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

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