Faster Sinkhorn's Algorithm with Small Treewidth

01/17/2023

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by   Zhao Song, et al.

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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^⊤ where A ∈ℝ^n × d, we proposed a faster Sinkhorn's Algorithm to approximate the OT distance when matrix A has treewidth τ. To approximate the OT distance, our algorithm improves the state-of-the-art results [Dvurechensky, Gasnikov, and Kroshnin ICML 2018] from O(ϵ^-2 n^2) time to O(ϵ^-2 n τ) time.