Regularized Divergences Between Covariance Operators and Gaussian Measures on Hilbert Spaces

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 regula...

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Bibliographic Details
Published in:Journal of theoretical probability 2021, Vol.34 (2), p.580-643
Main Author: Minh, Hà Quang
Format: Article
Language:English
Subjects:
Covariance
Euclidean geometry
Euclidean space
Hilbert space
Mathematical analysis
Mathematics
Mathematics and Statistics
Matrix methods
Operators (mathematics)
Probability Theory and Stochastic Processes
Regularization
Statistics
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Summary: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.
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-020-01003-2