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

Understand

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 is regularizing the $f$-divergence by a squared maximum mean discrepancy (MMD) associated with a characteristic kernel $K$.
• 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 $f$-divergences, particularly their gradients.