Faster Sinkhorn's Algorithm with Small Treewidth

TL;DR

This paper introduces a faster Sinkhorn's Algorithm for approximating optimal transport distances when the cost matrix has a specific structure, reducing computational complexity based on the matrix's treewidth.

Contribution

The paper presents a novel Sinkhorn's Algorithm that leverages small treewidth in the cost matrix to significantly improve approximation speed.

Findings

Reduces computational complexity from O(^{-2} n^2) to O(^{-2} n au) for matrices with small treewidth.

Provides a more efficient approximation method for OT distances in structured cost matrices.

Enhances scalability of OT computations in machine learning applications.

Abstract

Computing optimal transport (OT) distances such as the earth mover's distance is a fundamental problem in machine learning, statistics, and computer vision. In this paper, we study the problem of approximating the general OT distance between two discrete distributions of size $n$. Given the cost matrix $C=AA^\top$ where $A \in \mathbb{R}^{n \times d}$, we proposed a faster Sinkhorn's Algorithm to approximate the OT distance when matrix $A$ has treewidth $\tau$. To approximate the OT distance, our algorithm improves the state-of-the-art results [Dvurechensky, Gasnikov, and Kroshnin ICML 2018] from $\widetilde{O}(\epsilon^{-2} n^2)$ time to $\widetilde{O}(\epsilon^{-2} n \tau)$ time.