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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 factorsinto which the matrix is factorized. Low-rank matrix factorization is one ofthe basic methods used in data analysis for unsupervised learning of relevantfeatures and other types of dimensionality reduction. We present a framework tostudy the constrained low-rank matrix estimation for a general prior on thefactors, and a general output channel through which the matrix is observed. Wedraw a paralel with the study of vector-spin glass models - presenting aunifying way to study a number of problems considered previously in separatestatistical physics works. We present a number of applications for the problemin data analysis. We derive in detail a general form of the low-rankapproximate message passing (Low- RAMP) algorithm, that is known in statisticalphysics as the TAP equations. We thus unify the derivation of the TAP equationsfor models as different as the Sherrington-Kirkpatrick model, the restrictedBoltzmann machine, the Hopfield model or vector (xy, Heisenberg and other) spinglasses. The state evolution of the Low-RAMP algorithm is also derived, and isequivalent to the replica symmetric solution for the large class of vector-spinglass models. In the section devoted to result we study in detail phasediagrams and phase transitions for the Bayes-optimal inference in low-rankmatrix estimation. We present a typology of phase transitions and theirrelation to performance of algorithms such as the Low-RAMP or commonly usedspectral methods.

Constrained Low-rank Matrix Estimation: Phase Transitions, Approximate Message Passing and Applications