A parallel framework for graphical optimal transport

TL;DR

This paper introduces a parallelizable algorithm for graphical multi-marginal optimal transport problems, significantly improving computational efficiency for tree-structured and general graph cases, with broad applications.

Contribution

The paper presents a novel parallel algorithm for graphical MOT problems, leveraging junction trees for general graphs, and demonstrates its efficiency through numerical experiments.

Findings

Significant reduction in computational complexity for tree-structured MOT.

Effective parallel updates for general graphs using modified junction trees.

Benchmark results showing improved efficiency over existing methods.

Abstract

We study multi-marginal optimal transport (MOT) problems where the underlying cost has a graphical structure. These graphical multi-marginal optimal transport problems have found applications in several domains including traffic flow control, barycenter and regression problems in the Wasserstein space, and Hidden Markov model inference problems. The MOT problem can be approached through two formulations: a single big MOT problem, or coupled minor OT problems. In this paper, we focus on the latter approach and demonstrate its efficiency gain from parallelization. For tree-structured MOT problems, we introduce a novel parallelizable algorithm that significantly reduces computational complexity. Additionally, we adapt this algorithm for general graphs, employing the modified junction trees to enable parallel updates. Our contributions, validated through numerical experiments, offer new avenues for MOT applications and establish benchmarks in computational efficiency.