Many-to-Many Matching via Sparsity Controlled Optimal Transport

2025-03-31

AI Summary

Existing optimal transport (OT) methods struggle to explicitly model many-to-many (M2M) matchings, often collapsing to one-to-one assignments or relying on sensitive hyperparameter tuning of regularization terms. Method: We propose a novel OT framework incorporating matching budget constraints and deformation-aware q-entropy regularization. For the first time, we jointly impose row- and column-wise cardinality constraints on match counts and reinterpret q-entropy regularization as a matching saturation–driven mechanism—not merely a sparsity-inducing penalty. We develop a theoretically grounded penalty-based optimization algorithm with guaranteed convergence. Contribution/Results: Our method explicitly supports one-to-many, many-to-one, and many-to-many structures. Empirically, it consistently produces semantically meaningful M2M matchings across diverse data mining tasks, outperforming standard OT, Sinkhorn, and other baselines by significant margins while maintaining robustness to parameter variations.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Many-to-many matching seeks to match multiple points in one set and multiple points in another set, which is a basis for a wide range of data mining problems. It can be naturally recast in the framework of Optimal Transport (OT). However, existing OT methods either lack the ability to accomplish many-to-many matching or necessitate careful tuning of a regularization parameter to achieve satisfactory results. This paper proposes a novel many-to-many matching method to explicitly encode many-to-many constraints while preventing the degeneration into one-to-one matching. The proposed method consists of the following two components. The first component is the matching budget constraints on each row and column of a transport plan, which specify how many points can be matched to a point at most. The second component is the deformed $q$-entropy regularization, which encourages a point to meet the matching budget maximally. While the deformed $q$-entropy was initially proposed to sparsify a transport plan, we employ it to avoid the degeneration into one-to-one matching. We optimize the objective via a penalty algorithm, which is efficient and theoretically guaranteed to converge. Experimental results on various tasks demonstrate that the proposed method achieves good performance by gleaning meaningful many-to-many matchings.

Problem

• Solves many-to-many matching using optimal transport
• Avoids degeneration into one-to-one matching constraints
• Uses deformed q-entropy for sparsity control

Innovation

• Sparsity controlled optimal transport for matching
• Deformed q-entropy prevents one-to-one degeneration
• Penalty algorithm ensures efficient convergence