  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.
