# Unifying AMP Algorithms for Rotationally-Invariant Models

arXiv: https://arxiv.org/abs/2412.01574 · HTML: https://mlsanity.com/paper/2412.01574/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 65 references that are in the map.
- First posted on arXiv 2 Dec 2024.

## Builds on (most cited first, 20 of 20)

- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models](https://mlsanity.com/paper/1708.03395/) (2017)
- [A modern maximum-likelihood theory for high-dimensional logistic regression](https://mlsanity.com/paper/1803.06964/) (2018)
- [A unifying tutorial on Approximate Message Passing](https://mlsanity.com/paper/2105.02180/) (2021)
- [Estimation of Low-Rank Matrices via Approximate Message Passing](https://mlsanity.com/paper/1711.01682/) (2017)
- [The estimation error of general first order methods](https://mlsanity.com/paper/2002.12903/) (2020)
- [Approximate Message Passing algorithms for rotationally invariant matrices](https://mlsanity.com/paper/2008.11892/) (2020)
- [Phase transitions and sample complexity in Bayes-optimal matrix factorization](https://mlsanity.com/paper/1402.1298/) (2014)
- [A Non-Asymptotic Framework for Approximate Message Passing in Spiked Models](https://mlsanity.com/paper/2208.03313/) (2022)
- [Approximate Message Passing for orthogonally invariant ensembles: Multivariate non-linearities and spectral initialization](https://mlsanity.com/paper/2110.02318/) (2021)
- [Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2112.04330/) (2021)
- [Inference with Deep Generative Priors in High Dimensions](https://mlsanity.com/paper/1911.03409/) (2019)
- [Algorithmic Analysis and Statistical Estimation of SLOPE via Approximate Message Passing](https://mlsanity.com/paper/1907.07502/) (2019)
- [Inference in Deep Networks in High Dimensions](https://mlsanity.com/paper/1706.06549/) (2017)
- [Optimization of Mean-field Spin Glasses](https://mlsanity.com/paper/2001.00904/) (2020)
- [Memory AMP](https://mlsanity.com/paper/2012.10861/) (2020)
- [Approximate message passing from random initialization with applications to mathbbZ2 synchronization](https://mlsanity.com/paper/2302.03682/) (2023)
- [Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise](https://mlsanity.com/paper/2405.20993/) (2024)
- [Approximate Message Passing for Multi-Layer Estimation in Rotationally Invariant Models](https://mlsanity.com/paper/2212.01572/) (2022)
- [Optimality of Approximate Message Passing Algorithms for Spiked Matrix Models with Rotationally Invariant Noise](https://mlsanity.com/paper/2405.18081/) (2024)

## Built on by (most cited first, 4 of 4)

- [The Effect of Optimal Self-Distillation in Noisy Gaussian Mixture Model](https://mlsanity.com/paper/2501.16226/) (2025)
- [Robust Random Graph Matching in Dense Graphs via an Approximate Message Passing Type Algorithm](https://mlsanity.com/paper/2412.16457/) (2024)
- [Asymptotic Analysis of Nonlinear One-Bit Precoding in Massive MIMO Systems via Approximate Message Passing](https://mlsanity.com/paper/2509.13955/) (2025)
- [Decoupled Descent: Exact Test Error Tracking Via Approximate Message Passing](https://mlsanity.com/paper/2604.27883/) (2026)

## Abstract (arXiv)

This paper presents a unified framework for constructing Approximate Message Passing (AMP) algorithms for rotationally-invariant models. By employing a general iterative algorithm template and reducing it to long-memory Orthogonal AMP (OAMP), we systematically derive the correct Onsager terms of AMP algorithms. This approach allows us to rederive an AMP algorithm introduced by Fan and Opper et al., while shedding new light on the role of free cumulants of the spectral law. The free cumulants arise naturally from a recursive centering operation, potentially of independent interest beyond the scope of AMP. To illustrate the flexibility of our framework, we introduce two novel AMP variants and apply them to estimation in spiked models.