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Gromov-Wasserstein Distances: Entropic Regularization, Duality, and Sample Complexity

Zhengxin ZhangZiv GoldfeldYoussef MrouehBharath K. Sriperumbudur
Dec 2022
摘要
The Gromov-Wasserstein (GW) distance quantifies dissimilarity between metricmeasure spaces and provides a meaningful figure of merit for applicationsinvolving heterogeneous data. While computational aspects of the GW distancehave been widely studied, a strong duality theory and fundamental statisticalquestions concerning empirical convergence rates remained obscure. This workcloses these gaps for the $(2,2)$-GW distance (namely, with quadratic cost)over Euclidean spaces of different dimensions $d_x$ and $d_y$. We consider boththe standard GW and the entropic GW (EGW) distances, derive their dual forms,and use them to analyze expected empirical convergence rates. The resultingrates are $n^{-2/\max\{d_x,d_y,4\}}$ (up to a log factor when$\max\{d_x,d_y\}=4$) and $n^{-1/2}$ for the two-sample GW and EGW problems,respectively, which matches the corresponding rates for standard and entropicoptimal transport distances. We also study stability of EGW in the entropicregularization parameter and establish approximation and continuity results forthe cost and optimal couplings. Lastly, the duality is leveraged to shed newlight on the open problem of the one-dimensional GW distance between uniformdistributions on $n$ points, illuminating why the identity and anti-identitypermutations may not be optimal. Our results serve as a first step towards acomprehensive statistical theory as well as computational advancements for GWdistances, based on the discovered dual formulation.
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