# Orthogonal Approximate Message Passing Algorithms for Rectangular Spiked Matrix Models with Rotationally Invariant Noise

arXiv: https://arxiv.org/abs/2602.03283 · HTML: https://mlsanity.com/paper/2602.03283/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 33 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Builds on (most cited first, 10 of 10)

- [Exact Recovery in the Stochastic Block Model](https://mlsanity.com/paper/1405.3267/) (2014)
- [Estimation of Low-Rank Matrices via Approximate Message Passing](https://mlsanity.com/paper/1711.01682/) (2017)
- [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)
- [Approximate Message Passing for orthogonally invariant ensembles: Multivariate non-linearities and spectral initialization](https://mlsanity.com/paper/2110.02318/) (2021)
- [Phase Transitions in Sparse PCA](https://mlsanity.com/paper/1503.00338/) (2015)
- [OptShrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage](https://mlsanity.com/paper/1306.6042/) (2013)
- [Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise](https://mlsanity.com/paper/2405.20993/) (2024)
- [Optimality of Approximate Message Passing Algorithms for Spiked Matrix Models with Rotationally Invariant Noise](https://mlsanity.com/paper/2405.18081/) (2024)
- [Orthogonal Approximate Message Passing with Optimal Spectral Initializations for Rectangular Spiked Matrix Models](https://mlsanity.com/paper/2512.19334/) (2025)

## Abstract (arXiv)

We propose an orthogonal approximate message passing (OAMP) algorithm for signal estimation in the rectangular spiked matrix model with general rotationally invariant (RI) noise. We establish a rigorous state evolution that exactly characterizes the high-dimensional dynamics of the algorithm. Building on this framework, we derive an optimal variant of OAMP that minimizes the predicted mean-squared error at each iteration. For the special case of i.i.d. Gaussian noise, the fixed point of the proposed OAMP algorithm coincides with that of the standard AMP algorithm. For general RI noise models, we conjecture that the optimal OAMP algorithm is statistically optimal within a broad class of iterative methods, and achieves Bayes-optimal performance in certain regimes.