Log-Determinant Divergences Revisited: Alpha-Beta and Gamma Log-Det Divergences
AbstractThis 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
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Cichocki, A.; Cruces, S.; Amari, S.-I. Log-Determinant Divergences Revisited: Alpha-Beta and Gamma Log-Det Divergences. Entropy 2015, 17, 2988-3034.
Cichocki A, Cruces S, Amari S-I. Log-Determinant Divergences Revisited: Alpha-Beta and Gamma Log-Det Divergences. Entropy. 2015; 17(5):2988-3034.Chicago/Turabian Style
Cichocki, Andrzej; Cruces, Sergio; Amari, Shun-ichi. 2015. "Log-Determinant Divergences Revisited: Alpha-Beta and Gamma Log-Det Divergences." Entropy 17, no. 5: 2988-3034.