##### Department of Mathematics,

University of California San Diego

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### CSME Seminar

## Jun Zhang

#### University of Michigan, Ann Arbor

## Information Geometry: Geometerization of Information and Statistical Inference

##### Abstract:

Information Geometry is the differential geometric study of the manifold of probability models, and promises to be a unifying geometric framework for investigating statistical inference, information theory, machine learning, etc. Instead of using metric for measuring distances on such manifolds, these applications often use â€œdivergence functionsâ€ for measuring proximity of two points (that do not impose symmetry and triangular inequality), for instance Kullback-Leibler divergence, Bregman divergence, f-divergence, etc. Divergence functions are tied to generalized entropy (for instance, Tsallis entropy, Renyi entropy, phi-entropy) and cross-entropy functions widely used in machine learning and information sciences. It turns out that divergence functions enjoy pleasant geometric properties â€“ they induce what is called â€œstatistical structureâ€ on a manifold M: a Riemannian metric g together with a pair of affine connections D, D*, such that D and D* are both Codazzi coupled to g while being conjugate to each other. Divergence functions also induce a natural symplectic structure on the product manifold MxM for which M with statistical structure is a Lagrange submanifold. In joint work with M. Leok, we shown how divergence functions allow us to decouple Hamiltonian and Lagrangian dynamics in geometric mechanics. We recently characterize (para-) holomorphicity of D, D* in the (para-)Hermitian setting, and show that statistical structures can be enhanced to (para-)Hermitian and (para-)Kahler manifolds. The surprisingly rich geometric structures and properties of a statistical manifold open up the intriguing possibility of geometrizing statistical inference, information, and machine learning in string-theoretic languages.

Host: Melvin Leok

### April 25, 2019

### 11:00 AM

### AP&M 2402

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