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, 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 Ω(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 = Θ(n) and µ = Θ(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 ≥_log ⁿ _n(₈ ¹ _e + o(1)). The total running time of the algorithm is O(n² log 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 Ω(n_i) ≤ K_i ≤ o(n_i) and K₁ K₂. A sharp information-theoretic condition for the weak recovery of both clusters is also identified.