# Orthogonal Approximate Message Passing with Optimal Spectral Initializations for Rectangular Spiked Matrix Models

arXiv: https://arxiv.org/abs/2512.19334 · HTML: https://mlsanity.com/paper/2512.19334/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 76 references that are in the map.
- First posted on arXiv 22 Dec 2025.

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

- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Exact Recovery in the Stochastic Block Model](https://mlsanity.com/paper/1405.3267/) (2014)
- [Spectral redemption: clustering sparse networks](https://mlsanity.com/paper/1306.5550/) (2013)
- [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)
- [Fundamental Limits of Weak Recovery with Applications to Phase Retrieval](https://mlsanity.com/paper/1708.05932/) (2017)
- [Graph-based Approximate Message Passing Iterations](https://mlsanity.com/paper/2109.11905/) (2021)
- [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)
- [Phase Transitions of Spectral Initialization for High-Dimensional Nonconvex Estimation](https://mlsanity.com/paper/1702.06435/) (2017)
- [Approximate Message Passing with Spectral Initialization for Generalized Linear Models](https://mlsanity.com/paper/2010.03460/) (2020)
- [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)
- [Phase Transitions in Sparse PCA](https://mlsanity.com/paper/1503.00338/) (2015)
- [Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models](https://mlsanity.com/paper/2008.03326/) (2020)
- [OptShrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage](https://mlsanity.com/paper/1306.6042/) (2013)
- [PCA Initialization for Approximate Message Passing in Rotationally Invariant Models](https://mlsanity.com/paper/2106.02356/) (2021)
- [Memory AMP](https://mlsanity.com/paper/2012.10861/) (2020)
- [Precise Asymptotics for Spectral Methods in Mixed Generalized Linear Models](https://mlsanity.com/paper/2211.11368/) (2022)
- [Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise](https://mlsanity.com/paper/2405.20993/) (2024)
- [On Universality of Non-Separable Approximate Message Passing Algorithms](https://mlsanity.com/paper/2506.23010/) (2025)
- [When are the most informative components for inference also the principal components?](https://mlsanity.com/paper/1302.1232/) (2013)

## Built on by (most cited first, 1 of 1)

- [Orthogonal Approximate Message Passing Algorithms for Rectangular Spiked Matrix Models with Rotationally Invariant Noise](https://mlsanity.com/paper/2602.03283/) (2026)

## 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 precisely characterizes the algorithm's high-dimensional dynamics and enables the construction of iteration-wise optimal denoisers. Within this framework, we accommodate spectral initializations under minimal assumptions on the empirical noise spectrum. In the rectangular setting, where a single rank-one component typically generates multiple informative outliers, we further propose a procedure for combining these outliers under mild non-Gaussian signal assumptions. For general RI noise models, the predicted performance of the proposed optimal OAMP algorithm agrees with replica-symmetric predictions for the associated Bayes-optimal estimator, and we conjecture that it is statistically optimal within a broad class of iterative estimation methods.