{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T19:09:49Z","timestamp":1792004989897,"version":"4.4.0"},"reference-count":47,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2015,5,8]],"date-time":"2015-05-08T00:00:00Z","timestamp":1431043200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2019s 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, R\u00e9nyi, 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.<\/jats:p>","DOI":"10.3390\/e17052988","type":"journal-article","created":{"date-parts":[[2015,5,8]],"date-time":"2015-05-08T10:56:41Z","timestamp":1431082601000},"page":"2988-3034","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":55,"title":["Log-Determinant Divergences Revisited: Alpha-Beta and Gamma Log-Det Divergences"],"prefix":"10.3390","volume":"17","author":[{"given":"Andrzej","family":"Cichocki","sequence":"first","affiliation":[{"name":"Laboratory for Advanced Brain Signal Processing, Brain Science Institute, RIKEN, 2-1 Hirosawa, Wako, 351-0198 Saitama, Japan"},{"name":"Systems Research Institute, Intelligent Systems Laboratory, Newelska 6, 01-447 Warsaw, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4121-7137","authenticated-orcid":false,"given":"Sergio","family":"Cruces","sequence":"additional","affiliation":[{"name":"Dpto de Teor\u00eda de la Se\u00f1al y Comunicaciones, University of Seville, Camino de los Descubrimientos s\/n, 41092 Seville, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shun-ichi","family":"Amari","sequence":"additional","affiliation":[{"name":"Laboratory for Mathematical Neuroscience, RIKEN BSI, Wako, 351-0198 Saitama, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,5,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2131","DOI":"10.3390\/e16042131","article-title":"Information geometry of positive measures and positive-definite matrices: Decomposable dually flat structure","volume":"16","author":"Amari","year":"2014","journal-title":"Entropy"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1016\/j.sigpro.2012.09.003","article-title":"Divergence measures for statistical data processing\u2014An annotated bibliography","volume":"93","author":"Basseville","year":"2013","journal-title":"Signal Process"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Weickert, J., and Hagen, H. 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