Entropic Gromov-Wasserstein Distances: Stability, Algorithms, and Distributional Limits

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.