Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces

• URL: http://arxiv.org/abs/2402.04613v2
• Date: Sat, 9 Mar 2024 22:58:57 GMT
• Title: Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces
• Authors: Sebastian Neumayer, Viktor Stein, Gabriele Steidl, Nicolaj Rux
• Abstract summary: We regularize the $f$-divergence by a squared maximum mean discrepancy associated with a characteristic kernel $K$. We exploit well-known results on envelopes in Hilbert spaces to prove properties of the MMD-regularized $f$-divergences. We provide proof-of-the-concept numerical examples for $f$-divergences with both infinite and finite recession constant.
• Score: 1.3654846342364308
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Most commonly used $f$-divergences of measures, e.g., the Kullback-Leibler divergence, are subject to limitations regarding the support of the involved measures. A remedy consists of regularizing the $f$-divergence by a squared maximum mean discrepancy (MMD) associated with a characteristic kernel $K$. In this paper, we use the so-called kernel mean embedding to show that the corresponding regularization can be rewritten as the Moreau envelope of some function in the reproducing kernel Hilbert space associated with $K$. Then, we exploit well-known results on Moreau envelopes in Hilbert spaces to prove properties of the MMD-regularized $f$-divergences and, in particular, their gradients. Subsequently, we use our findings to analyze Wasserstein gradient flows of MMD-regularized $f$-divergences. Finally, we consider Wasserstein gradient flows starting from empirical measures. We provide proof-of-the-concept numerical examples for $f$-divergences with both infinite and finite recession constant.

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