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

We study in this paper computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple nonoverlapping) signal submatrix (of magnitude λ and size km×kn) embedded in a large noise matrix (of size m × n), the goal is to optimal identify the support of the signal submatrix computationally and statistically. Two transition thresholds for the signal-to-noise ratio λ/σ are established in terms of m, n, km and kn. The first threshold, SNRc, corresponds to the computational boundary. We introduce a new linear time spectral algorithm that identifies the submatrix with high probability when the signal strength is above the threshold SNRc. Below this threshold, it is shown that no polynomial time algorithm can succeed in identifying the submatrix, under the hidden clique hypothesis. The second threshold, SNRs, captures the statistical boundary, below which no method can succeed in localization with probability going to one in the minimax sense. The exhaustive search method successfully finds the submatrix above this threshold. In marked contrast to submatrix detection and sparse PCA, the results show an interesting phenomenon that SNRc is always significantly larger than SNRs, which implies an essential gap between statistical optimality and computational efficiency for submatrix localization

Information-theoretic bounds and phase transitions in clustering, sparse PCA, and submatrix localization

We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp information-theoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these "planted models" and null models where the signal matrix is zero. Our bounds differ by at most a factor of root two when the rank is large (in the clustering and submatrix localization problems, when the number of clusters or blocks is large) or the signal matrix is very sparse. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information- theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured 'hard but detectable' regime for community detection in sparse graphs.Comment: For sparse PCA and submatrix localization, we determine the information-theoretic threshold exactly in the limit where the number of blocks is large or the signal matrix is very sparse based on a conditional second moment method, closing the factor of root two gap in the first versio