As a valid metric of metric-measure spaces, Gromov-Wasserstein (GW) distance has shown the potential for matching problems of structured data like point clouds and graphs. However, its application in practice is limited due to the high computational complexity. To overcome this challenge, we propose a novel importance sparsification method, called \textsc{Spar-GW}, to approximate GW distance efficiently. In particular, instead of considering a dense coupling matrix, our method leverages a simple but effective sampling strategy to construct a sparse coupling matrix and update it with few computations. The proposed \textsc{Spar-GW} method is applicable to the GW distance with arbitrary ground cost, and it reduces the complexity from $O(n^4)$ to $O(n^{2+\delta})$ for an arbitrary small $\delta>0$. Theoretically, the convergence and consistency of the proposed estimation for GW distance are established under mild regularity conditions. In addition, this method can be extended to approximate the variants of GW distance, including the entropic GW distance, the fused GW distance, and the unbalanced GW distance. Experiments show the superiority of our \textsc{Spar-GW} to state-of-the-art methods in both synthetic and real-world tasks.

翻译：Gromov-Wasserstein(GW)距离作为计量空间的一个有效衡量标准,显示了将点云和图形等结构数据问题匹配起来的可能性。然而,由于计算复杂程度高,其实际应用有限。为了克服这一挑战,我们建议了一种新型的强调分解方法,称为\ textsc{Spar-GW},以高效地接近GW距离。特别是,我们的方法不是考虑密集的组合矩阵,而是利用简单而有效的取样战略来构建一个稀薄的组合矩阵,并以很少的计算加以更新。提议的计算方法适用于GW距离的任意地面成本,其复杂性从O(n4)美元降低到$O(n ⁇ 2 ⁇ delta}),任意的小美元。理论上说,拟议GW距离估算的趋同性和一致性是在较轻的常规条件下确定的。此外,这一方法可以扩展为GW距离变量的近似值,包括GW距离的实际距离、GW的距离和GW世界的距离。