近似线性时间上的二次运输度量
Approximating the Quadratic Transportation Metric in Near-Linear Time
摘要：
计算两个点云之间的二次传输度量（也称为$2$ -Wasserstein距离或均方根距离），或者更一般地，两个离散分布，是机器学习，统计学，计算机图形学和理论计算机科学中的基本问题。由于Agarwal和Sharathkumar在2014年的一系列工作，一个复杂的几何算法达到了高潮，该算法在时间$\tilde{O}(n^{3/2})$中运行，其中$n$是点数。然而，获得更快的算法已被证明是困难的，因为已知$2$ -Wasserstein距离具有较差的草图和嵌入属性，这限制了几何方法的有效性。在本文中，我们给出了一个非常简单的确定性算法与$\tilde{O}(n)$运行时使用一个完全不同的方法的基础上熵正则化，近似Sinkhorn缩放，和高斯核矩阵的低秩近似。我们给我们的算法的维数和精度的近似明确的依赖。
英文摘要：
Computing the quadratic transportation metric (also called the $2$-Wasserstein distance or root mean square distance) between two point clouds, or, more generally, two discrete distributions, is a fundamental problem in machine learning, statistics, computer graphics, and theoretical computer science. A long line of work has culminated in a sophisticated geometric algorithm due to Agarwal and Sharathkumar in 2014, which runs in time $\tilde{O}(n^{3/2})$, where $n$ is the number of points. However, obtaining faster algorithms has proven difficult since the $2$-Wasserstein distance is known to have poor sketching and embedding properties, which limits the effectiveness of geometric approaches. In this paper, we give an extremely simple deterministic algorithm with $\tilde{O}(n)$ runtime by using a completely different approach based on entropic regularization, approximate Sinkhorn scaling, and low-rank approximations of Gaussian kernel matrices. We give explicit dependence of our algorithm on the dimension and precision of the approximation.
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