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

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.