In this work, we present the Bregman Alternating Projected Gradient (BAPG) method, a single-loop algorithm that offers an approximate solution to the Gromov-Wasserstein (GW) distance. We introduce a novel relaxation technique that balances accuracy and computational efficiency, albeit with some compromises in the feasibility of the coupling map. Our analysis is based on the observation that the GW problem satisfies the Luo-Tseng error bound condition, which relates to estimating the distance of a point to the critical point set of the GW problem based on the optimality residual. This observation allows us to provide an approximation bound for the distance between the fixed-point set of BAPG and the critical point set of GW. Moreover, under a mild technical assumption, we can show that BAPG converges to its fixed point set. The effectiveness of BAPG has been validated through comprehensive numerical experiments in graph alignment and partition tasks, where it outperforms existing methods in terms of both solution quality and wall-clock time.

翻译：本文提出Bregman交替投影梯度（BAPG）方法，这是一种用于近似求解Gromov-Wasserstein（GW）距离的单环算法。我们引入了一种新的松弛技术，在耦合映射可行性方面做出适当权衡的同时，平衡了计算精度与效率。我们的分析基于GW问题满足Luo-Tseng误差界条件这一观察，该条件通过最优性残差表征任意点到GW问题临界点集的距离估计。这一观察使我们能够给出BAPG不动点集与GW临界点集之间距离的近似界。此外，在温和的技术假设下，我们证明了BAPG收敛至其不动点集。通过在图对齐与图分割任务中的全面数值实验，BAPG在解质量与计算时间两方面均优于现有方法，其有效性得到了充分验证。