Computational and Statistical Boundaries for Submatrix Localization in a Large Noisy Matrix

The interplay between computational efficiency and statistical accuracy in high-dimensional inference has drawn increasing attention in the literature. In this paper, we study computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple non-overlapping) signal submatrix (of magnitude λ and size km × kn) contaminated with a noise matrix (of size m × n), we establish two transition thresholds for the signal to noise λ/σ ratio in terms of m, n, km, and kn. The first threshold, sf SNRc, corresponds to the computational boundary. Below this threshold, it is shown that no polynomial time algorithm can succeed in identifying the submatrix, under the textithidden clique hypothesis. We introduce adaptive linear time spectral algorithms that identify the submatrix with high probability when the signal strength is above the threshold sf SNRc. The second threshold, sf SNRs, captures the statistical boundary, below which no method can succeed with probability going to one in the minimax sense. The exhaustive search method successfully finds the submatrix above this threshold. The results show an interesting phenomenon that sf SNRc is always significantly larger than sf SNRs, which implies an essential gap between statistical optimality and computational efficiency for submatrix localization.