# Precise Asymptotics for Spectral Methods in Mixed Generalized Linear Models

arXiv: https://arxiv.org/abs/2211.11368 · HTML: https://mlsanity.com/paper/2211.11368/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Nov 2022.

## Built on by (most cited first, 7 of 7)

- [Approximate message passing from random initialization with applications to mathbbZ2 synchronization](https://mlsanity.com/paper/2302.03682/) (2023)
- [Statistical-Computational Tradeoffs in Mixed Sparse Linear Regression](https://mlsanity.com/paper/2303.02118/) (2023)
- [Spectral Estimators for Structured Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2308.14507/) (2023)
- [Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methods](https://mlsanity.com/paper/2405.13912/) (2024)
- [Mixed Regression via Approximate Message Passing](https://mlsanity.com/paper/2304.02229/) (2023)
- [Orthogonal Approximate Message Passing with Optimal Spectral Initializations for Rectangular Spiked Matrix Models](https://mlsanity.com/paper/2512.19334/) (2025)
- [Inferring Change Points in High-Dimensional Regression via Approximate Message Passing](https://mlsanity.com/paper/2404.07864/) (2024)

## Abstract (arXiv)

In a mixed generalized linear model, the goal is to learn multiple signals from unlabeled observations: each sample comes from exactly one signal, but it is not known which one. We consider the prototypical problem of estimating two statistically independent signals in a mixed generalized linear model with Gaussian covariates. Spectral methods are a popular class of estimators which output the top two eigenvectors of a suitable data-dependent matrix. However, despite the wide applicability, their design is still obtained via heuristic considerations, and the number of samples n needed to guarantee recovery is super-linear in the signal dimension d. In this paper, we develop exact asymptotics on spectral methods in the challenging proportional regime in which n, d grow large and their ratio converges to a finite constant. This allows us optimize the design of the spectral method, and combine it with a simple linear estimator, to minimize the estimation error. Our characterization exploits a mix of tools from random matrices, free probability and the theory of approximate message passing algorithms. Numerical simulations for mixed linear regression and phase retrieval demonstrate the advantage enabled by our analysis over existing designs of spectral methods.