Achieving Exact Cluster Recovery Threshold via Semidefinite Programming: Extensions
Abstract
Resolving a conjecture of Abbe, Bandeira and Hall, the authors have recently shown that the semidefinite programming (SDP) relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community structure under the binary stochastic block model of two equal-sized clusters. The same was shown for the case of a single cluster and outliers. Extending the proof techniques, in this paper it is shown that SDP relaxations also achieve the sharp recovery threshold in the following cases: (1) Binary stochastic block model with two clusters of sizes proportional to network size but not necessarily equal; (2) Stochastic block model with a fixed number of equal-sized clusters; (3) Binary censored block model with the background graph being Erd\H{o}s-R\'enyi. Furthermore, a sufficient condition is given for an SDP procedure to achieve exact recovery for the general case of a fixed number of clusters plus outliers. These results demonstrate the versatility of SDP relaxation as a simple, general purpose, computationally feasible methodology for community detection.
Keywords
Cite
@article{arxiv.1502.07738,
title = {Achieving Exact Cluster Recovery Threshold via Semidefinite Programming: Extensions},
author = {Bruce Hajek and Yihong Wu and Jiaming Xu},
journal= {arXiv preprint arXiv:1502.07738},
year = {2016}
}
Comments
This paper was accepted to IEEE Transactions on Information Theory on April 25, 2016. The material was presented in part at the 2015 49th Asilomar Conference on Signals, Systems and Computers and the 2015 IEEE Information Theory Workshop. This work was also in part presented at the Workshop on Community Detection, February 26-27, Institut Henri Poincar\'e, Paris