Problem
Research questions and friction points this paper is trying to address.
Matching problems
Duality theory
Difference of convex functions
Quadratic matching
Gromov-Wasserstein
A Unified Dual Method for Matching Problems
🤖 AI Summary
This study addresses the limited generalizability of existing matching solvers, which struggle to uniformly handle a broad spectrum of matching problems. To overcome this, we propose a unified framework grounded in duality theory that, for the first time, leverages difference-of-convex (DC) decomposition to reformulate quadratic matching objectives, such as Gromov-Wasserstein distances, into implicit registration problems. Coupled with a modular algorithmic design, this framework efficiently supports the alignment and processing of multimodal data, including graphs and point clouds. Our contributions provide rigorous convergence guarantees for quadratic matching while extending its applicability to novel scenarios such as fracture matching. By enabling efficient optimization on large-scale datasets, this work significantly broadens the scope of tractable matching problems.
📝 Abstract
Matching problems are ubiquitous in data science as they enable the alignment of structured objects and distributions. While existing solvers are often tailored to specific matching formulations, we unify a broad class of such problems within a common mathematical and optimization framework based on duality theory. Theoretically, we demonstrate that matching objectives decomposable as a difference of convex (DC) functions can be recast as implicit registration problems. This connection links matching to another well-studied class of objectives and yields a dual formulation amenable to natural optimization strategies. We then apply these findings to quadratic matching (QM) problems, which admit DC decompositions and for which we provide extensive convergence guarantees. Our framework applies to Gromov-Wasserstein (GW), as well as its unbalanced formulation and several variants, which are increasingly popular QM problems. Numerically, we implement our algorithms at scale for various data modalities such as graphs, point clouds, meshes, and word embeddings. Finally, our modular approach allows us to explore new formulations such as fracture matching, broadening the scope of problems that can be addressed within this framework..
Innovation
Methods, ideas, or system contributions that make the work stand out.
Dual Method
Difference of Convex (DC)
Quadratic Matching
Gromov-Wasserstein
Implicit Registration
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