Continuum-marginal optimal transport: a mesh-free kernel method

TL;DR

This paper introduces a mesh-free kernel method for continuum-marginal optimal transport, enabling efficient recovery of velocity fields from probability marginals without spatial discretization.

Contribution

It proposes a novel, practical mesh-free solver embedding the continuity equation in a reproducing kernel Hilbert space, applicable to both deterministic and stochastic optimal transport problems.

Findings

Achieves accurate drift recovery in synthetic experiments

Ensures marginal consistency in the results

Applicable to the Nelson stochastic optimal transport problem

Abstract

In this paper we study continuum-marginal optimal transport. Given a time-continuous family of probability marginals, the problem is to recover the minimum-energy velocity field whose flow reproduces every marginal. This problem is the continuum limit of the classical two-marginal Benamou--Brenier formulation, and also the deterministic limit of the Nelson problem of stochastic optimal transport. We propose a practical mesh-free solver for this problem. The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial discretization. The velocity is parametrized by any linear-in-parameters dictionary or neural network, and is optimized by mini-batch stochastic methods. Synthetic experiments confirm that the method achieves accurate drift recovery and marginal consistency. The same computational framework also applies to the stochastic Nelson problem.