Optimal Transportation by Orthogonal Coupling Dynamics

TL;DR

This paper introduces a new projection-based gradient descent method for solving optimal transport problems, offering improved computational efficiency and novel numerical schemes for Wasserstein distances.

Contribution

It proposes a novel dynamics framework based on conditional expectation and opinion dynamics, enabling efficient computation of optimal transport maps.

Findings

Recovers random maps with favorable computational performance

Provides theoretical insights into the dynamics of optimal transport

Lays groundwork for new numerical schemes for Wasserstein distances

Abstract

Many numerical and learning algorithms rely on the solution of the Monge-Kantorovich problem and Wasserstein distances, which provide appropriate distributional metrics. While the natural approach is to treat the problem as an infinite-dimensional linear programming, such a methodology limits the computational performance due to the polynomial scaling with respect to the sample size along with intensive memory requirements. We propose a novel alternative framework to address the Monge-Kantorovich problem based on a projection type gradient descent scheme. The dynamics builds on the notion of the conditional expectation, where the connection with the opinion dynamics is leveraged to devise efficient numerical schemes. We demonstrate that the resulting dynamics recovers random maps with favourable computational performance. Along with the theoretical insight, the proposed dynamics paves the way for innovative approaches to construct numerical schemes for computing optimal transport maps as well as Wasserstein distances.