A numerical algorithm with linear complexity for Multi-marginal Optimal Transport with $L^1$ Cost

TL;DR

This paper introduces a novel recursive tensor-vector product algorithm that significantly reduces the computational complexity of solving multi-marginal optimal transport problems with L1 cost from exponential to linear, enabling faster solutions.

Contribution

The authors develop a recursive tensor-vector product method that drastically improves computational efficiency for MMOT with L1 cost, reducing complexity from exponential to linear.

Findings

The new algorithm achieves linear computational complexity $O(N)$ for MMOT with L1 cost.

Numerical experiments show the method is several orders of magnitude faster than traditional Sinkhorn algorithms.

The approach effectively handles moderate-scale MMOT problems with support sizes of $N \\geq 1000$.

Abstract

Numerically solving multi-marginal optimal transport (MMOT) problems is computationally prohibitive, even for moderate-scale instances involving $l\ge4$ marginals with support sizes of $N\ge1000$. The cost in MMOT is represented as a tensor with $N^l$ elements. Even accessing each element once incurs a significant computational burden. In fact, many algorithms require direct computation of tensor-vector products, leading to a computational complexity of $O(N^l)$ or beyond. In this paper, inspired by our previous work [$Comm. \ Math. \ Sci.$, 20 (2022), pp. 2053 - 2057], we observe that the costly tensor-vector products in the Sinkhorn Algorithm can be computed with a recursive process by separating summations and dynamic programming. Based on this idea, we propose a fast tensor-vector product algorithm to solve the MMOT problem with $L^1$ cost, achieving a miraculous reduction in the computational cost of the entropy regularized solution to $O(N)$. Numerical experiment results confirm such high performance of this novel method which can be several orders of magnitude faster than the original Sinkhorn algorithm.