# Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise

arXiv: https://arxiv.org/abs/2405.20993 · HTML: https://mlsanity.com/paper/2405.20993/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 80 references that are in the map.
- First posted on arXiv 31 May 2024.

## Builds on (most cited first, 22 of 22)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [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)
- [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)
- [Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula](https://mlsanity.com/paper/1606.04142/) (2016)
- [Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (or : How to Prove Kabashima's Replica Formula)](https://mlsanity.com/paper/2006.06581/) (2020)
- [MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel](https://mlsanity.com/paper/1507.03857/) (2015)
- [The Computer Science and Physics of Community Detection: Landscapes, Phase Transitions, and Hardness](https://mlsanity.com/paper/1702.00467/) (2017)
- [Message-passing algorithms for synchronization problems over compact groups](https://mlsanity.com/paper/1610.04583/) (2016)
- [Fundamental limits of detection in the spiked Wigner model](https://mlsanity.com/paper/1806.09588/) (2018)
- [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)
- [Model, sample, and epoch-wise descents: exact solution of gradient flow in the random feature model](https://mlsanity.com/paper/2110.11805/) (2021)
- [Universality of Approximate Message Passing Algorithms](https://mlsanity.com/paper/2003.10431/) (2020)
- [PCA Initialization for Approximate Message Passing in Rotationally Invariant Models](https://mlsanity.com/paper/2106.02356/) (2021)
- [Multi-Layer Generalized Linear Estimation](https://mlsanity.com/paper/1701.06981/) (2017)
- [Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures](https://mlsanity.com/paper/2009.12939/) (2020)
- [Rank-one matrix estimation: analytic time evolution of gradient descent dynamics](https://mlsanity.com/paper/2105.12257/) (2021)
- [Gradient flow in the gaussian covariate model: exact solution of learning curves and multiple descent structures](https://mlsanity.com/paper/2212.06757/) (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, 6 of 6)

- [On the phase diagram of extensive-rank symmetric matrix denoising beyond rotational invariance](https://mlsanity.com/paper/2411.01974/) (2024)
- [Unifying AMP Algorithms for Rotationally-Invariant Models](https://mlsanity.com/paper/2412.01574/) (2024)
- [Sparse corruption in low-rank matrix inference: the PCA benchmark](https://mlsanity.com/paper/2511.11927/) (2025)
- [Orthogonal Approximate Message Passing with Optimal Spectral Initializations for Rectangular Spiked Matrix Models](https://mlsanity.com/paper/2512.19334/) (2025)
- [Computational and statistical lower bounds for low-rank estimation under general inhomogeneous noise](https://mlsanity.com/paper/2510.08541/) (2025)
- [Orthogonal Approximate Message Passing Algorithms for Rectangular Spiked Matrix Models with Rotationally Invariant Noise](https://mlsanity.com/paper/2602.03283/) (2026)

## Abstract (arXiv)

We consider a prototypical problem of Bayesian inference for a structured spiked model: a low-rank signal is corrupted by additive noise. While both information-theoretic and algorithmic limits are well understood when the noise is a Gaussian Wigner matrix, the more realistic case of structured noise still proves to be challenging. To capture the structure while maintaining mathematical tractability, a line of work has focused on rotationally invariant noise. However, existing studies either provide sub-optimal algorithms or are limited to special cases of noise ensembles. In this paper, using tools from statistical physics (replica method) and random matrix theory (generalized spherical integrals) we establish the first characterization of the information-theoretic limits for a noise matrix drawn from a general trace ensemble. Remarkably, our analysis unveils the asymptotic equivalence between the rotationally invariant model and a surrogate Gaussian one. Finally, we show how to saturate the predicted statistical limits using an efficient algorithm inspired by the theory of adaptive Thouless-Anderson-Palmer (TAP) equations.