  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.
