Wasserstein gradient flows for Moreau envelopes of f-divergences in reproducing kernel Hilbert spaces

Commonly used [Formula: see text]-divergences of measures, e.g., the Kullback–Leibler divergence, are subject to limitations regarding the support of the involved measures. A remedy is regularizing the [Formula: see text]-divergence by a squared maximum mean discrepancy (MMD) associated with a characteristic kernel [Formula: see text]. We use the kernel mean embedding to show that this regularization can be rewritten as the Moreau envelope of some function on the associated reproducing kernel Hilbert space. Then, we exploit well-known results on Moreau envelopes in Hilbert spaces to analyze the MMD-regularized [Formula: see text]-divergences, particularly their gradients. Subsequently, we use our findings to analyze Wasserstein gradient flows of MMD-regularized [Formula: see text]-divergences. We provide proof-of-the-concept numerical examples for flows starting from empirical measures. Here, we cover [Formula: see text]-divergences with infinite and finite recession constants. Lastly, we extend our results to the tight variational formulation of [Formula: see text]-divergences and numerically compare the resulting flows.