Fast entropic regularized optimal transport using semidiscrete cost approximation

Evgeny Tenetov, Gershon Wolansky, Ron Kimmel

Research output: Contribution to journalArticlepeer-review


The optimal transportation theory was successfully applied to different tasks on geometric domains as images and triangle meshes. In these applications the transport problem is defined on a Riemannian manifold with geodesic distance d(x, y). Usually, the cost function used is the geodesic distance d or the squared geodesic distance d2. These choices result in the 1-Wasserstein distance, also known as the earth mover's distance (EMD), or the 2-Wasserstein distance. The entropy regularized optimal transport problem can be solved using the Bregman projection algorithm. This algorithm can be implemented using only matrix multiplications of matrix exp(-C/ϵ) (pointwise exponent) and pointwise vector multiplications, where C is a cost matrix, and e is the regularization parameter. In this paper, we obtain a low-rank decomposition of this matrix and exploit it to accelerate the Bregman projection algorithm. Our low-rank decomposition is based on the semidiscrete approximation of the cost function, which is valid for a large family of cost functions, including dp(x,y), where p ≥ 1. Our method requires the calculation of only a small portion of the distances.

Original languageEnglish
Pages (from-to)A3400-A3422
JournalSIAM Journal on Scientific Computing
Issue number5
StatePublished - 2018


  • Bregman projection
  • Convex optimization
  • Entropy regularization
  • Geodesic distance
  • Kullback-Leibler
  • Low-rank approximations
  • Optimal transport
  • Wasserstein barycenter

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics


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