Interaction-Force Transport Gradient Flows

TL;DR

This paper introduces a novel gradient flow framework combining unbalanced optimal transport and interaction forces via reproducing kernels, with theoretical convergence guarantees and applications to MMD minimization.

Contribution

It proposes the interaction-force transport (IFT) gradient flows and a particle-based optimization algorithm, bridging unbalanced transport and kernel interactions with proven convergence.

Findings

Proposes IFT gradient flows with convergence guarantees.

Develops a particle-based optimization algorithm for sampling.

Demonstrates improved empirical results in MMD minimization.

Abstract

This paper presents a new gradient flow dissipation geometry over non-negative and probability measures. This is motivated by a principled construction that combines the unbalanced optimal transport and interaction forces modeled by reproducing kernels. Using a precise connection between the Hellinger geometry and the maximum mean discrepancy (MMD), we propose the interaction-force transport (IFT) gradient flows and its spherical variant via an infimal convolution of the Wasserstein and spherical MMD tensors. We then develop a particle-based optimization algorithm based on the JKO-splitting scheme of the mass-preserving spherical IFT gradient flows. Finally, we provide both theoretical global exponential convergence guarantees and improved empirical simulation results for applying the IFT gradient flows to the sampling task of MMD-minimization. Furthermore, we prove that the spherical IFT gradient flow enjoys the best of both worlds by providing the global exponential convergence guarantee for both the MMD and KL energy.