  The Gromov-Wasserstein (GW) distance, rooted in optimal transport (OT)
theory, provides a natural framework for aligning heterogeneous datasets. Alas,
statistical estimation of the GW distance suffers from the curse of
dimensionality and its exact computation is NP hard. To circumvent these
issues, entropic regularization has emerged as a remedy that enables parametric
estimation rates via plug-in and efficient computation using Sinkhorn
iterations. Motivated by further scaling up entropic GW (EGW) alignment methods
to data dimensions and sample sizes that appear in modern machine learning
applications, we propose a novel neural estimation approach. Our estimator
parametrizes a minimax semi-dual representation of the EGW distance by a neural
network, approximates expectations by sample means, and optimizes the resulting
empirical objective over parameter space. We establish non-asymptotic error
bounds on the EGW neural estimator of the alignment cost and optimal plan. Our
bounds characterize the effective error in terms of neural network (NN) size
and the number of samples, revealing optimal scaling laws that guarantee
parametric convergence. The bounds hold for compactly supported distributions,
and imply that the proposed estimator is minimax-rate optimal over that class.
Numerical experiments validating our theory are also provided.
