Multi-Layer Generalized Linear Estimation
This provides a theoretical framework for multi-layer estimation problems, with potential applications in signal processing and machine learning, though it appears incremental as an extension of AMP methods to multi-layer settings.
The paper tackles the problem of reconstructing signals from multi-layered non-linear measurements by developing the ML-AMP algorithm and analyzing its performance with state evolution equations, deriving asymptotic free energy and minimal achievable reconstruction error. It demonstrates applications in compressed sensing, perceptron learning with structured matrices, and latent variable estimation in auto-encoders.
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.