# Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices

arXiv: https://arxiv.org/abs/2203.00821 · HTML: https://mlsanity.com/paper/2203.00821/

## Key facts

- Cited by 1 papers in the map (data through 7 Oct 2026).
- Builds on 6 of its 25 references that are in the map.
- First posted on arXiv 2 Mar 2022.

## Builds on (most cited first, 6 of 6)

- [Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models](https://mlsanity.com/paper/1807.00891/) (2018)
- [Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula](https://mlsanity.com/paper/1606.04142/) (2016)
- [Fundamental limits of detection in the spiked Wigner model](https://mlsanity.com/paper/1806.09588/) (2018)
- [Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matrices](https://mlsanity.com/paper/1811.12804/) (2018)
- [Detection limits in the high-dimensional spiked rectangular model](https://mlsanity.com/paper/1802.07309/) (2018)
- [Detection of Signal in the Spiked Rectangular Models](https://mlsanity.com/paper/2104.13517/) (2021)

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

- [Detection problems in the spiked matrix models](https://mlsanity.com/paper/2301.05331/) (2023)

## Abstract (arXiv)

We consider the problem of detecting the presence of a signal in a rank-one spiked Wigner model. For general non-Gaussian noise, assuming that the signal is drawn from the Rademacher prior, we prove that the log likelihood ratio (LR) of the spiked model against the null model converges to a Gaussian when the signal-to-noise ratio is below a certain threshold. The threshold is optimal in the sense that the reliable detection is possible by a transformed principal component analysis (PCA) above it. From the mean and the variance of the limiting Gaussian for the log-LR, we compute the limit of the sum of the Type-I error and the Type-II error of the likelihood ratio test. We also prove similar results for a rank-one spiked IID model where the noise is asymmetric but the signal is symmetric.