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Semidefinite Programs for Exact Recovery of a Hidden Community

B. Hajek,Yihong Wu,Jiaming Xu

Published 2016 in Annual Conference Computational Learning Theory

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 $i,j$, $A_{ij} \sim P$ if $i, j$ are both in the community and $A_{ij} \sim Q$ 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 ($P=\mathcal{N}(\mu,1)$ and $Q=\mathcal{N}(0,1)$ with $\mu>0$), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If $K=\omega( n /\log n)$, SDP attains the information-theoretic recovery limits with sharp constants; (2) If $K=\Theta(n/\log n)$, SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If $K=o(n/\log n)$ and $K \to \infty$, 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.

PUBLICATION RECORD

  • Publication year

    2016

  • Venue

    Annual Conference Computational Learning Theory

  • Publication date

    2016-02-20

  • Fields of study

    Mathematics, Computer Science

  • Identifiers
    arXiv 1602.06410
  • External record

    Open on Semantic Scholar

  • Source metadata

    Semantic Scholar

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