On Unbalanced Optimal Transport: An Analysis of Sinkhorn Algorithm
Abstract
We provide a computational complexity analysis for the Sinkhorn algorithm that solves the entropic regularized Unbalanced Optimal Transport (UOT) problem between two measures of possibly different masses with at most $n$ components. We show that the complexity of the Sinkhorn algorithm for finding an $\varepsilon$approximate solution to the UOT problem is of order $\widetilde{\mathcal{O}}(n^2/ \varepsilon)$, which is nearlinear time. To the best of our knowledge, this complexity is better than the complexity of the Sinkhorn algorithm for solving the Optimal Transport (OT) problem, which is of order $\widetilde{\mathcal{O}}(n^2/\varepsilon^2)$. Our proof technique is based on the geometric convergence of the Sinkhorn updates to the optimal dual solution of the entropic regularized UOT problem and some properties of the primal solution. It is also different from the proof for the complexity of the Sinkhorn algorithm for approximating the OT problem since the UOT solution does not have to meet the marginal constraints.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.03293
 Bibcode:
 2020arXiv200203293P
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Optimization and Control;
 Statistics  Machine Learning
 EPrint:
 The first two authors contributed equally to this work