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Entropy 2013, 15(11), 4732-4747; doi:10.3390/e15114732
Article

Group Invariance of Information Geometry on q-Gaussian Distributions Induced by Beta-Divergence

1,*  and 2
1 Department of Electrical and Electronics Engineering, University of Fukui, Fukui 910-8507, Japan 2 The Institute of Statistical Mathematics and the Graduate University of Advanced Studies, Tachikawa Tokyo 190-8562, Japan
* Author to whom correspondence should be addressed.
Received: 6 October 2013 / Revised: 26 October 2013 / Accepted: 29 October 2013 / Published: 4 November 2013
(This article belongs to the collection Advances in Applied Statistical Mechanics)
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Abstract

We demonstrate that the q-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this property via the correspondence between information geometry induced by a deformed potential on the space and the one induced by what we call β-divergence defined on the q-exponential family with q = β + 1. The results are fundamental in robust multivariate analysis using the q-Gaussian family.
Keywords: multivariate q-Gaussian family; β-divergence; information geometry; GL(n;R)-invariance; V -potential multivariate q-Gaussian family; β-divergence; information geometry; GL(n; R)-invariance; V -potential
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Ohara, A.; Eguchi, S. Group Invariance of Information Geometry on q-Gaussian Distributions Induced by Beta-Divergence. Entropy 2013, 15, 4732-4747.

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