A Parallel Method for Earth Mover’s Distance

Abstract

We propose a new algorithm to approximate the Earth Mover’s distance (EMD). Our main idea is motivated by the theory of optimal transport, in which EMD can be reformulated as a familiar \(L_1\) type minimization. We use a regularization which gives us a unique solution for this \(L_1\) type problem. The new regularized minimization is very similar to problems which have been solved in the fields of compressed sensing and image processing, where several fast methods are available. In this paper, we adopt a primal-dual algorithm designed there, which uses very simple updates at each iteration and is shown to converge very rapidly. Several numerical examples are provided.

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Correspondence to Wuchen Li.

Additional information

This work is partially supported by ONR Grants N000141410683, N000141210838 and DOE Grant DE-SC00183838.

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Li, W., Ryu, E.K., Osher, S. et al. A Parallel Method for Earth Mover’s Distance. J Sci Comput 75, 182–197 (2018). https://doi.org/10.1007/s10915-017-0529-1

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Keywords

  • Earth Mover’s distance
  • Optimal transport
  • Compressed sensing
  • Primal-dual algorithm
  • \(L_1\) regularization