A k-fold homogeneity test for metric spaces

A. Munk, L. A. Rodríguez Ramírez, F. Steimkamp

We develop a statistical framework to match finitely many objects modeled as metric measure spaces (mm-spaces). The method relies on the second lower bound (SLB) of the Gromov–Wasserstein distance, capturing differences in distributions of pairwise distances within each space. To distinguish objects, we define a surrogate SLB barycenter, explicitly computed from their distance distributions. For K spaces with n samples each, the statistic is efficiently computed in O(K·n²) operations. We establish its asymptotic distribution and finite-sample bounds, enabling statistical inference. Applications include an asymptotic test for pose-invariant discrimination and a classification approach using SLB barycenters. The methods are validated through simulations and applied to comparing protein domain structures using the SCOPe database.

Keywords: Gromov-Wasserstein, Optimal Transport, Second Lower Bound

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