Continuum-marginal optimal transport: a mesh-free kernel method
For researchers in optimal transport and related fields, this provides a practical solver for a previously computationally challenging problem, though it is an incremental extension of existing kernel methods.
This paper tackles the continuum-marginal optimal transport problem, proposing a mesh-free solver that embeds the weak continuity equation in a reproducing kernel Hilbert space. The method achieves accurate drift recovery and marginal consistency in synthetic experiments.
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.