# Submatrix localization via message passing; Bruce Hajek, Yihong Wu, Jiaming Xu

The principal submatrix localization problem deals with

recovering a $K\times K$ principal submatrix of elevated mean

$\mu$ in a large $n\times n$ symmetric matrix subject to

additive standard Gaussian noise, or more generally, mean zero,

variance one, subgaussian noise. This problem serves as a

prototypical example for community detection, in which the

community corresponds to the support of the submatrix. The main

result of this paper is that in the regime $Ømega(\sqrt{n}) \leq

K \leq o(n)$, the support of the submatrix can be weakly

recovered (with $o(K)$ misclassification errors on average) 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 Deshpande and Montanari previously obtained

for $K=\Theta(\sqrt{n})$ and $\mu=\Theta(1).$ In addition, the

algorithm can be combined with a voting procedure to achieve the

information-theoretic limit of exact recovery with sharp

constants for all $K \geq \frac{n}{\log n} (\frac{1}{8e} +

o(1))$. The total running time of the algorithm is $O(n^2\log

n)$. Another version of the submatrix localization problem,

known as noisy biclustering, 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 passing algorithm and its

analysis are adapted to the bicluster problem assuming

$Ømega(\sqrt{n_i}) \leq K_i \leq o(n_i)$ and $K_1\asymp K_2.$ A

sharp information-theoretic condition for the weak recovery of

both clusters is also identified.

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