Semidefinite Programs for Exact Recovery of a Hidden Community

Bruce Hajek, Yihong Wu, Jiaming Xu
29th Annual Conference on Learning Theory, PMLR 49:1051-1095, 2016.

Abstract

We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality K from an n \times n symmetric data matrix A, where for distinct indices i,j, A_ij ∼P if i, j are both in the community and A_ij ∼Q otherwise, for two known probability distributions P and Q. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case (P=\rm Bern(p) and Q=\rm Bern(q) with p>q) and the Gaussian case (P=\mathcalN(μ,1) and Q=\mathcalN(0,1) with μ>0), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If K=ω( n /\log n), SDP attains the information-theoretic recovery limits with sharp constants; (2) If K=Θ(n/\log n), SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If K=o(n/\log n) and K \to ∞, SDP is order-wise suboptimal. The same critical scaling for K is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of n vertices partitioned into multiple communities of equal size K. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix.

Cite this Paper

BibTeX
@InProceedings{pmlr-v49-hajek16, title = {Semidefinite Programs for Exact Recovery of a Hidden Community}, author = {Hajek, Bruce and Wu, Yihong and Xu, Jiaming}, booktitle = {29th Annual Conference on Learning Theory}, pages = {1051--1095}, year = {2016}, editor = {Feldman, Vitaly and Rakhlin, Alexander and Shamir, Ohad}, volume = {49}, series = {Proceedings of Machine Learning Research}, address = {Columbia University, New York, New York, USA}, month = {23--26 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v49/hajek16.pdf}, url = {https://proceedings.mlr.press/v49/hajek16.html}, abstract = {We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality K from an n \times n symmetric data matrix A, where for distinct indices i,j, A_ij ∼P if i, j are both in the community and A_ij ∼Q otherwise, for two known probability distributions P and Q. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case (P=\rm Bern(p) and Q=\rm Bern(q) with p>q) and the Gaussian case (P=\mathcalN(μ,1) and Q=\mathcalN(0,1) with μ>0), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If K=ω( n /\log n), SDP attains the information-theoretic recovery limits with sharp constants; (2) If K=Θ(n/\log n), SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If K=o(n/\log n) and K \to ∞, SDP is order-wise suboptimal. The same critical scaling for K is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of n vertices partitioned into multiple communities of equal size K. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix. } }
Endnote
%0 Conference Paper %T Semidefinite Programs for Exact Recovery of a Hidden Community %A Bruce Hajek %A Yihong Wu %A Jiaming Xu %B 29th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2016 %E Vitaly Feldman %E Alexander Rakhlin %E Ohad Shamir %F pmlr-v49-hajek16 %I PMLR %P 1051--1095 %U https://proceedings.mlr.press/v49/hajek16.html %V 49 %X We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality K from an n \times n symmetric data matrix A, where for distinct indices i,j, A_ij ∼P if i, j are both in the community and A_ij ∼Q otherwise, for two known probability distributions P and Q. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case (P=\rm Bern(p) and Q=\rm Bern(q) with p>q) and the Gaussian case (P=\mathcalN(μ,1) and Q=\mathcalN(0,1) with μ>0), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If K=ω( n /\log n), SDP attains the information-theoretic recovery limits with sharp constants; (2) If K=Θ(n/\log n), SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If K=o(n/\log n) and K \to ∞, SDP is order-wise suboptimal. The same critical scaling for K is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of n vertices partitioned into multiple communities of equal size K. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix.
RIS
TY - CPAPER TI - Semidefinite Programs for Exact Recovery of a Hidden Community AU - Bruce Hajek AU - Yihong Wu AU - Jiaming Xu BT - 29th Annual Conference on Learning Theory DA - 2016/06/06 ED - Vitaly Feldman ED - Alexander Rakhlin ED - Ohad Shamir ID - pmlr-v49-hajek16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 49 SP - 1051 EP - 1095 L1 - http://proceedings.mlr.press/v49/hajek16.pdf UR - https://proceedings.mlr.press/v49/hajek16.html AB - We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality K from an n \times n symmetric data matrix A, where for distinct indices i,j, A_ij ∼P if i, j are both in the community and A_ij ∼Q otherwise, for two known probability distributions P and Q. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case (P=\rm Bern(p) and Q=\rm Bern(q) with p>q) and the Gaussian case (P=\mathcalN(μ,1) and Q=\mathcalN(0,1) with μ>0), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If K=ω( n /\log n), SDP attains the information-theoretic recovery limits with sharp constants; (2) If K=Θ(n/\log n), SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If K=o(n/\log n) and K \to ∞, SDP is order-wise suboptimal. The same critical scaling for K is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of n vertices partitioned into multiple communities of equal size K. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix. ER -
APA
Hajek, B., Wu, Y. & Xu, J.. (2016). Semidefinite Programs for Exact Recovery of a Hidden Community. 29th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 49:1051-1095 Available from https://proceedings.mlr.press/v49/hajek16.html.

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