# Multi-Layer Generalized Linear Estimation

## TL;DR

This paper introduces the Multi-Layer Approximate Message Passing algorithm for reconstructing signals from complex multi-layered non-linear measurements, analyzing its performance and theoretical limits.

## Contribution

It develops a novel ML-AMP algorithm, derives state evolution equations, and explores applications in compressed sensing, perceptron learning, and latent variable estimation.

## Key findings

- ML-AMP effectively reconstructs signals in multi-layer non-linear measurement models.
- Derived state evolution equations predict algorithm performance accurately.
- Identified theoretical limits for minimal reconstruction error.

## Abstract

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.