Semi-discrete Gromov-Wasserstein distances: Existence of Gromov-Monge Maps and Statistical Theory
Keywords: Gromov-Wasserstein, semi-discrete Gromov-Wasserstein, Gromov-Monge maps, sample complexity, limit distributions
TL;DR: We study the Gromov-Wasserstein problem in the semi-discrete setting, providing a simple condition guaranteeing the existence of Gromov-Monge maps. We equally address the questions of sample complexity and limit distributions.
Abstract: The Gromov-Wasserstein (GW) distance serves as a discrepancy measure between metric measure spaces. Despite recent theoretical developments, its structural properties, such as existence of optimal maps, remain largely unaccounted for. In this work, we analyze the semi-discrete regime for the GW problem wherein one measure is finitely supported. Notably, we derive a primitive condition which guarantees the existence of optimal maps. This condition also enables us to derive the asymptotic distribution of the empirical semi-discrete GW distance under proper centering and scaling. As a complement to this asymptotic result, we also derive expected empirical convergence rates. As is the case with the standard Wasserstein distance, the rate we derive in the semi-discrete GW case, $n^{-\frac{1}{2}}$, is dimension-independent which is in stark contrast to the curse of dimensionality rate obtained in general.
Submission Number: 63
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