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.

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One possibility to circumvent the distinguishability problem and to compare objects in a pose invariant manner is to model them as metric spaces X = (X, d X ) and Y = (Y, d Y ) and to consider them as elements of the set of isometry classes of compact metric spaces M (two compact metric spaces are in the same class if and only if there exists an isometry between them). It is well known that M can be turned into a metric space by equipping it with the Gromov-Hausdorff distancewhere supp (π) denotes the support of the measure π and C(µ, ν) denotes the set of all couplings of µ and ν, i.e., the set of all measures π on Z × Z such thatx,x ∈X, y,y ∈Y s.t. (x,y),(x ,y )∈supp(π)Based on this, the ultrametric Gromov-Wasserstein distance of order p ∈ [1, ∞], is defined asdis ult p (π).(3.7) Main ResultsThe main contributions of Paper B are the proofs that both Strum's ultrametric Gromov-Wasserstein distance and the ultrametric Gromov-Wasserstein distance constitute metrics on U w , the derivation of several properties of these metrics, a polynomial time algorithm for the calculation of u sturm GW,∞ = u GW,∞ , the derivation of polynomial time computable lower bounds for u GW,p , 1 ≤ p < ∞, as well as an empirical illustration of the distance captured by u GW,p for synthetic and real data.Properties of the Metrics: Recall that a metric space (X, d X ) is called p-metric space for p ∈ [1, ∞) if it fulfills the subsequent stronger version of the triangle inequality:for all x, x , x ∈ X. Note that ultrametric spaces can be understood as a limit case of p-metric spaces (p → ∞). Our first main result is the fact that both u sturm GW,p and u GW,p are p-metrics on U w that induce different topologies than d sturm GW,p and d GW,p . Further, we show that 2 −1/p u GW,p ≤ u sturm GW,p , that u sturm GW,p and u GW,p are topologically equivalent for 1 ≤ p < ∞ and that u sturm GW,∞ = u GW,∞ .Computational Aspects: Similar as for the ultrametric Gromov-Hausdorff distance, we derive Discussion and Related WorkBy construction u sturm GW,p and u GW,p are closely related to d sturm GW,p and d GW,p as well as the ultrametric Gromov-Hausdorff distance u GH . Sturm's Gromov-Wasserstein distance is studied in Sturm (2006Sturm ( , 2012 and the Gromov-Wasserstein distance is investigated by ; Chowdhury and Mémoli (2019). Furthermore, the ultrametric Gromov-Hausdorff distance u GH is introduced in Zarichnyi (2005), its theoretical properties are studied in Qiu (2009) and a polynomial time algorithm for its computation is devised in Mémoli et al. (2021b). Additionally, Mémoli and Wan (2019) study a variant of the Gromov-Hausdorff metric on the collection of all p-metric spaces, 1 ≤ p ≤ ∞, that coincides for p = ∞ with u GH . Similar in spirit to our work are Evans (2007), who describes some variants of the Gromov-Hausdorff distance between metric trees, and Greven et al. (2009), who introduces metric measure space representations of trees and a certain Gromov-Prokhorov type of metric on the collection of these representations.In Kloeckner (2015), th...

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