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.
Consider an Erdos---Renyi random graph in which each edge is present independently with probability @math 1 2, except for a subset @math CN of the vertices that form a clique (a completely connected subgraph). We consider the problem of identifying the clique, given a realization of such a random graph. The algorithm of (ANALCO. SIAM, pp 67---75, 2011) provably identifies the clique @math CN in linear time, provided @math |CN|?1.261N. Spectral methods can be shown to fail on cliques smaller than @math N. In this paper we describe a nearly linear-time algorithm that succeeds with high probability for @math |CN|?(1+?)N e for any @math ?>0. This is the first algorithm that provably improves over spectral methods. We further generalize the hidden clique problem to other background graphs (the standard case corresponding to the complete graph on @math N vertices). For large-girth regular graphs of degree @math (Δ+1) we prove that so-called local algorithms succeed if @math |CN|?(1+?)N eΔ and fail if @math |CN|≤(1-?)N eΔ.