pyBregMan: A Python library for Bregman Manifolds

Abstract

A Bregman manifold is a synonym for a dually flat space in information geometry which admits as a canonical divergence a Bregman divergence. Bregman manifolds are induced by smooth strictly convex functions like the cumulant or partition functions of regular exponential families, the negative entropy of mixture families, or the characteristic functions of regular cones just to list a few such convex Bregman generators. We describe the design of pyBregMan, a library which implements generic operations on Bregman manifolds and instantiate several common Bregman manifolds used in information sciences. At the core of the library is the notion of Legendre-Fenchel duality inducing a canonical pair of dual potential functions and dual Bregman divergences. The library also implements the Fisher-Rao manifolds of categorical/multinomial distributions and multivariate normal distributions. To demonstrate the use of the pyBregMan kernel manipulating those Bregman and Fisher-Rao manifolds, the library also provides several core algorithms for various applications in statistics, machine learning, information fusion, and so on.