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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.
Supplementary Materials
In the supplementary materials we give the full, technical proofs of our main results and we provide additional details for the examples considered in the main document. Furthermore, we include a more general consideration of distributions of Euclidean distances of a certain class of metric measure spaces and we present additional material on our simulation results. For the sake of completeness, several technical auxiliary results conclude the supplementary materials.
Disclosure Statement
The authors report there are no competing interests to declare.
Funding
Acknowledgments
We are grateful to F. Mémoli, V. Liebscher and M. Wendler for interesting discussions and helpful comments and to C. Brécheteau for providing code for the application of the DTM-test. We thank two anonymous referees and an associate editor for their constructive comments which improved the article.