Efficient Approximation of Gromov-Wasserstein Distance Using Importance Sparsification

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...

Full description

Saved in:
Bibliographic Details
Published in:Journal of computational and graphical statistics Vol. 32; no. 4; pp. 1512 - 1523
Main Authors: Li, Mengyu, Yu, Jun, Xu, Hongteng, Meng, Cheng
Format: Journal Article
Language:English
Published: Alexandria Taylor & Francis 02.10.2023
Taylor & Francis Ltd
Subjects:
Complexity
Coupling
Element-wise sampling
Importance sampling
Sinkhorn-scaling algorithm
Structured data
Unbalanced Gromov-Wasserstein distance
ISSN:1061-8600, 1537-2715
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 + δ ) for an arbitrary small δ > 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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1061-8600
1537-2715
DOI:10.1080/10618600.2023.2165500