Article
Log-Determinant Divergences Revisited: Alpha-Beta and Gamma Log-Det Divergences
Author/s | Cichocki, Andrzej
Cruces Álvarez, Sergio Antonio Amari, Shun-ichi |
Department | Universidad de Sevilla. Departamento de Teoría de la Señal y Comunicaciones |
Publication Date | 2015-05-08 |
Deposit Date | 2016-12-01 |
Published in |
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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) ... 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. |
Citation | Cichocki, A., Cruces, S.A. y Shun-ichi, A. (2015). Log-Determinant Divergences Revisited: Alpha-Beta and Gamma Log-Det Divergences. Entropy, 17 (5), 2988-3034. |
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