) Fourier(Min,+) o Log o Laplace(+,X). In the second part, we will apply Koszul Information Geometry to compute the median of a set of Toeplitz and Toeplitz-Block-Toeplitz Hermitian Positive Definite Matrices (THPD and TBTHPD matrices). This median is defined geometrically as Frechet's median in metric space and solved by Weiszfeld/Karcher flow. In a first step, the problem is solved for HPD matrices without Toeplitz constraints introduced as a particular case of median computation in Cartan-Siegel Homogenous Bounded Domains where Information Geometry metric is introduced in the framework of Symplectic geometry. Unfortunately, the Weiszfeld/Karcher flow doesn't preserve Toeplitz structure of HPD matrices. To solve this drawback, we use the Trench/Verblunsky theorem or Partial Iwasawa Decomposition proving the diffeomorphism of THPD matrices with product space RxDn (where Dn is unit Poincaré polydisk) through Complex Auto-Regressive (CAR) model. For this new parametrization, we give the hessian metric given by Koszul Information Geometry. For the new Karcher Flow in Poincaré disk, the point driven by the flow is fixed at the origin of the unit disk and then others points are moved from Karcher drift through unit disk automorphism. Inverse automorphism of this drift at each step provides the median point in unit disk coordinates. This is done by using the polar decomposition of points in the unit disk where at each step, the drift is only deduced from the polar phases. To extend the problem for TBTHPD matrices, we have introduced the Berger Fibration and Mostow Decomposition to extend the Karcher flow in Siegel unit Disk. For TBTHPD matrices, we used matrix extension of Trench/Verblunsky theorem, given the diffeomorphism of TBTHPD matrix with product space HPDxSDn (where SDn is the unit Siegel polydisk). As Siegel disk SD can be fibered by associating geodesically to each point, one point on its Shilov Boundary, this Fibration, given by Mostow Decomposition can be interpreted as matrix extension of Poincare disk polar decomposition for the Siegel disk. Finally, we observe that respectively the Frechet's median in unit Poincare disk/Siegel disk is equivalent to conformal Douady-Earle/Busemann Barycenters of same points geodesically pushed on their Shilov's boundaries. References: 1. Koszul J.L., Variétés localement plates et convexité, Osaka J. Math. , n°2, p.285-290, 1965 2. Koszul J.L., Domaines bornées homogènes et orbites de groupes de transformations affines, Bull. Soc. Math. France 89, pp. 515-533., 1961 3. Koszul J.L., Ouverts convexes homogènes des espaces affines,Math. 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