Information Divergence

Shinto Eguchi () and Osamu Komori ()
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Shinto Eguchi: Institute of Statistical Mathematic
Osamu Komori: Seikei University

Chapter Chapter 2 in Minimum Divergence Methods in Statistical Machine Learning, 2022, pp 19-69 from Springer

Abstract: Abstract The general framework of information divergenceInformation divergence is introduced presenting several useful examples. We observe that the properties associated with the information divergenceInformation divergence directly reflect the information geometry discussed in Chap. 1 . This chapter focuses on two types of information divergenceInformation divergence, called the W-divergenceW-divergence and the U-divergenceU-divergence, where W and U denote generator functions. In general, the information divergenceInformation divergence is associated with a generative geometry, in which the Riemannian metricRiemannian metric and linear connectionsLinear connections are defined in a differential geometric formulation. The generalized Pythagorean theoremGeneralized Pythagorean theorem is given by two geodesic paths with respect to U-divergenceU-divergence. Afterwards, we extend the m-geodesic and e-geodesic paths to the $$\varphi $$ φ -path $$\varphi $$ φ -path with a generator function $$\varphi $$ φ in virtue of the Kolgomorov-Nagumo (KN) mean. The $$\varphi $$ φ -path $$\varphi $$ φ -path is a geodesic path with respect to a parallel transportParallel transports associated with $$\varphi $$ φ . Finally, we consider these geometric objects in a space of positive-definite matrices and apply them to the problem of human color perception in a regression framework.

Date: 2022
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DOI: 10.1007/978-4-431-56922-0_2

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