(De)-regularized Maximum Mean Discrepancy Gradient Flow

Understand

We introduce a (de)-regularization of the Maximum Mean Discrepancy (DrMMD) and its Wasserstein gradient flow.

• Existing gradient flows that transport samples from source distribution to target distribution with only target samples, either lack tractable numerical implementation ($f$-divergence flows) or require strong assumptions, and modifications such as noise injection, to ensure convergence (Maximum Mean Discrepancy flows).
• In contrast, DrMMD flow can simultaneously (i) guarantee near-global convergence for a broad class of targets in both continuous and discrete time, and (ii) be implemented in closed form using only samples.
• The former is achieved by leveraging the connection between the DrMMD and the $\chi^2$-divergence, while the latter comes by treating DrMMD as MMD with a de-regularized kernel.