Maximum Mean Discrepancy Gradient Flow

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.

References in corpus (11)

• MMD GAN: Towards Deeper Understanding of Moment Matching Network
• Generative Moment Matching Networks
• The Cramer Distance as a Solution to Biased Wasserstein Gradients
• User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient
• Training generative neural networks via Maximum Mean Discrepancy optimization
• Stein Variational Gradient Descent as Gradient Flow
• On Graduated Optimization for Stochastic Non-Convex Problems
• Gradient Estimators for Implicit Models
• An Interpolating Distance between Optimal Transport and Fisher-Rao
• Analysis of Langevin Monte Carlo via convex optimization
• Global convergence of neuron birth-death dynamics

Cited by in corpus (17)

• On the geometry of Stein variational gradient descent
• Statistical and Topological Properties of Sliced Probability Divergences
• Particle-based Energetic Variational Inference
• Mean-Field Neural ODEs via Relaxed Optimal Control
• Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics
• Optimal quantisation of probability measures using maximum mean discrepancy
• Neural Tangent Kernel Maximum Mean Discrepancy
• Refining Deep Generative Models via Discriminator Gradient Flow
• Dataset Dynamics via Gradient Flows in Probability Space
• Ridge Regression with Over-Parametrized Two-Layer Networks Converge to Ridgelet Spectrum
• The Wasserstein Proximal Gradient Algorithm
• Frank-Wolfe Methods in Probability Space
• On the Convergence of Gradient Descent in GANs: MMD GAN As a Gradient Flow
• On the Optimization Landscape of Maximum Mean Discrepancy
• Dual Training of Energy-Based Models with Overparametrized Shallow Neural Networks
• Dynamic Flows on Curved Space Generated by Labeled Data
• Particle Dynamics for Learning EBMs