Submatrix localization via message passing

The principal submatrix localization problem deals with recovering a K× K principal submatrix of elevated mean μ in a large n× n symmetric matrix subject to additive standard Gaussian 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 Omega(√n) ≤ K ≤ 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 λ = μ2K2/n, the signal-to-noise ratio, exceeds 1/e. This extends a result by Deshpande and Montanari previously obtained for K=Theta(√n). In addition, the algorithm can be extended to provide exact recovery whenever information-theoretically possible and achieve the information limit of exact recovery as long as K ≥ fracnlog n (frac18e + o(1)). The total running time of the algorithm is O(n2log n). Another version of the submatrix localization problem, known as noisy biclustering, aims to recover a K1× K2 submatrix of elevated mean μ in a large n1× n2 Gaussian matrix. The optimized message passing algorithm and its analysis are adapted to the bicluster problem assuming Omega(√ni) ≤ Ki ≤ o(ni) and K1asymp K2. A sharp information-theoretic condition for the weak recovery of both clusters is also identified.