Efficient Approximation of Gromov-Wasserstein Distance Using Importance Sparsification

TL;DR

This paper introduces Spar-GW, a sparsification method that significantly accelerates the computation of Gromov-Wasserstein distance using importance sampling, making it practical for large structured data matching.

Contribution

The paper proposes Spar-GW, a novel importance sparsification technique that reduces GW distance computation complexity while maintaining theoretical guarantees and extending to variants.

Findings

Spar-GW reduces complexity from O(n^4) to O(n^{2+})

Theoretical convergence and consistency are established for the method

Experiments demonstrate superior performance over state-of-the-art methods

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 \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.