Duality and Sample Complexity for the Gromov-Wasserstein Distance

The Gromov-Wasserstein (GW) distance, rooted in optimal transport (OT) theory, quantifies dissimilarity between metric measure spaces and provides a framework for aligning heterogeneous datasets. While computational aspects of the GW problem have been widely studied, a duality theory and fundamental statistical questions concerning empirical convergence rates remained obscure. This work closes these gaps for the quadratic GW distance over Euclidean spaces of different dimensions $d_x$ and $d_y$. We derive a dual form that represents the GW distance in terms of the well-understood OT problem. This enables employing proof techniques from statistical OT based on regularity analysis of dual potentials and empirical process theory, using which we establish the first GW empirical convergence rates. The derived two-sample rate is $n^{-2/\max\{\min\{d_x,d_y\},4\}}$ (up to a log factor when $\min\{d_x,d_y\}=4$), which matches the corresponding rates for OT. We also provide matching lower bounds, thus establishing sharpness of the derived rates. Lastly, the duality is leveraged to shed new light on the open problem of the one-dimensional GW distance between uniform distributions on $n$ points, illuminating why the identity and anti-identity permutations may not be optimal. Our results serve as a first step towards a comprehensive statistical theory as well as computational advancements for GW distances, based on the discovered dual formulations.