A Sparse Multi-Scale Algorithm for Dense Optimal Transport

TL;DR

This paper introduces a multi-scale algorithm for dense optimal transport that leverages geometric structure to improve scalability, reducing runtime and memory use for large problems.

Contribution

It proposes a framework for verifying global optimality locally and constructs a multi-scale algorithm that uses sparse problems, compatible with existing solvers.

Findings

Significant reduction in run-time

Lower memory requirements

Effective for large dense problems

Abstract

Discrete optimal transport solvers do not scale well on dense large problems since they do not explicitly exploit the geometric structure of the cost function. In analogy to continuous optimal transport we provide a framework to verify global optimality of a discrete transport plan locally. This allows construction of an algorithm to solve large dense problems by considering a sequence of sparse problems instead. The algorithm lends itself to being combined with a hierarchical multi-scale scheme. Any existing discrete solver can be used as internal black-box.Several cost functions, including the noisy squared Euclidean distance, are explicitly detailed. We observe a significant reduction of run-time and memory requirements.