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The principal submatrix localization problem deals with recovering a $K\timesK$ principal submatrix of elevated mean $\mu$ in a large $n\times n$ symmetricmatrix subject to additive standard Gaussian noise. This problem serves as aprototypical example for community detection, in which the communitycorresponds to the support of the submatrix. The main result of this paper isthat in the regime $\Omega(\sqrt{n}) \leq K \leq o(n)$, the support of thesubmatrix can be weakly recovered (with $o(K)$ misclassification errors onaverage) by an optimized message passing algorithm if $\lambda = \mu^2K^2/n$,the signal-to-noise ratio, exceeds $1/e$. This extends a result by Deshpandeand Montanari previously obtained for $K=\Theta(\sqrt{n}).$ In addition, thealgorithm can be extended to provide exact recovery wheneverinformation-theoretically possible and achieve the information limit of exactrecovery as long as $K \geq \frac{n}{\log n} (\frac{1}{8e} + o(1))$. The totalrunning time of the algorithm is $O(n^2\log n)$.  Another version of the submatrix localization problem, known as noisybiclustering, aims to recover a $K_1\times K_2$ submatrix of elevated mean$\mu$ in a large $n_1\times n_2$ Gaussian matrix. The optimized message passingalgorithm and its analysis are adapted to the bicluster problem assuming$\Omega(\sqrt{n_i}) \leq K_i \leq o(n_i)$ and $K_1\asymp K_2.$ A sharpinformation-theoretic condition for the weak recovery of both clusters is alsoidentified.

Submatrix localization via message passing