The Gromov-Wasserstein (GW) distance quantifies discrepancy between metric measure spaces, but suffers from computational hardness. The entropic Gromov-Wasserstein (EGW) distance serves as a computationally efficient proxy for the GW distance. Recently, it was shown that the quadratic GW and EGW distances admit variational forms that tie them to the well-understood optimal transport (OT) and entropic OT (EOT) problems. By leveraging this connection, we derive two notions of stability for the EGW problem with the quadratic or inner product cost. The first stability notion enables us to establish convexity and smoothness of the objective in this variational problem. This results in the first efficient algorithms for solving the EGW problem that are subject to formal guarantees in both the convex and non-convex regimes. The second stability notion is used to derive a comprehensive limit distribution theory for the empirical EGW distance and, under additional conditions, asymptotic normality, bootstrap consistency, and semiparametric efficiency thereof.

翻译：Gromov-Wasserstein（GW）距离量化了度量度量空间之间的差异，但存在计算困难。熵化Gromov-Wasserstein（EGW）距离作为GW距离的计算高效代理。近期研究表明，二次型GW和EGW距离具有变分形式，将其与已被充分理解的最优传输（OT）和熵化OT（EOT）问题相关联。通过利用这一关联，我们推导出二次型或内积成本下EGW问题的两种稳定性概念。第一种稳定性概念使我们能够建立该变分问题中目标的凸性和光滑性。这产生了首个在凸与非凸区域均具有形式化保证的EGW问题高效算法。第二种稳定性概念用于推导经验EGW距离的全面极限分布理论，并在附加条件下推导其渐近正态性、自举一致性和半参数效率。