Low-rank tensor approximations for solving multi-marginal optimal transport problems

TL;DR

This paper enhances the efficiency of multi-marginal optimal transport solutions by combining tensor network duals with low-rank approximations, significantly reducing computation time in practical applications like image color transfer.

Contribution

It introduces a novel method that integrates tensor network duals with low-rank approximations to accelerate multi-marginal optimal transport computations.

Findings

Achieved over 96% reduction in computation time for color transfer.

Demonstrated effectiveness of the method on multi-image color transfer tasks.

Validated the approach's potential for large-scale optimal transport problems.

Abstract

By adding entropic regularization, multi-marginal optimal transport problems can be transformed into tensor scaling problems, which can be solved numerically using the multi-marginal Sinkhorn algorithm. The main computational bottleneck of this algorithm is the repeated evaluation of marginals. Recently, it has been suggested that this evaluation can be accelerated when the application features an underlying graphical model. In this work, we accelerate the computation further by combining the tensor network dual of the graphical model with additional low-rank approximations. We provide an example for the color transfer between several images, in which these additional low-rank approximations save more than 96% of the computation time.