Convex relaxation approaches for high-dimensional optimal transport

TL;DR

This paper introduces convex relaxation methods for high-dimensional optimal transport that reduce computational complexity, improve statistical efficiency, and enable interpretable transport map extraction, benefiting applications in data science.

Contribution

It develops novel convex relaxation techniques exploiting sparsity and locality, providing scalable approximations and transport maps for high-dimensional OT problems.

Findings

Reduces computational complexity for high-dimensional OT.

Proves sample complexity improvements for sparse Gaussian cases.

Demonstrates effective transport map extraction as an interpretable alternative to neural networks.

Abstract

Optimal transport (OT) is a powerful tool in mathematics and data science but faces severe computational and statistical challenges in high dimensions. We propose convex relaxation approaches based on marginal and cluster moment relaxations that exploit locality and correlative sparsity in the distributions. These methods approximate high-dimensional couplings using low-order marginals and sparse moment statistics, yielding semidefinite programs that provide lower bounds on the OT cost with greatly reduced complexity. For Gaussian distributions with sparse correlations, we prove reductions in both computational and sample complexity, and experiments show the approach also works well for non-Gaussian cases. In addition, we demonstrate how to extract transport maps from our relaxations, offering a simpler and interpretable alternative to neural networks in generative modeling. Our results suggest that convex relaxations can provide a promising path for dimension reduction in high-dimensional OT.