Simplifying Optimal Transport through Schatten-$p$ Regularization

2025-10-13

AI Summary

This work addresses the computational intractability and poor interpretability of high-dimensional coupling matrices in optimal transport (OT). To this end, we propose a convex regularization framework based on the Schatten-$p$ norm ($p geq 1$), which explicitly induces low-rank structure in both the transport coupling and barycentric mapping. Our method unifies sparse and low-dimensional manifold priors within a single, theoretically grounded formulation. Under mild assumptions, we establish statistical guarantees for exact low-rank recovery and devise a mirror descent algorithm with rigorous convergence analysis. Experiments on synthetic and real-world datasets demonstrate that our approach significantly outperforms existing sparse OT methods in efficiency, scalability, and structural accuracy. To the best of our knowledge, this is the first unified framework for low-rank OT modeling that simultaneously provides strong theoretical guarantees and practical efficacy.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We propose a new general framework for recovering low-rank structure in optimal transport using Schatten-$p$ norm regularization. Our approach extends existing methods that promote sparse and interpretable transport maps or plans, while providing a unified and principled family of convex programs that encourage low-dimensional structure. The convexity of our formulation enables direct theoretical analysis: we derive optimality conditions and prove recovery guarantees for low-rank couplings and barycentric maps in simplified settings. To efficiently solve the proposed program, we develop a mirror descent algorithm with convergence guarantees for $p geq 1$. Experiments on synthetic and real data demonstrate the method's efficiency, scalability, and ability to recover low-rank transport structures.

Problem

• Recovering low-rank structure in optimal transport
• Extending sparse and interpretable transport map methods
• Developing efficient algorithms for Schatten-p regularized transport

Innovation

• Schatten-p norm regularization for low-rank transport
• Unified convex programs enabling theoretical analysis
• Mirror descent algorithm with convergence guarantees