Optimal Non-Convex Exact Recovery in Stochastic Block Model via Projected Power Method

Peng Wang, Huikang Liu, Zirui Zhou, Anthony Man-Cho So
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:10828-10838, 2021.

Abstract

In this paper, we study the problem of exact community recovery in the symmetric stochastic block model, where a graph of $n$ vertices is randomly generated by partitioning the vertices into $K \ge 2$ equal-sized communities and then connecting each pair of vertices with probability that depends on their community memberships. Although the maximum-likelihood formulation of this problem is discrete and non-convex, we propose to tackle it directly using projected power iterations with an initialization that satisfies a partial recovery condition. Such an initialization can be obtained by a host of existing methods. We show that in the logarithmic degree regime of the considered problem, the proposed method can exactly recover the underlying communities at the information-theoretic limit. Moreover, with a qualified initialization, it runs in $\mO(n\log^2n/\log\log n)$ time, which is competitive with existing state-of-the-art methods. We also present numerical results of the proposed method to support and complement our theoretical development.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-wang21o, title = {Optimal Non-Convex Exact Recovery in Stochastic Block Model via Projected Power Method}, author = {Wang, Peng and Liu, Huikang and Zhou, Zirui and So, Anthony Man-Cho}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {10828--10838}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/wang21o/wang21o.pdf}, url = {https://proceedings.mlr.press/v139/wang21o.html}, abstract = {In this paper, we study the problem of exact community recovery in the symmetric stochastic block model, where a graph of $n$ vertices is randomly generated by partitioning the vertices into $K \ge 2$ equal-sized communities and then connecting each pair of vertices with probability that depends on their community memberships. Although the maximum-likelihood formulation of this problem is discrete and non-convex, we propose to tackle it directly using projected power iterations with an initialization that satisfies a partial recovery condition. Such an initialization can be obtained by a host of existing methods. We show that in the logarithmic degree regime of the considered problem, the proposed method can exactly recover the underlying communities at the information-theoretic limit. Moreover, with a qualified initialization, it runs in $\mO(n\log^2n/\log\log n)$ time, which is competitive with existing state-of-the-art methods. We also present numerical results of the proposed method to support and complement our theoretical development.} }
Endnote
%0 Conference Paper %T Optimal Non-Convex Exact Recovery in Stochastic Block Model via Projected Power Method %A Peng Wang %A Huikang Liu %A Zirui Zhou %A Anthony Man-Cho So %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-wang21o %I PMLR %P 10828--10838 %U https://proceedings.mlr.press/v139/wang21o.html %V 139 %X In this paper, we study the problem of exact community recovery in the symmetric stochastic block model, where a graph of $n$ vertices is randomly generated by partitioning the vertices into $K \ge 2$ equal-sized communities and then connecting each pair of vertices with probability that depends on their community memberships. Although the maximum-likelihood formulation of this problem is discrete and non-convex, we propose to tackle it directly using projected power iterations with an initialization that satisfies a partial recovery condition. Such an initialization can be obtained by a host of existing methods. We show that in the logarithmic degree regime of the considered problem, the proposed method can exactly recover the underlying communities at the information-theoretic limit. Moreover, with a qualified initialization, it runs in $\mO(n\log^2n/\log\log n)$ time, which is competitive with existing state-of-the-art methods. We also present numerical results of the proposed method to support and complement our theoretical development.
APA
Wang, P., Liu, H., Zhou, Z. & So, A.M.. (2021). Optimal Non-Convex Exact Recovery in Stochastic Block Model via Projected Power Method. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:10828-10838 Available from https://proceedings.mlr.press/v139/wang21o.html.

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