Improved Complexity Analysis of the Sinkhorn and Greenkhorn Algorithms for Optimal Transport

TL;DR

This paper provides a tighter complexity analysis for the vanilla Sinkhorn and Greenkhorn algorithms, showing they operate in $O(n^2 orm{C}_ty^2 log n / ^2)$ time for -accuracy, improving upon previous bounds.

Contribution

It establishes the first tight complexity bounds for the vanilla Sinkhorn and Greenkhorn algorithms, using dual variable equicontinuity analysis.

Findings

Complexity of vanilla Sinkhorn and Greenkhorn is $O(n^2 orm{C}_ty^2 log n / ^2)$.

Analysis based on equicontinuity of dual variables.

Results improve understanding of algorithm efficiency for optimal transport.

Abstract

The Sinkhorn algorithm is a widely used method for solving the optimal transport problem, and the Greenkhorn algorithm is one of its variants. While there are modified versions of these two algorithms whose computational complexities are $O({n^2\|C\|_\infty^2\log n}/{\varepsilon^2})$ to achieve an $\varepsilon$-accuracy, to the best of our knowledge, the existing complexities for the vanilla versions are $O({n^2\|C\|_\infty^3\log n}/{\varepsilon^3})$. In this paper we fill this gap and show that the complexities of the vanilla Sinkhorn and Greenkhorn algorithms are indeed $O({n^2\|C\|_\infty^2\log n}/{\varepsilon^2})$. The analysis relies on the equicontinuity of the dual variables for the discrete entropic regularized optimal transport problem, which is of independent interest.