Hông Vân Lê

Information geometry and sufficient statistics

Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari–Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. This leads to the question how the geometric structures behave under such sufficient statistics. While this is well studied in the finite sample size case, in the infinite case, we encounter technical problems concerning the appropriate topologies. Here, we introduce notions of parametrized measure models and tensor fields on them that exhibit the right behavior under statistical transformations. Within this framework, we can then handle the topological issues and show that the Fisher metric and the Amari–Chentsov tensor on statistical models in the class of symmetric 2-tensor fields and 3-tensor fields can be uniquely (up to a constant) characterized by their invariance under sufficient statistics, thereby achieving a full generalization of the original result of Chentsov to infinite sample sizes. More generally, we decompose Markov morphisms between statistical models in terms of statistics. In particular, a monotonicity result for the Fisher information naturally follows.

Cohomology theories on locally conformal symplectic manifolds

In this note we introduce primitive cohomology groups of locally conformal symplectic manifolds

Poisson Smooth Structures on Stratified Symplectic Spaces

(M^{2n}, \omega, \theta)

Smooth structures on pseudomanifolds with isolated conical singularities

. We study the relation between the primitive cohomology groups and the Lichnerowicz-Novikov cohomology groups of

Lower Bounds for the Circuit Size of Partially Homogeneous Polynomials

(M^{2n}, \omega, \theta)

Invariant Geometric Structures on Statistical Models

, using and extending the technique of spectral sequences developed by Di Pietro and Vinogradov for symplectic manifolds. We discuss related results by many peoples, e.g. Bouche, Lychagin, Rumin, Tseng-Yau, in light of our spectral sequences. We calculate the primitive cohomology groups of a

Compact symplectic manifolds of low cohomogeneity

(2n+2)

McLean’s second variation formula revisited

-dimensional locally conformal symplectic nilmanifold as well as those of a l.c.s. solvmanifold. We show that the l.c.s. solvmanifold is a mapping torus of a contactomorphism, which is not isotopic to the identity.

The Cramér-Rao inequality on singular statistical models

In this paper we introduce the notion of a smooth structure on a stratified space, the notion of a Poisson smooth structure and the notion of a weakly symplectic smooth structure on a stratified symplectic space, refining the concept of a stratified symplectic Poisson algebra introduced by Sjamaar and Lerman. We show that these smooth spaces possess several important properties, e.g., the existence of smooth partitions of unity. Furthermore, under mild conditions many properties of a symplectic manifold can be extended to a symplectic stratified space provided with a smooth Poisson structure, e.g., the existence and uniqueness of a Hamiltonian flow, the isomorphism between the Brylinski-Poisson homology and the de Rham homology, and the existence of a Lefschetz decomposition on a symplectic stratified space. We give many examples of stratified symplectic spaces possessing a Poisson smooth structure which is also weakly symplectic.

Finite Information Geometry

In this note we introduce the notion of a smooth structure on a conical pseudomanifold M in terms of C∞-rings of smooth functions on M. For a finitely generated smooth structure C∞(M) we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of M, and the notion of characteristic classes of M. We prove the vanishing of a Nash vector field at a singular point for a special class of Euclidean smooth structures on M. We introduce the notion of a conical symplectic form on M and show that it is smooth with respect to a Euclidean smooth structure on M. If a conical symplectic structure is also smooth with respect to a compatible Poisson smooth structure C∞(M), we show that its Brylinski–Poisson homology groups coincide with the de Rham homology groups of M. We show nontrivial examples of these smooth conical symplectic-Poisson pseudomanifolds.