Non-Convex Joint Community Detection and Group Synchronization via Generalized Power Method
Sijin Chen, Xiwei Cheng, Anthony Man-Cho So
Abstract
This paper proposes a Generalized Power Method (GPM) to simultaneously solve the joint problem of community detection and group synchronization in a direct non-convex manner, in contrast to the existing method of semidefinite programming (SDP). Under a natural extension of stochastic block model (SBM), our theoretical analysis proves that the proposed algorithm is able to exactly recover the ground truth in $O(n\log^2 n)$ time for problems of size $n$, sharply outperforming the $O(n^{3.5})$ runtime of SDP. Moreover, we give a lower bound of model parameters as a sufficient condition for the exact recovery of GPM. The new bound breaches the information-theoretic limit for pure community detection under SBM, thus demonstrating the superiority of our simultaneous optimization algorithm over any two-stage method that performs the two tasks in succession. We also conduct numerical experiments on GPM and SDP to corroborate our theoretical analysis.
BibTeX
@InProceedings{pmlr-v238-chen24e,
title = {Non-Convex Joint Community Detection and Group Synchronization via Generalized Power Method},
author = {Chen, Sijin and Cheng, Xiwei and Man-Cho So, Anthony},
booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics},
pages = {2899--2907},
year = {2024},
editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen},
volume = {238},
series = {Proceedings of Machine Learning Research},
month = {02--04 May},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v238/chen24e/chen24e.pdf},
url = {https://proceedings.mlr.press/v238/chen24e.html},
abstract = {This paper proposes a Generalized Power Method (GPM) to simultaneously solve the joint problem of community detection and group synchronization in a direct non-convex manner, in contrast to the existing method of semidefinite programming (SDP). Under a natural extension of stochastic block model (SBM), our theoretical analysis proves that the proposed algorithm is able to exactly recover the ground truth in $O(n\log^2 n)$ time for problems of size $n$, sharply outperforming the $O(n^{3.5})$ runtime of SDP. Moreover, we give a lower bound of model parameters as a sufficient condition for the exact recovery of GPM. The new bound breaches the information-theoretic limit for pure community detection under SBM, thus demonstrating the superiority of our simultaneous optimization algorithm over any two-stage method that performs the two tasks in succession. We also conduct numerical experiments on GPM and SDP to corroborate our theoretical analysis.}
}