Optimal Transportation and Alignment Between Gaussian Measures

TL;DR

This paper advances the theory and computation of optimal transport and Gromov-Wasserstein alignment for Gaussian measures, providing closed-form solutions and bounds, and demonstrating applications in data science tasks.

Contribution

It offers new closed-form solutions and bounds for Gaussian OT and IGW alignment, including uncentered cases and barycenters, and introduces an efficient algorithm for Gaussian multimarginal OT.

Findings

Closed-form solutions for centered Gaussian IGW alignment.

Analytic bounds for uncentered Gaussian IGW alignment.

Efficient algorithm for Gaussian multimarginal OT.

Abstract

Optimal transport (OT) and Gromov-Wasserstein (GW) alignment provide interpretable geometric frameworks for comparing, transforming, and aggregating heterogeneous datasets -- tasks ubiquitous in data science and machine learning. Because these frameworks are computationally expensive, large-scale applications often rely on closed-form solutions for Gaussian distributions under quadratic cost. This work provides a comprehensive treatment of Gaussian, quadratic cost OT and inner product GW (IGW) alignment, closing several gaps in the literature to broaden applicability. First, we treat the open problem of IGW alignment between uncentered Gaussians on separable Hilbert spaces by giving a closed-form expression up to a quadratic optimization over unitary operators, for which we derive tight analytic upper and lower bounds. If at least one Gaussian measure is centered, the solution reduces to a fully closed-form expression, which we further extend to an analytic solution for the IGW barycenter between centered Gaussians. We also present a reduction of Gaussian multimarginal OT with pairwise quadratic costs to a tractable optimization problem and provide an efficient algorithm to solve it using a rank-deficiency constraint. To demonstrate utility, we apply our results to knowledge distillation and heterogeneous clustering on synthetic and real-world datasets.