Gromov-Wasserstein Distance based Object Matching: Asymptotic Inference

In this paper, we aim to provide a statistical theory for object matching based on the Gromov-Wasserstein distance. 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 an empirical version of a $\beta$-trimmed lower bound of the Gromov-Wasserstein distance. We derive for $\beta\in[0,1/2)$ distributional limits of this test statistic. To this end, we introduce a novel $U$-type process indexed in $\beta$ and show its weak convergence. Finally, the theory developed is investigated in Monte Carlo simulations and applied to structural protein comparisons.Comment: For a version with the complete supplement see [v2

Gromov-Wasserstein Distance based Object Matching: Asymptotic Inference

In this paper, we aim to provide a statistical theory for object matching based on the Gromov-Wasserstein distance. 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 an empirical version of a $\beta$-trimmed lower bound of the Gromov-Wasserstein distance. We derive for $\beta\in[0,1/2)$ distributional limits of this test statistic. To this end, we introduce a novel $U$-type process indexed in $\beta$ and show its weak convergence. Finally, the theory developed is investigated in Monte Carlo simulations and applied to structural protein comparisons

Distribution of Distances based Object Matching: Asymptotic Inference

In this paper, 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.</p