On the Acceleration of the Sinkhorn and Greenkhorn Algorithms for Optimal Transport

We propose and analyze a novel approach to accelerate the Sinkhorn and Greenkhorn algorithms for solving the entropic regularized optimal transport (OT) problems. Focusing on the discrete setting where the probability distributions have at most $n$ atoms, and letting $\varepsilon \in \left(0, 1\right)$ denote the tolerance, we introduce accelerated algorithms that have complexity bounds of $\widetilde{\mathcal{O}}\left(n^{5/2}/\varepsilon^{3/2}\right)$. This improves on the known complexity bound of $\widetilde{\mathcal{O}} \left(n^{2}/\varepsilon^2\right)$ for the Sinkhorn and Greenkhorn algorithms. We also present two hybrid algorithms that use the new accelerated algorithms to initialize the Sinkhorn and Greenkhorn algorithms, and we establish complexity bounds of $\widetilde{\mathcal{O}}\left(n^{7/3}/\varepsilon\right)$ for these hybrid algorithms. We provide an extensive experimental comparison on both synthetic and real datasets to explore the relative advantages of the new algorithms.