Gromov-Wasserstein Distance based Object Matching: Asymptotic Inference

Abstract

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 β-trimmed lower bound of the Gromov-Wasserstein distance. We derive for β ∈ [0, 1/2) distributional limits of this test statistic. To this end, we introduce a novel U -type process indexed in β and show its weak convergence. Finally, the theory developed is investigated in Monte Carlo simulations and applied to structural protein comparisons.

Quantized Gromov-Wasserstein

“…The formulation in Equation ( 3) is the one used more often in computational settings [26], and we focus on this case throughout the current work. After initial theoretical development, large scale applications of GW to machine learning and graphics problems were later explored in [25,29], and a vast literature on GW distance has since developed, focusing on both theoretical [18,7,9,8,28,35] and applications-driven [1,6,10] aspects.…”

Statistical analysis of random objects via metric measure Laplacians

“…The locations 1 of these atoms in the three-dimensional space constitute the nodes and this set of points is metrised with the Euclidean distance. The mass is either chosen uniform as in [48] or proportional to the inverse of the temperature factor, as the temperature factor encodes the uncertainty in the measurements. In the sequel, we will often abbreviate metric measure space to mm-space as originating in [18].…”

“…Indeed, in general, (1.2) constitutes a quadratic program and is as such NP-hard [29]. In practice, one then resorts either to regularised versions of the problem [30,35] or to lower bounds [25,23,28,27,48] on the latter distance. In the first case, there is no general proof of convergence of the proposed projected gradient descent and the quantity returned is not necessarily the optimum in (1.2).…”

“…In particular, the fact that these lower bounds are zero does not imply that the mm-spaces are isomorphic. Still, such signatures can convey important information and help discriminate between two mm-spaces, see [15] for an application in structural chemistry and [48] for an application in protein matching.…”

Entropy-Transport distances between unbalanced metric measure spaces

“…The flexibility of such a framework allows to unify the treatment of a series of problems stemming from various fields of science and technology, e.g. chemistry [24], data science [33], multi-omics data alignment [13], computer vision [40], language processing [1], graph [46] and shape [42,49] matching, barycenters & shape analysis [34], generative networks [6,11], machine learning [47]. The theory of metric measure spaces has been flourishing in pure Mathematics as well, providing a unified setting to investigate concentration of measure phenomena [26,41], the theory of Ricci limit spaces [8,19] and, more generally, synthetic notions of Ricci curvature lower bounds [4,30,43,44].…”