  This paper considers probabilistic estimation of a low-rank matrix from
non-linear element-wise measurements of its elements. We derive the
corresponding approximate message passing (AMP) algorithm and its state
evolution. Relying on non-rigorous but standard assumptions motivated by
statistical physics, we characterize the minimum mean squared error (MMSE)
achievable information theoretically and with the AMP algorithm. Unlike in
related problems of linear estimation, in the present setting the MMSE depends
on the output channel only trough a single parameter - its Fisher information.
We illustrate this striking finding by analysis of submatrix localization, and
of detection of communities hidden in a dense stochastic block model. For this
example we locate the computational and statistical boundaries that are not
equal for rank larger than four.
