发表机构
Beijing Normal University; BNU-HKBU United International College（北京师范大学; 北师港浸大）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究矩阵多面体上的正则化耦合映射，涵盖两类正则化，为图匹配与最优传输提供互补算法工具，正则项决定几何结构，外部任务决定精度标准。
AI 中文摘要
线性分配、图匹配和最优传输这三个经典问题具有共同结构：它们在矩阵多面体上寻求最优的矩阵值耦合。本文研究正则化耦合映射，其包含两类正则化：熵/KL正则化，该方法对输入得分矩阵取指数，再通过KL投影将其投影到相关矩阵多面体上；Frobenius/欧氏正则化，该方法对缩放后的输入矩阵执行欧氏投影，投影至同一矩阵多面体。我们探究这两种几何如何为图匹配（CSGO、ASM、FRAM）和最优传输（IP-EOT、PSN）提供互补的算法工具。正则化耦合映射在图匹配中作为方向生成器，在最优传输中作为解的表示，其中正则项决定几何结构，外部任务决定精度标准。
英文摘要
Three classical problems-linear assignment, graph matching, and optimal transport-share a common structure: they seek an optimal matrix-valued coupling on a matrix polytope. We study regularized coupling maps that encompass two families of regularization: entropy/KL regularization, which exponentiates the input score matrix and projects it onto the relevant matrix polytope via the KL-projection; and the Frobenius/Euclidean regularization, which performs the Euclidean projection of the scaled input matrix onto the same polytope. We trace how these two geometries yield complementary algorithmic tools for graph matching (CSGO, ASM, FRAM) and optimal transport (IP-EOT, PSN). The regularized coupling map serves as a direction generator in graph matching and as a solution representation in optimal transport, with the regularizer determining the geometry and the outer task determining the precision standard.
Comments22 pages, 7 figures, 2 tables. An invited manuscript for the Proceedings of the International Consortium of Chinese (ICCM)
