# The price of ignorance: how much does it cost to forget noise structure in low-rank matrix estimation? arXiv: https://arxiv.org/abs/2205.10009 ยท HTML: https://mlsanity.com/paper/2205.10009/ ## Key facts - Cited by 9 papers in the map (data through 14 Oct 2026). - Builds on 27 of its 93 references that are in the map. - First posted on arXiv 20 May 2022. ## Builds on (most cited first, 27 of 27) - [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) - [Entropy and mutual information in models of deep neural networks](https://mlsanity.com/paper/1805.09785/) (2018) - [A unifying tutorial on Approximate Message Passing](https://mlsanity.com/paper/2105.02180/) (2021) - [Variational Bayes for high-dimensional linear regression with sparse priors](https://mlsanity.com/paper/1904.07150/) (2019) - [Estimation of Low-Rank Matrices via Approximate Message Passing](https://mlsanity.com/paper/1711.01682/) (2017) - [Bayesian linear regression with sparse priors](https://mlsanity.com/paper/1403.0735/) (2014) - [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) - [Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (or : How to Prove Kabashima's Replica Formula)](https://mlsanity.com/paper/2006.06581/) (2020) - [Phase transitions and sample complexity in Bayes-optimal matrix factorization](https://mlsanity.com/paper/1402.1298/) (2014) - [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) - [Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2112.04330/) (2021) - [Inference with Deep Generative Priors in High Dimensions](https://mlsanity.com/paper/1911.03409/) (2019) - [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) - [Statistical Mechanics of High-Dimensional Inference](https://mlsanity.com/paper/1601.04650/) (2016) - [Empirical Bayes PCA in high dimensions](https://mlsanity.com/paper/2012.11676/) (2020) - [Variational Inference in high-dimensional linear regression](https://mlsanity.com/paper/2104.12232/) (2021) - [Glassy nature of the hard phase in inference problems](https://mlsanity.com/paper/1805.05857/) (2018) - [Performance of Bayesian linear regression in a model with mismatch](https://mlsanity.com/paper/2107.06936/) (2021) - [Mismatched Estimation of rank-one symmetric matrices under Gaussian noise](https://mlsanity.com/paper/2107.08927/) (2021) ## Built on by (most cited first, 9 of 9) - [Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methods](https://mlsanity.com/paper/2405.13912/) (2024) - [On the phase diagram of extensive-rank symmetric matrix denoising beyond rotational invariance](https://mlsanity.com/paper/2411.01974/) (2024) - [Bayes-optimal limits in structured PCA, and how to reach them](https://mlsanity.com/paper/2210.01237/) (2022) - [Approximate Message Passing for Multi-Layer Estimation in Rotationally Invariant Models](https://mlsanity.com/paper/2212.01572/) (2022) - [Sparse corruption in low-rank matrix inference: the PCA benchmark](https://mlsanity.com/paper/2511.11927/) (2025) - [Gradient flow on extensive-rank positive semi-definite matrix denoising](https://mlsanity.com/paper/2303.09474/) (2023) - [Unrolled denoising networks provably learn optimal Bayesian inference](https://mlsanity.com/paper/2409.12947/) (2024) - [Mismatched estimation of non-symmetric rank-one matrices corrupted by structured noise](https://mlsanity.com/paper/2302.03306/) (2023) - [Computational and statistical lower bounds for low-rank estimation under general inhomogeneous noise](https://mlsanity.com/paper/2510.08541/) (2025) ## Abstract (arXiv) We consider the problem of estimating a rank-1 signal corrupted by structured rotationally invariant noise, and address the following question: how well do inference algorithms perform when the noise statistics is unknown and hence Gaussian noise is assumed? While the matched Bayes-optimal setting with unstructured noise is well understood, the analysis of this mismatched problem is only at its premises. In this paper, we make a step towards understanding the effect of the strong source of mismatch which is the noise statistics. Our main technical contribution is the rigorous analysis of a Bayes estimator and of an approximate message passing (AMP) algorithm, both of which incorrectly assume a Gaussian setup. The first result exploits the theory of spherical integrals and of low-rank matrix perturbations; the idea behind the second one is to design and analyze an artificial AMP which, by taking advantage of the flexibility in the denoisers, is able to "correct" the mismatch. Armed with these sharp asymptotic characterizations, we unveil a rich and often unexpected phenomenology. For example, despite AMP is in principle designed to efficiently compute the Bayes estimator, the former is outperformed by the latter in terms of mean-square error. We show that this performance gap is due to an incorrect estimation of the signal norm. In fact, when the SNR is large enough, the overlaps of the AMP and the Bayes estimator coincide, and they even match those of optimal estimators taking into account the structure of the noise.