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arXiv · 1810.10046

Approximating the Quadratic Transportation Metric in Near-Linear Time

Jason Altschuler·Francis Bach·Alessandro Rudi·Jonathan Weed
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Abstract

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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Jason Altschuler, Francis Bach, Alessandro Rudi, Jonathan Weed. 2018-10-23. Approximating the Quadratic Transportation Metric in Near-Linear Time. https://arxiv.org/abs/1810.10046

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