Projections with logarithmic divergences

TL;DR

This paper explores the geometric structure of statistical manifolds induced by logarithmic divergences, revealing a dual foliation and applying it to develop a generalized PCA method.

Contribution

It constructs a dual foliation of the manifold induced by logarithmic divergence, extending dual flatness concepts and introducing a new generalized PCA approach.

Findings

Established a dual foliation of the statistical manifold

Extended the concept of dual flatness to logarithmic divergence

Formulated a new $L^{(eta)}$-PCA method

Abstract

In information geometry, generalized exponential families and statistical manifolds with curvature are under active investigation in recent years. In this paper we consider the statistical manifold induced by a logarithmic $L^{(\alpha)}$-divergence which generalizes the Bregman divergence. It is known that such a manifold is dually projectively flat with constant negative sectional curvature, and is closely related to the $\mathcal{F}^{(\alpha)}$-family, a generalized exponential family introduced by the second author. Our main result constructs a dual foliation of the statistical manifold, i.e., an orthogonal decomposition consisting of primal and dual autoparallel submanifolds. This decomposition, which can be naturally interpreted in terms of primal and dual projections with respect to the logarithmic divergence, extends the dual foliation of a dually flat manifold studied by Amari. As an application, we formulate a new $L^{(\alpha)}$-PCA problem which generalizes the exponential family PCA.