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Phase diagram of matrix compressed sensing

In the problem of matrix compressed sensing we aim to recover a low-rank matrix from few of its element-wise linear projections. 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 recently introduced P-BiG-AMP algorithm. We show the existence of different types of phase transitions, their implications for the solvability of the problem, and we compare the results of the theoretical analysis to the 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.