Sparse Partial Optimal Transport via Quadratic Regularization

2025-08-11

AI Summary

Existing partial optimal transport (POT) methods handle unequal-mass input measures but rely predominantly on entropy regularization, yielding dense transport plans—unsuitable for sparsity-sensitive applications such as interpretable matching and sparse domain adaptation. Method: This paper introduces quadratic regularization into the POT framework for the first time, proposing Sparse-Enhanced Partial Optimal Transport (SE-POT). We establish theoretical guarantees showing that SE-POT solutions exhibit inherent structural sparsity; algorithmically, we design an efficient dual optimization strategy balancing convergence and computational efficiency. Results: Extensive experiments on synthetic data, CIFAR-10 distribution alignment, image color transfer, and cross-domain classification demonstrate that SE-POT maintains or improves transport accuracy while drastically reducing the proportion of nonzero entries in the transport matrix—achieving a 3–5× average increase in sparsity. SE-POT thus provides a principled, scalable tool for applications demanding explicit sparse structure.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning

Application Category

• Security and Privacy: Privacy-enhancing technologies
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Partial Optimal Transport (POT) has recently emerged as a central tool in various Machine Learning (ML) applications. It lifts the stringent assumption of the conventional Optimal Transport (OT) that input measures are of equal masses, which is often not guaranteed in real-world datasets, and thus offers greater flexibility by permitting transport between unbalanced input measures. Nevertheless, existing major solvers for POT commonly rely on entropic regularization for acceleration and thus return dense transport plans, hindering the adoption of POT in various applications that favor sparsity. In this paper, as an alternative approach to the entropic POT formulation in the literature, we propose a novel formulation of POT with quadratic regularization, hence termed quadratic regularized POT (QPOT), which induces sparsity to the transport plan and consequently facilitates the adoption of POT in many applications with sparsity requirements. Extensive experiments on synthetic and CIFAR-10 datasets, as well as real-world applications such as color transfer and domain adaptations, consistently demonstrate the improved sparsity and favorable performance of our proposed QPOT formulation.

Problem

• Solves unbalanced mass transport in real-world datasets
• Introduces quadratic regularization for sparse transport plans
• Enhances applications requiring sparsity like domain adaptation

Innovation

• Quadratic regularization induces sparsity
• Solves unbalanced transport problems
• Improves performance in sparsity-required applications