Regularized Divergences Between Covariance Operators and Gaussian Measures on Hilbert Spaces

Hà Quang Minh ()
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Hà Quang Minh: RIKEN Center for Advanced Intelligence Project

Journal of Theoretical Probability, 2021, vol. 34, issue 2, 580-643

Abstract: Abstract This work presents an infinite-dimensional generalization of the correspondence between the Kullback–Leibler and Rényi divergences between Gaussian measures on Euclidean space and the Alpha Log-Determinant divergences between symmetric, positive definite matrices. Specifically, we present the regularized Kullback–Leibler and Rényi divergences between covariance operators and Gaussian measures on an infinite-dimensional Hilbert space, which are defined using the infinite-dimensional Alpha Log-Determinant divergences between positive definite trace class operators. We show that, as the regularization parameter approaches zero, the regularized Kullback–Leibler and Rényi divergences between two equivalent Gaussian measures on a Hilbert space converge to the corresponding true divergences. The explicit formulas for the divergences involved are presented in the most general Gaussian setting.

Keywords: Gaussian measures; Hilbert space; Covariance operators; Kullback–Leibler divergence; Rényi divergence; Regularized divergences; 28C20; 60G15; 47B65; 15A15 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-020-01003-2

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