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Regularized Divergences Between Covariance Operators and Gaussian Measures on Hilbert Spaces

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  • Hà Quang Minh

    (RIKEN Center for Advanced Intelligence Project)

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.

Suggested Citation

  • Hà Quang Minh, 2021. "Regularized Divergences Between Covariance Operators and Gaussian Measures on Hilbert Spaces," Journal of Theoretical Probability, Springer, vol. 34(2), pages 580-643, June.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:2:d:10.1007_s10959-020-01003-2
    DOI: 10.1007/s10959-020-01003-2
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    Cited by:

    1. Hà Quang Minh, 2023. "Entropic Regularization of Wasserstein Distance Between Infinite-Dimensional Gaussian Measures and Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 36(1), pages 201-296, March.

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