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The Gromov-Wasserstein (GW) distance, rooted in optimal transport (OT)theory, quantifies dissimilarity between metric measure spaces and provides aframework for aligning heterogeneous datasets. While computational aspects ofthe GW problem have been widely studied, a duality theory and fundamentalstatistical questions concerning empirical convergence rates remained obscure.This work closes these gaps for the quadratic GW distance over Euclidean spacesof different dimensions $d_x$ and $d_y$. We treat both the standard and theentropically regularized GW distance, and derive dual forms that represent themin terms of the well-understood OT and entropic OT (EOT) problems,respectively. This enables employing proof techniques from statistical OT basedon regularity analysis of dual potentials and empirical process theory, usingwhich we establish the first GW empirical convergence rates. The derivedtwo-sample rates are $n^{-2/\max\{\min\{d_x,d_y\},4\}}$ (up to a log factorwhen $\min\{d_x,d_y\}=4$) for standard GW and $n^{-1/2}$ for EGW, which matchesthe corresponding rates for standard and entropic OT. The parametric rate forEGW is evidently optimal, while for standard GW we provide matching lowerbounds, which establish sharpness of the derived rates. We also study stabilityof EGW in the entropic regularization parameter and prove approximation andcontinuity results for the cost and optimal couplings. Lastly, the duality isleveraged to shed new light on the open problem of the one-dimensional GWdistance between uniform distributions on $n$ points, illuminating why theidentity and anti-identity permutations may not be optimal. Our results serveas a first step towards a comprehensive statistical theory as well ascomputational advancements for GW distances, based on the discovered dualformulations.

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