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.

翻译：Gromov-Wasserstein（GW）距离根植于最优传输（OT）理论，为对齐异构数据集提供了自然框架。然而，GW距离的统计估计受到维度灾难的影响，且其精确计算属于NP难问题。为规避这些困难，熵正则化作为补救方法应运而生，可通过插件法实现参数化估计速率，并借助Sinkhorn迭代进行高效计算。为进一步将熵化GW（EGW）对齐方法扩展到现代机器学习应用中常见的数据维度和样本规模，我们提出一种新型神经估计方法。该估计器通过神经网络参数化EGW距离的极小极大半对偶表示，利用样本均值近似期望，并在参数空间上优化所得经验目标函数。我们建立了对齐代价和最优传输方案的EGW神经估计的非渐近误差界。这些误差界通过神经网络（NN）规模与样本数刻画有效误差，揭示了保证参数化收敛的最优标度律。该误差界适用于紧支撑分布，并表明所提估计器在该分布类上达到极小极大速率最优。本文还提供了验证理论结果的数值实验。