A Homotopy Algorithm for Optimal Transport

TL;DR

This paper introduces a homotopy algorithm for optimal transport that efficiently traces a solution path by transforming the problem through a series of rotations, aiming for faster computation in high-dimensional datasets.

Contribution

The paper presents a novel homotopy algorithm utilizing subspace rotations and eigenvalue decomposition to solve optimal transport problems more efficiently than existing methods.

Findings

Achieves complexity bound of O(n^2 log n)

Transforms the problem into an easier form via target distribution change

Provides a discretized solution path using eigenvalue decomposition

Abstract

The optimal transport problem has many applications in machine learning, physics, biology, economics, etc. Although its goal is very clear and mathematically well-defined, finding its optimal solution can be challenging for large datasets in high-dimensional space. Here, we propose a homotopy algorithm that first transforms the problem into an easy form, by changing the target distribution. It then transforms the problem back to the original form through a series of iterations, tracing a path of solutions until it finds the optimal solution for the original problem. We define the homotopy path as a subspace rotation based on the orthogonal Procrustes problem, and then we discretize the homotopy path using eigenvalue decomposition of the rotation matrix. Our goal is to provide an algorithm with complexity bound $\mathcal{O}(n^2 \log(n))$, faster than the existing methods in the literature.