This article is an extended version of previous work of the authors [40, 41] on low-rank matrix estimation in the presence of constraints on the factors into which the matrix is factorized. Low-rank matrix factorization is one of the basic methods used in data analysis for unsupervised learning of relevant features and other types of dimensionality reduction. We present a framework to study the constrained low-rank matrix estimation for a general prior on the factors, and a general output channel through which the matrix is observed. We draw a paralel with the study of vector-spin glass models - presenting a unifying way to study a number of problems considered previously in separate statistical physics works. We present a number of applications for the problem in data analysis. We derive in detail a general form of the low-rank approximate message passing (Low- RAMP) algorithm, that is known in statistical physics as the TAP equations. We thus unify the derivation of the TAP equations for models as different as the Sherrington-Kirkpatrick model, the restricted Boltzmann machine, the Hopfield model or vector (xy, Heisenberg and other) spin glasses. The state evolution of the Low-RAMP algorithm is also derived, and is equivalent to the replica symmetric solution for the large class of vector-spin glass models. In the section devoted to result we study in detail phase diagrams and phase transitions for the Bayes-optimal inference in low-rank matrix estimation. We present a typology of phase transitions and their relation to performance of algorithms such as the Low-RAMP or commonly used spectral methods.
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.