Structured Transforms Across Spaces with Cost-Regularized Optimal Transport

TL;DR

This paper introduces a cost-regularized optimal transport framework for matching probability measures across different spaces, enabling structured transformations with regularizers and demonstrating applications in spatial transcriptomics.

Contribution

It develops a novel cost-regularized OT approach for cross-space measure matching, incorporating structure-inducing regularizers and a proximal algorithm for unaligned data.

Findings

Effective in matching measures across different Euclidean spaces

Enables structured transformations like sparsity

Demonstrated on spatial transcriptomics data

Abstract

Matching a source to a target probability measure is often solved by instantiating a linear optimal transport (OT) problem, parameterized by a ground cost function that quantifies discrepancy between points. When these measures live in the same metric space, the ground cost often defaults to its distance. When instantiated across two different spaces, however, choosing that cost in the absence of aligned data is a conundrum. As a result, practitioners often resort to solving instead a quadratic Gromow-Wasserstein (GW) problem. We exploit in this work a parallel between GW and cost-regularized OT, the regularized minimization of a linear OT objective parameterized by a ground cost. We use this cost-regularized formulation to match measures across two different Euclidean spaces, where the cost is evaluated between transformed source points and target points. We show that several quadratic OT problems fall in this category, and consider enforcing structure in linear transform (e.g. sparsity), by introducing structure-inducing regularizers. We provide a proximal algorithm to extract such transforms from unaligned data, and demonstrate its applicability to single-cell spatial transcriptomics/multiomics matching tasks.