## Abstract

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 d^{2}. 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 d^{p}(x,y), where p ≥ 1. Our method requires the calculation of only a small portion of the distances.

Original language | English |
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Pages (from-to) | A3400-A3422 |

Journal | SIAM Journal on Scientific Computing |

Volume | 40 |

Issue number | 5 |

DOIs | |

State | Published - 2018 |

## Keywords

- 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