Distribution of Distances based Object Matching: Asymptotic Inference

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 (beta-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 t 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.

Automated summary

In this article, a statistical theory for object matching is proposed based on a lower bound of the Gromov-Wasserstein distance. The method models general objects as metric measure spaces and introduces a simple and efficiently computable asymptotic statistical test for pose invariant object discrimination. The theory is validated through Monte Carlo simulations and applied to structural protein comparisons.