Maximum Mean Discrepancy Gradient Flow

TL;DR

This paper develops a Wasserstein gradient flow for the maximum mean discrepancy (MMD), analyzes its convergence, proposes a regularization method with theoretical and empirical support, and provides practical implementation details.

Contribution

It introduces a new Wasserstein gradient flow for MMD, analyzes convergence conditions, and proposes a noise-based regularization method with theoretical and empirical validation.

Findings

Convergence conditions for the MMD gradient flow towards a global optimum.

A regularization technique involving noise injection improves flow stability.

The flow can be practically implemented using simple closed-form expressions.

Abstract

We construct a Wasserstein gradient flow of the maximum mean discrepancy (MMD) and study its convergence properties. The MMD is an integral probability metric defined for a reproducing kernel Hilbert space (RKHS), and serves as a metric on probability measures for a sufficiently rich RKHS. We obtain conditions for convergence of the gradient flow towards a global optimum, that can be related to particle transport when optimizing neural networks. We also propose a way to regularize this MMD flow, based on an injection of noise in the gradient. This algorithmic fix comes with theoretical and empirical evidence. The practical implementation of the flow is straightforward, since both the MMD and its gradient have simple closed-form expressions, which can be easily estimated with samples.