This note addresses the property frequently mentioned in the literature that the Gromov-Wasserstein (GW) distance is NP-hard. We provide the details on the non-convex nature of the GW optimization problem that imply NP-hardness of the GW distance between finite spaces for any instance of an input data. We further illustrate the non-convexity of the problem with several explicit examples.

翻译：本文探讨了文献中常提及的Gromov-Wasserstein（GW）距离具有NP难性的性质。我们详细阐述了GW优化问题的非凸特性，该特性意味着对于任意输入数据实例，有限空间之间的GW距离均具有NP难性。此外，我们通过若干具体示例进一步说明了该问题的非凸性。