## 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 language | English |
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Pages (from-to) | 2988-3034 |

Number of pages | 47 |

Journal | Entropy |

Volume | 17 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |

## 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)