To achieve greater accuracy, hypergraph matching algorithms require exponential increases in computational resources. Recent kd-tree-based approximate nearest neighbor (ANN) methods, despite the sparsity of their compatibility tensor, still require exhaustive calculations for large-scale graph matching. This work utilizes CUR tensor decomposition and introduces a novel cascaded second and third-order hypergraph matching framework (CURSOR) for efficient hypergraph matching. A CUR-based second-order graph matching algorithm is used to provide a rough match, and then the core of CURSOR, a fiber-CUR-based tensor generation method, directly calculates entries of the compatibility tensor by leveraging the initial second-order match result. This significantly decreases the time complexity and tensor density. A probability relaxation labeling (PRL)-based matching algorithm, especially suitable for sparse tensors, is developed. Experiment results on large-scale synthetic datasets and widely-adopted benchmark sets demonstrate the superiority of CURSOR over existing methods. The tensor generation method in CURSOR can be integrated seamlessly into existing hypergraph matching methods to improve their performance and lower their computational costs.

翻译：为获得更高精度，超图匹配算法需要指数级增长的计算资源。尽管近期的基于kd树的近似最近邻方法利用兼容性张量的稀疏性，但在大规模图匹配中仍需穷举计算。本文利用CUR张量分解，提出了一种新颖的级联二阶与三阶超图匹配框架CURSOR，用于高效超图匹配。首先，基于CUR的二阶图匹配算法提供粗略匹配结果；随后，CURSOR的核心——基于纤维CUR的张量生成方法——利用初始二阶匹配结果直接计算兼容性张量的元素。这显著降低了时间复杂度和张量密度。此外，开发了一种特别适用于稀疏张量的基于概率松弛标记的匹配算法。大规模合成数据集及广泛采用的基准数据集上的实验结果表明，CURSOR优于现有方法。CURSOR中的张量生成方法可无缝集成到现有超图匹配方法中，以提升其性能并降低计算成本。