# Optimality of Approximate Message Passing Algorithms for Spiked Matrix Models with Rotationally Invariant Noise

arXiv: https://arxiv.org/abs/2405.18081 · HTML: https://mlsanity.com/paper/2405.18081/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 May 2024.

## Built on by (most cited first, 3 of 3)

- [Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise](https://mlsanity.com/paper/2405.20993/) (2024)
- [Unifying AMP Algorithms for Rotationally-Invariant Models](https://mlsanity.com/paper/2412.01574/) (2024)
- [Orthogonal Approximate Message Passing Algorithms for Rectangular Spiked Matrix Models with Rotationally Invariant Noise](https://mlsanity.com/paper/2602.03283/) (2026)

## Abstract (arXiv)

We study the problem of estimating a rank one signal matrix from an observed matrix generated by corrupting the signal with additive rotationally invariant noise. We develop a new class of approximate message-passing algorithms for this problem and provide a simple and concise characterization of their dynamics in the high-dimensional limit. At each iteration, these algorithms exploit prior knowledge about the noise structure by applying a non-linear matrix denoiser to the eigenvalues of the observed matrix and prior information regarding the signal structure by applying a non-linear iterate denoiser to the previous iterates generated by the algorithm. We exploit our result on the dynamics of these algorithms to derive the optimal choices for the matrix and iterate denoisers. We show that the resulting algorithm achieves the smallest possible asymptotic estimation error among a broad class of iterative algorithms under a fixed iteration budget.