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2012 | OriginalPaper | Buchkapitel

Divergence Measures and Means of Symmetric Positive-Definite Matrices

verfasst von : Maher Moakher

Erschienen in: New Developments in the Visualization and Processing of Tensor Fields

Verlag: Springer Berlin Heidelberg

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Abstract

The importance of symmetric positive-definite matrices can hardly be exaggerated as they play fundamental roles in many disciplines such as mathematics, numerical analysis, probability and statistics, engineering, and biological and social sciences. On the other hand, in the last few years there has been a renewed interest in developing the theory of means of symmetric positive-definite matrices. In this work we present several divergence functions for measuring closeness between symmetric positive-definite matrices. We then study the invariance properties of these divergence functions as well as the matrix means based on them. We show that the means based on the various divergence functions of a finite collection of symmetric positive-definite matrices are bounded below by their harmonic mean and above by their arithmetic mean. Furthermore, the means based on the studied divergence functions of two symmetric positive-definite matrices are given in closed forms. In particular, we show that the mean based on the Bhattacharyya divergence function of a pair of symmetric positive-definite matrices coincides with their geometric mean.

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Titel: Divergence Measures and Means of Symmetric Positive-Definite Matrices
verfasst von: Maher Moakher
Verlag: Springer Berlin Heidelberg
Buch: New Developments in the Visualization and Processing of Tensor Fields
Print ISBN: 978-3-642-27342-1

Electronic ISBN: 978-3-642-27343-8

Copyright-Jahr: 2012
DOI: https://doi.org/10.1007/978-3-642-27343-8_16

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