Spectral Graph Matching and Regularized Quadratic Relaxations: Algorithm and Theory
Zhou Fan, Cheng Mao, Yihong Wu, Jiaming Xu
Abstract
Graph matching, also known as network alignment, aims at recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. To tackle this task, we propose a spectral method, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), which first constructs a similarity matrix as a weighted sum of outer products between all pairs of eigenvectors of the two graphs, and then outputs a matching by a simple rounding procedure. For a universality class of correlated Wigner models, GRAMPA achieves exact recovery of the latent matching between two graphs with edge correlation $1 - 1/\mathrm{polylog}(n)$ and average degree at least $\mathrm{polylog}(n)$. This matches the state-of-the-art guarantees for polynomial-time algorithms established for correlated Erdős-Rényi graphs, and significantly improves over existing spectral methods. The superiority of GRAMPA is also demonstrated on a variety of synthetic and real datasets, in terms of both statistical accuracy and computational efficiency.
BibTeX
@InProceedings{pmlr-v119-fan20a,
title = {Spectral Graph Matching and Regularized Quadratic Relaxations: Algorithm and Theory},
author = {Fan, Zhou and Mao, Cheng and Wu, Yihong and Xu, Jiaming},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
pages = {2985--2995},
year = {2020},
editor = {III, Hal Daumé and Singh, Aarti},
volume = {119},
series = {Proceedings of Machine Learning Research},
month = {13--18 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v119/fan20a/fan20a.pdf},
url = {https://proceedings.mlr.press/v119/fan20a.html},
abstract = {Graph matching, also known as network alignment, aims at recovering the latent vertex correspondence between two unlabeled, edge-correlated weighted graphs. To tackle this task, we propose a spectral method, GRAph Matching by Pairwise eigen-Alignments (GRAMPA), which first constructs a similarity matrix as a weighted sum of outer products between all pairs of eigenvectors of the two graphs, and then outputs a matching by a simple rounding procedure. For a universality class of correlated Wigner models, GRAMPA achieves exact recovery of the latent matching between two graphs with edge correlation $1 - 1/\mathrm{polylog}(n)$ and average degree at least $\mathrm{polylog}(n)$. This matches the state-of-the-art guarantees for polynomial-time algorithms established for correlated Erdős-Rényi graphs, and significantly improves over existing spectral methods. The superiority of GRAMPA is also demonstrated on a variety of synthetic and real datasets, in terms of both statistical accuracy and computational efficiency.}
}