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Kullback-Leibler and Renyi divergences in reproducing kernel Hilbert space and Gaussian process settings
In this work, we present formulations for regularized Kullback-Leibler and R\'enyi divergences via the Alpha Log-Determinant (Log-Det) divergences between positive Hilbert-Schmidt operators on Hilbert spaces in two different settings, namely (i) covariance operators and Gaussian measures defined on reproducing kernel Hilbert spaces (RKHS); and (ii) Gaussian processes with squared integrable sample paths.
Finite sample approximations of exact and entropic Wasserstein distances between covariance operators and Gaussian processes
Using this representation, we show that the Sinkhorn divergence between two centered Gaussian processes can be consistently and efficiently estimated from the divergence between their corresponding normalized finite-dimensional covariance matrices, or alternatively, their sample covariance operators.
Convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings
Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space, convergence in the 2-Sinkhorn divergence is {\it strictly weaker} than convergence in the exact 2-Wasserstein distance.
Entropic regularization of Wasserstein distance between infinite-dimensional Gaussian measures and Gaussian processes
This work studies the entropic regularization formulation of the 2-Wasserstein distance on an infinite-dimensional Hilbert space, in particular for the Gaussian setting.
Infinite-dimensional Log-Determinant divergences II: Alpha-Beta divergences
This work presents a parametrized family of divergences, namely Alpha-Beta Log- Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space.
Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces
This paper introduces a novel mathematical and computational framework, namely {\it Log-Hilbert-Schmidt metric} between positive definite operators on a Hilbert space.
