Structured Matching via Cost-Regularized Unbalanced Optimal Transport

TL;DR

This paper introduces CR-UOT, a flexible unbalanced optimal transport framework that adapts the ground cost to better match heterogeneous data, improving alignment in complex applications like single-cell omics.

Contribution

We propose cost-regularized unbalanced optimal transport (CR-UOT), enabling variable ground costs and better handling of heterogeneous data in measure matching problems.

Findings

CR-UOT effectively matches measures across different spaces.

The method improves alignment of single-cell omics data.

Algorithms with entropic regularization are developed for CR-UOT.

Abstract

Unbalanced optimal transport (UOT) provides a flexible way to match or compare nonnegative finite Radon measures. However, UOT requires a predefined ground transport cost, which may misrepresent the data's underlying geometry. Choosing such a cost is particularly challenging when datasets live in heterogeneous spaces, often motivating practitioners to adopt Gromov-Wasserstein formulations. To address this challenge, we introduce cost-regularized unbalanced optimal transport (CR-UOT), a framework that allows the ground cost to vary while allowing mass creation and removal. We show that CR-UOT incorporates unbalanced Gromov-Wasserstein type problems through families of inner-product costs parameterized by linear transformations, enabling the matching of measures or point clouds across Euclidean spaces. We develop algorithms for such CR-UOT problems using entropic regularization and demonstrate that this approach improves the alignment of heterogeneous single-cell omics profiles, especially when many cells lack direct matches.