# Multi-Layer Generalized Linear Estimation

arXiv: https://arxiv.org/abs/1701.06981 · HTML: https://mlsanity.com/paper/1701.06981/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jan 2017.

## Built on by (most cited first, 12 of 12)

- [Learning Gaussian Mixtures with Generalised Linear Models: Precise Asymptotics in High-dimensions](https://mlsanity.com/paper/2106.03791/) (2021)
- [Graph-based Approximate Message Passing Iterations](https://mlsanity.com/paper/2109.11905/) (2021)
- [Deterministic equivalent and error universality of deep random features learning](https://mlsanity.com/paper/2302.00401/) (2023)
- [Asymptotics of Learning with Deep Structured (Random) Features](https://mlsanity.com/paper/2402.13999/) (2024)
- [Deep learning via message passing algorithms based on belief propagation](https://mlsanity.com/paper/2110.14583/) (2021)
- [High-dimensional learning of narrow neural networks](https://mlsanity.com/paper/2409.13904/) (2024)
- [Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise](https://mlsanity.com/paper/2405.20993/) (2024)
- [Memory Approximate Message Passing](https://mlsanity.com/paper/2106.02237/) (2021)
- [Neural-prior stochastic block model](https://mlsanity.com/paper/2303.09995/) (2023)
- [Approximate Message Passing for Multi-Layer Estimation in Rotationally Invariant Models](https://mlsanity.com/paper/2212.01572/) (2022)
- [Sufficient-Statistic Memory AMP](https://mlsanity.com/paper/2112.15327/) (2021)
- [Bayes optimal learning in high-dimensional linear regression with network side information](https://mlsanity.com/paper/2306.05679/) (2023)

## Abstract (arXiv)

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.