A Linear Complexity Algorithm for Optimal Transport Problem with Log-type Cost

TL;DR

This paper extends a linear-time dynamic programming approach to solve optimal transport problems with log-type costs, including applications like Sinkhorn ranking and reflector problems, demonstrating efficiency through numerical simulations.

Contribution

It develops a novel linear complexity algorithm for optimal transport with log-type costs, broadening the scope of efficient solutions beyond Wasserstein-1.

Findings

Linear-time matrix-vector multiplication for log-type costs

Effective application to Sinkhorn ranking and reflector problems

Numerical simulations confirm efficiency and accuracy

Abstract

In [Q. Liao et al., Commun. Math. Sci., 20(2022)], a linear-time Sinkhorn algorithm is developed based on dynamic programming, which significantly reduces the computational complexity involved in solving optimal transport problems. However, this algorithm is specifically designed for the Wasserstein-1 metric. We are curious whether the preceding dynamic programming framework can be extended to tackle optimal transport problems with different transport costs. Notably, two special kinds of optimal transport problems, the Sinkhorn ranking and the far-field reflector and refractor problems, are closely associated with the log-type transport costs. Interestingly, by employing series rearrangement and dynamic programming techniques, it is feasible to perform the matrix-vector multiplication within the Sinkhorn iteration in linear time for this type of cost. This paper provides a detailed exposition of its implementation and applications, with numerical simulations demonstrating the effectiveness and efficiency of our methods.