Log-determinant divergences revisited: Alpha-Beta and Gamma log-det divergences

Andrzej Cichocki, Sergio Cruces, Shun ichi Amari

Research output: Contribution to journalReview articlepeer-review

36 Citations (Scopus)

Abstract

This work reviews and extends a family of log-determinant (log-det) divergences for symmetric positive definite (SPD) matrices and discusses their fundamental properties. We show how to use parameterized Alpha-Beta (AB) and Gamma log-det divergences to generate many well-known divergences; in particular, we consider the Stein's loss, the S-divergence, also called Jensen-Bregman LogDet (JBLD) divergence, Logdet Zero (Bhattacharyya) divergence, Affine Invariant Riemannian Metric (AIRM), and other divergences. Moreover, we establish links and correspondences between log-det divergences and visualise them on an alpha-beta plane for various sets of parameters. We use this unifying framework to interpret and extend existing similarity measures for semidefinite covariance matrices in finite-dimensional Reproducing Kernel Hilbert Spaces (RKHS). This paper also shows how the Alpha-Beta family of log-det divergences relates to the divergences of multivariate and multilinear normal distributions. Closed form formulas are derived for Gamma divergences of two multivariate Gaussian densities; the special cases of the Kullback-Leibler, Bhattacharyya, Rényi, and Cauchy-Schwartz divergences are discussed. Symmetrized versions of log-det divergences are also considered and briefly reviewed. Finally, a class of divergences is extended to multiway divergences for separable covariance (or precision) matrices.

Original languageEnglish
Pages (from-to)2988-3034
Number of pages47
JournalEntropy
Volume17
Issue number5
DOIs
Publication statusPublished - 2015
Externally publishedYes

Keywords

  • Affine Invariant Riemannian Metric (AIRM)
  • Alpha-Beta Log-Det divergences
  • Burg's matrix divergence
  • Gamma divergences
  • Generalized divergences for symmetric positive definite (covariance) matrices
  • Geodesic distance
  • Hilbert projective metric and their extensions
  • Jeffrey's KL divergence
  • Jensen-Bregman LogDet (JBLD)
  • LogDet Zero divergence
  • Riemannian metric
  • S-divergence
  • Similarity measures
  • Stein's loss
  • Symmetrized KL Divergence Metric (KLDM)

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