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Entropy 2015, 17(5), 2988-3034; doi:10.3390/e17052988

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

1
Laboratory for Advanced Brain Signal Processing, Brain Science Institute, RIKEN, 2-1 Hirosawa, Wako, 351-0198 Saitama, Japan
2
Systems Research Institute, Intelligent Systems Laboratory, Newelska 6, 01-447 Warsaw, Poland
3
Dpto de Teoría de la Señal y Comunicaciones, University of Seville, Camino de los Descubrimientos s/n, 41092 Seville, Spain
4
Laboratory for Mathematical Neuroscience, RIKEN BSI, Wako, 351-0198 Saitama, Japan
*
Authors to whom correspondence should be addressed.
Academic Editor: Raúl Alcaraz Martínez
Received: 19 December 2014 / Revised: 18 March 2015 / Accepted: 5 May 2015 / Published: 8 May 2015
(This article belongs to the Section Information Theory)
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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. View Full-Text
Keywords: Similarity measures; generalized divergences for symmetric positive definite (covariance) matrices; Stein’s loss; Burg’s matrix divergence; Affine Invariant Riemannian Metric (AIRM); Riemannian metric; geodesic distance; Jensen-Bregman LogDet (JBLD); S-divergence; LogDet Zero divergence; Jeffrey’s KL divergence; symmetrized KL Divergence Metric (KLDM); Alpha-Beta Log-Det divergences; Gamma divergences; Hilbert projective metric and their extensions Similarity measures; generalized divergences for symmetric positive definite (covariance) matrices; Stein’s loss; Burg’s matrix divergence; Affine Invariant Riemannian Metric (AIRM); Riemannian metric; geodesic distance; Jensen-Bregman LogDet (JBLD); S-divergence; LogDet Zero divergence; Jeffrey’s KL divergence; symmetrized KL Divergence Metric (KLDM); Alpha-Beta Log-Det divergences; Gamma divergences; Hilbert projective metric and their extensions
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Cichocki, A.; Cruces, S.; Amari, S.-I. Log-Determinant Divergences Revisited: Alpha-Beta and Gamma Log-Det Divergences. Entropy 2015, 17, 2988-3034.

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