匹配问题的统一对偶方法
A Unified Dual Method for Matching Problems
AI总结：
本文提出统一对偶框架，将匹配问题转化为隐式配准，应用于二次匹配及Gromov-Wasserstein变体，提供收敛保证，并大规模实现于多种数据模态，拓展断裂匹配等新问题。
AI中文摘要：
匹配问题在数据科学中无处不在，因为它们能够实现结构化对象和分布的对齐。虽然现有的求解器通常针对特定的匹配公式进行定制，但我们基于对偶理论，在共同的数学和优化框架内统一了此类问题的一大类。理论上，我们证明了可分解为凸函数差（DC）的匹配目标可以重新表述为隐式配准问题。这种联系将匹配与另一类研究充分的目标联系起来，并产生了适合自然优化策略的对偶公式。然后，我们将这些发现应用于二次匹配（QM）问题，这些问题允许DC分解，并且我们为其提供了广泛的收敛保证。我们的框架适用于Gromov-Wasserstein（GW）及其不平衡公式和几种变体，这些是日益流行的QM问题。在数值上，我们针对各种数据模态（如图、点云、网格和词嵌入）大规模实现了我们的算法。最后，我们的模块化方法使我们能够探索新的公式，如断裂匹配，拓宽了可以在该框架内解决的问题范围。
英文摘要：
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..
