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原文摘要

We analyze Sobolev descent, a regularized variant of maximum mean discrepancy (MMD) gradient flow based on a gradient penalty on the MMD witness function. This regularization mitigates the non-convexity of the MMD objective, and yields provable \emph{global} convergence guarantees in MMD, in both continuous and discrete time. A surprising appeal is that our convergence analysis does not rely on isoperimetric assumptions on the target distribution. Instead, it is based on a regularity condition on the difference between kernel mean embeddings. We extend Sobolev descent such that it is applicable in both sampling (from an unnormalized target distribution)---using Stein kernels---and generative modeling settings, unlike previous works, where a gradient flow is suitable for only generative modeling or sampling but not both. We empirically validate the method on a broad range of tasks across both settings.