Optimal Transportation and Alignment Between Gaussian Measures

2025-12-03

AI Summary

This paper investigates optimal transport (OT) under quadratic cost and inner-product Gromov–Wasserstein (IGW) alignment for Gaussian measures on separable Hilbert spaces. Addressing the long-standing open problem of lacking closed-form solutions for IGW alignment of *non-centered* Gaussians, we derive the first analytical expression for the IGW distance and propose tight upper and lower bounds. We further generalize the IGW barycenter to the non-centered setting and obtain its closed-form solution. Additionally, we formulate Gaussian multi-marginal OT as a quadratic optimization problem constrained by unitary operators, yielding a tractable reduced-form characterization and an efficient low-rank algorithm. Theoretical contributions fill several fundamental gaps in the literature on Gaussian OT and IGW theory. Empirical evaluations demonstrate substantial improvements in computational efficiency and performance for knowledge distillation and heterogeneous clustering tasks.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Closed-form solution for uncentered Gaussian IGW alignment
• Analytic barycenter solution for centered Gaussian IGW
• Efficient algorithm for Gaussian multimarginal optimal transport

Innovation

• Closed-form IGW alignment for uncentered Gaussians
• Analytic solution for IGW barycenter of centered Gaussians
• Efficient multimarginal OT algorithm with rank constraint