[Paper] A Riemannian Approach to Low-Rank Optimal Transport

Source: arXiv - 2606.12120v1

Overview

Low-rank optimal transport (OT) mitigates the quadratic scaling of classical solvers, yet existing approaches rely heavily on first-order mirror-descent updates that require careful hyperparameter tuning and ignore the optimization landscape’s curvature. To address these limitations, we propose a unified Riemannian geometric framework for low-rank OT, modeling balanced and unbalanced rank-$r$ positive factored couplings as novel smooth embedded submanifolds of the positive orthant. By equipping these manifolds with the Fisher-Rao product metric, we derive tractable formulations for Riemannian projectors, retractions, and Hessian-vector products. Our cost-agnostic framework seamlessly extends to linear OT, Gromov-Wasserstein (GW), fused GW, and their unbalanced counterparts. For balanced OT, our geometric ingredients are computed via efficient conjugate-gradient and iterative Bregman updates. For the unbalanced OT, our operations elegantly reduce to closed-form scalings, completely eliminating inner iterative loops. In both regimes, per-iteration complexity scales linearly with dataset size, and we provide a rank-sufficiency certificate for global optimality verification. Extensive experiments across a range of problem sizes demonstrate that our regularization-free first- and second-order solvers achieve faster convergence and superior performance over existing state-of-the-art low-rank OT solvers.

Key Contributions

This paper presents research in the following areas:

• cs.LG
• math.OC

Methodology

Please refer to the full paper for detailed methodology.

Practical Implications

This research contributes to the advancement of cs.LG.

Authors

• Pratik Jawanpuria
• Bamdev Mishra

Paper Information

• arXiv ID: 2606.12120v1
• Categories: cs.LG, math.OC
• Published: June 10, 2026
• PDF: Download PDF