Distribution of Distances based Object Matching: Asymptotic Inference

Abstract

In this article, we aim to provide a statistical theory for object matching based on a lower bound of the Gromov-Wasserstein distance related to the distribution of (pairwise) distances of the considered objects. To this end, we model general objects as metric measure spaces. Based on this, we propose a simple and efficiently computable asymptotic statistical test for pose invariant object discrimination. This is based on a (β-trimmed) empirical version of the afore-mentioned lower bound. We derive the distributional limits of this test statistic for the trimmed and untrimmed case. For this purpose, we introduce a novel U-type process indexed in β and show its weak convergence. The theory developed is investigated in Monte Carlo simulations and applied to structural protein comparisons. Supplementary materials for this article are available online.