Convergence of empirical Gromov-Wasserstein distance

TL;DR

This paper analyzes the convergence rates of the empirical Gromov-Wasserstein distance estimator, extending previous results to unbounded supports and establishing matching lower bounds and deviation inequalities.

Contribution

It extends convergence rate results for empirical GW distance to unbounded supports with finite moments and provides deviation inequalities and minimax lower bounds.

Findings

Convergence rate of n^{-2/min{d_x,d_y}} established for unbounded supports.

Matching minimax lower bounds up to logarithmic factors.

Deviation inequalities show high probability bounds for empirical GW error.

Abstract

We study rates of convergence for estimation of the Gromov-Wasserstein (GW) distance. For two marginals supported on compact subsets of $\R^{d_x}$ and $\R^{d_y}$, respectively, with $\min \{ d_x,d_y \} > 4$, prior work established the rate $n^{-\frac{2}{\min\{d_x,d_y\}}}$ in $L^1$ for the plug-in empirical estimator based on $n$ i.i.d. samples. We extend this fundamental result to marginals with unbounded supports, assuming only finite polynomial moments. Our proof techniques for the upper bounds can be adapted to obtain sample complexity results for penalized Wasserstein alignment that encompasses the GW distance and Wasserstein Procrustes. Furthermore, we establish matching minimax lower bounds (up to logarithmic factors) for estimating the GW distance. Finally, we establish deviation inequalities for the error of empirical GW in cases where two marginals have compact supports, exponential tails, or finite polynomial moments. The deviation inequalities yield that the same rate $n^{-\frac{2}{\min\{d_x,d_y\}}}$ holds for empirical GW also with high probability.