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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 (CC BY 3.0).
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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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