Harmonic exponential families on homogeneous spaces
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Harmonic exponential families on homogeneous spaces Koichi Tojo1
· Taro Yoshino2
Received: 30 November 2019 / Revised: 27 August 2020 / Accepted: 4 September 2020 © Springer Nature Singapore Pte Ltd. 2020
Abstract Exponential families play an important role in the field of information geometry. By definition, there are infinitely many exponential families. However, only a small part of them are widely used. We want to give a framework to deal with these “good” families. In the light of the observation that the sample spaces of most of them are homogeneous spaces of certain Lie groups, we propose a method to construct exponential families on homogeneous spaces G/H by taking advantage of representation theory. Families obtained by this method are G-invariant exponential families. Then the following question naturally arises: are any G-invariant exponential families on G/H obtained by this method? We give an affirmative answer to this question. More precisely, any G-invariant exponential family on G/H can be realized as a subfamily of a family obtained by our method. Keywords Exponential family · Representation theory · Homogeneous space · Harmonic analysis Mathematics Subject Classification 62H10 · 62H11 · 20G05 · 22F30
Contents 1 Introduction . . . 1.1 Background . 1.2 G/H -method 2 Preliminary . . . . 2.1 Notation . . .
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Koichi Tojo [email protected] Taro Yoshino [email protected]
1
RIKEN Center for Advanced Intelligence Project, Nihonbashi 1-chome Mitsui Building, 15th floor, 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
2
Graduate School of Mathematical Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
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Information Geometry 2.2 Exponential family . . . . . . . . . . . . . . . . . . 2.3 Continuity of Laplace transform . . . . . . . . . . . 2.4 Relative G-invariance and strong quasi G-invariance 3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . 3.1 G-invariance of harmonic exponential family . . . . 3.2 Harmonicity of G-invariant exponential family . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . Radon measure . . . . . . . . . . . . . . . . . . . . . . . Radon–Nikodým derivative . . . . . . . . . . . . . . . . Measure on G-space . . . . . . . . . . . . . . . . . . . . Measure on homogeneous space . . . . . . . . . . . . . . Elementary notes . . . . . . . . . . . . . . . . . . . . . . Continuity of parameter map . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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