Efficient Approximation of Gromov-Wasserstein Distance Using Importance Sparsification

Abstract

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 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 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 Spar-GW to state-of-the-art methods in both synthetic and real-world tasks. Supplementary materials for this article are available online.

KEYWORDS:

• Element-wise sampling
• Importance sampling
• Sinkhorn-scaling algorithm
• Unbalanced Gromov-Wasserstein distance

Supplementary Materials

Appendix: contains the importance sparsification algorithm for approximating the fused Gromov-Wasserstein distance; complete proofs of theoretical results; and additional experiments to evaluate the approximation accuracy, time cost, and memory consumption of the proposed method. (appendix.pdf, a pdf file)

Code: contains Python code that implements the proposed method and reproduces the numerical results. A readme file is included describing the contents. (code.zip, a zip file)

Acknowledgments

We appreciate the Editor, Associate Editor, and two anonymous reviewers for their constructive comments that helped improve the work. This research was supported by Public Computing Cloud, Renmin University of China.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Mengyu Li is supported by the Outstanding Innovative Talents Cultivation Funded Programs 2021 of Renmin University of China. The authors would like to acknowledge the support from National Natural Science Foundation of China grant no.12101606, No.12001042, No.12271522, and Renmin University of China research fund program for young scholars, and Beijing Institute of Technology research fund program for young scholars.