Computational difficulty of quadratic matching and the Gromov-Wasserstein distance has led to various approximation and relaxation schemes. One of such methods, relying on the notion of distance profiles, has been widely used in practice, but its theoretical understanding is limited. By delving into the statistical complexity of the previously proposed method based on distance profiles, we show that it suffers from the curse of dimensionality unless we make certain assumptions on the underlying metric measure spaces. Building on this insight, we propose and analyze a modified matching procedure that can be used to robustly match points under a certain probabilistic setting. We demonstrate the performance of the proposed methods using simulations and real data applications to complement the theoretical findings. As a result, we contribute to the literature by providing theoretical underpinnings of the matching procedures based on distance invariants like distance profiles, which have been widely used in practice but rarely analyzed theoretically.

翻译：二次匹配与Gromov-Wasserstein距离的计算困难性催生了多种近似与松弛方案。其中依赖距离剖面概念的方法已在实践中广泛应用，但其理论认知仍有限。通过深入分析先前提出的基于距离剖面方法的统计复杂性，我们证明该方法会遭受维度灾难，除非对底层度量测度空间施加特定假设。基于此洞见，我们提出并分析了一种改进的匹配流程，可在特定概率设定下实现鲁棒的点匹配。我们通过仿真实验与真实数据应用展示了所提方法的性能，以补充理论发现。最终，本研究通过为基于距离不变量（如距离剖面）的匹配流程提供理论基础作出贡献——这类方法在实践中已被广泛使用，却鲜有理论分析。