A deep learning approach to multi-marginal optimal transport via Hilbert space embeddings of probability measures

TL;DR

This paper introduces a computationally efficient deep learning method for solving multi-marginal optimal transport problems using Hilbert space embeddings and maximum mean discrepancy, suitable for large-scale applications.

Contribution

It presents a novel numerical approach combining Hilbert space embeddings and penalization for multi-marginal optimal transport, enabling GPU acceleration and scalability.

Findings

Effective in synthetic data experiments

Computationally efficient and scalable

Suitable for large-scale problems

Abstract

We propose a numerical method for solving the multi-marginal Monge problem, which extends the classical Monge formulation to settings involving multiple target distributions. Our approach is based on the Hilbert space embedding of probability measures and employs a penalization technique using the maximum mean discrepancy to enforce marginal constraints. The method is designed to be computationally efficient, enabling GPU-based implementation suitable for large-scale problems. We confirm the effectiveness of the proposed method through numerical experiments using synthetic data.