Log-Determinant Divergences Revisited: Alpha--Beta and Gamma Log-Det Divergences

TL;DR

This paper reviews and extends a family of log-det divergences for symmetric positive definite matrices, establishing their properties, interrelations, and connections to Gaussian distribution divergences, including new formulas and multiway extensions.

Contribution

It introduces new links among log-det divergences, derives closed-form formulas for gamma divergences of Gaussian densities, and extends divergences to multiway covariance matrices.

Findings

Unified framework for various log-det divergences.

Closed-form formulas for Gaussian divergence measures.

Extension of divergences to multiway covariance matrices.

Abstract

In this paper, we review and extend a family of log-det divergences for symmetric positive definite (SPD) matrices and discuss their fundamental properties. We show how to generate from parameterized Alpha-Beta (AB) and Gamma Log-det divergences many well known divergences, for example, the Stein's loss, S-divergence, called also Jensen-Bregman LogDet (JBLD) divergence, the Logdet Zero (Bhattacharryya) divergence, Affine Invariant Riemannian Metric (AIRM) as well as some new divergences. Moreover, we establish links and correspondences among many log-det divergences and display them on alpha-beta plain for various set of parameters. Furthermore, this paper bridges these divergences and shows also their links to divergences of multivariate and multiway Gaussian distributions. Closed form formulas are derived for gamma divergences of two multivariate Gaussian densities including as special cases the Kullback-Leibler, Bhattacharryya, R\'enyi and Cauchy-Schwartz divergences. Symmetrized versions of the log-det divergences are also discussed and reviewed. A class of divergences is extended to multiway divergences for separable covariance (precision) matrices.