# Mismatched estimation of non-symmetric rank-one matrices corrupted by structured noise

arXiv: https://arxiv.org/abs/2302.03306 · HTML: https://mlsanity.com/paper/2302.03306/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 45 references that are in the map.
- First posted on arXiv 7 Feb 2023.

## Builds on (most cited first, 17 of 17)

- [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)
- [Estimation of Low-Rank Matrices via Approximate Message Passing](https://mlsanity.com/paper/1711.01682/) (2017)
- [Phase retrieval in high dimensions: Statistical and computational phase transitions](https://mlsanity.com/paper/2006.05228/) (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)
- [Approximate Message Passing with Spectral Initialization for Generalized Linear Models](https://mlsanity.com/paper/2010.03460/) (2020)
- [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)
- [Approximate Message Passing for orthogonally invariant ensembles: Multivariate non-linearities and spectral initialization](https://mlsanity.com/paper/2110.02318/) (2021)
- [All-or-nothing statistical and computational phase transitions in sparse spiked matrix estimation](https://mlsanity.com/paper/2006.07971/) (2020)
- [PCA Initialization for Approximate Message Passing in Rotationally Invariant Models](https://mlsanity.com/paper/2106.02356/) (2021)
- [A Random Matrix Perspective on Random Tensors](https://mlsanity.com/paper/2108.00774/) (2021)
- [When Random Tensors meet Random Matrices](https://mlsanity.com/paper/2112.12348/) (2021)
- [The price of ignorance: how much does it cost to forget noise structure in low-rank matrix estimation?](https://mlsanity.com/paper/2205.10009/) (2022)
- [Information-theoretic limits of a multiview low-rank symmetric spiked matrix model](https://mlsanity.com/paper/2005.08017/) (2020)
- [Bayes-optimal limits in structured PCA, and how to reach them](https://mlsanity.com/paper/2210.01237/) (2022)

## Abstract (arXiv)

We study the performance of a Bayesian statistician who estimates a rank-one signal corrupted by non-symmetric rotationally invariant noise with a generic distribution of singular values. As the signal-to-noise ratio and the noise structure are unknown, a Gaussian setup is incorrectly assumed. We derive the exact analytic expression for the error of the mismatched Bayes estimator and also provide the analysis of an approximate message passing (AMP) algorithm. The first result exploits the asymptotic behavior of spherical integrals for rectangular matrices and of low-rank matrix perturbations; the second one relies on the design and analysis of an auxiliary AMP. The numerical experiments show that there is a performance gap between the AMP and Bayes estimators, which is due to the incorrect estimation of the signal norm.