Approximating Optimal Transport via Low-rank and Sparse Factorization

TL;DR

This paper introduces a new approximation method for optimal transport that decomposes the transport plan into low-rank and sparse components, improving accuracy and efficiency in machine learning applications.

Contribution

It proposes a novel low-rank plus sparse factorization approach for approximating OT, addressing limitations of previous low-rank only methods.

Findings

The method effectively reduces approximation error in OT computation.

The augmented Lagrangian algorithm efficiently computes the transport plan.

The approach balances accuracy and computational efficiency.

Abstract

Optimal transport (OT) naturally arises in a wide range of machine learning applications but may often become the computational bottleneck. Recently, one line of works propose to solve OT approximately by searching the \emph{transport plan} in a low-rank subspace. However, the optimal transport plan is often not low-rank, which tends to yield large approximation errors. For example, when Monge's \emph{transport map} exists, the transport plan is full rank. This paper concerns the computation of the OT distance with adequate accuracy and efficiency. A novel approximation for OT is proposed, in which the transport plan can be decomposed into the sum of a low-rank matrix and a sparse one. We theoretically analyze the approximation error. An augmented Lagrangian method is then designed to efficiently calculate the transport plan.