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

TMLR Paper9432 Authors

Abstract:

In this paper we study \emph{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.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Tongzheng_Ren1

Submission Number: 9432