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.
翻译：移动器距离( OT) 等计算机最佳运输距离( OT) 是机器学习、统计和计算机视觉的根本问题 。 在本文中, 我们研究的是两个大小的离散分布之间的一般 OT 距离( $n美元 ) 。 根据成本矩阵 $C = A* attop$, 其中$A\ in\ mathbb{ R ⁇ n\ times d} $, 我们建议使用更快的 Sinkhorn ALgorithm 来接近 OT 距离, 当 $A$ 的矩阵有树枝 $\ tau 。 为了接近 OT 距离, 我们的算法改进了最先进的结果[ Dvurechensky, Gasnikov, 和 Kroshnin ICML 2018], 从$\ plite{ O} (\psilon}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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