title: Phase diagram of matrix compressed sensing.
abstract: In the problem of matrix compressed sensing, we aim to recover a low-rank matrix from a few noisy linear measurements. In this contribution, we analyze the asymptotic performance of a Bayes-optimal inference procedure for a model where the matrix to be recovered is a product of random matrices. The results that we obtain using the replica method describe the state evolution of the Parametric Bilinear Generalized Approximate Message Passing (P-BiG-AMP) algorithm, recently introduced in J. T. Parker and P. Schniter [IEEE J. Select. Top. Signal Process. 10, 795 (2016)1932-455310.1109/JSTSP.2016.2539123]. We show the existence of two different types of phase transition and their implications for the solvability of the problem, and we compare the results of our theoretical analysis to the numerical performance reached by P-BiG-AMP. Remarkably, the asymptotic replica equations for matrix compressed sensing are the same as those for a related but formally different problem of matrix factorization.
authors.affiliation: Laboratoire de Physique Statistique, CNRS, PSL Universités et Ecole Normale Supérieure, 75005, Paris, France and Institut de Physique Théorique, CNRS, CEA, Université Paris-Saclay, 91191, Gif-sur-Yvette, France.
authors.forename: Christophe
authors.initials: C
authors.lastname: Schülke
authors.affiliation: Department of Electrical and Computer Engineering, The Ohio State University, Columbus, Ohio 43210, USA.
authors.forename: Philip
authors.initials: P
authors.lastname: Schniter
authors.affiliation: Institut de Physique Théorique, CNRS, CEA, Université Paris-Saclay, 91191, Gif-sur-Yvette, France.
authors.forename: Lenka
authors.initials: L
authors.lastname: Zdeborová