Faster Sinkhorn's Algorithm with Small Treewidth

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 $C=A{A}^{\top }$ 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 $\tilde{O}({\epsilon }^{-2}{n}^2)$ time to $\tilde{O}({\epsilon }^{-2}n\tau )$ time.