  We consider the problem of reconstructing a signal from multi-layered
(possibly) non-linear measurements. Using non-rigorous but standard methods
from statistical physics we present the Multi-Layer Approximate Message Passing
(ML-AMP) algorithm for computing marginal probabilities of the corresponding
estimation problem and derive the associated state evolution equations to
analyze its performance. We also give the expression of the asymptotic free
energy and the minimal information-theoretically achievable reconstruction
error. Finally, we present some applications of this measurement model for
compressed sensing and perceptron learning with structured matrices/patterns,
and for a simple model of estimation of latent variables in an auto-encoder.
