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.

翻译：最常用的测度f-散度（例如Kullback-Leibler散度）在涉及测度的支撑集方面存在局限性。一种解决方案是通过与特征核K相关的平方最大平均差异（MMD）对f-散度进行正则化。本文利用所谓的核均值嵌入，证明相应的正则化可以重写为与K相关的再生核希尔伯特空间中某个函数的Moreau包络。然后，我们利用希尔伯特空间中Moreau包络的已知结果来证明MMD正则化f-散度及其梯度的性质。随后，我们利用这些发现分析MMD正则化f-散度的Wasserstein梯度流。最后，我们考虑从经验测度出发的Wasserstein梯度流。我们为具有无限和有限衰退常数的f-散度提供了概念验证数值示例。