# Detection problems in the spiked matrix models

arXiv: https://arxiv.org/abs/2301.05331 · HTML: https://mlsanity.com/paper/2301.05331/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 49 references that are in the map.
- First posted on arXiv 12 Jan 2023.

## Builds on (most cited first, 15 of 15)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [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)
- [MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel](https://mlsanity.com/paper/1507.03857/) (2015)
- [A goodness-of-fit test for stochastic block models](https://mlsanity.com/paper/1412.4857/) (2014)
- [The spiked matrix model with generative priors](https://mlsanity.com/paper/1905.12385/) (2019)
- [Fundamental limits of detection in the spiked Wigner model](https://mlsanity.com/paper/1806.09588/) (2018)
- [Hypothesis Testing for Automated Community Detection in Networks](https://mlsanity.com/paper/1311.2694/) (2013)
- [Estimation of the number of spiked eigenvalues in a covariance matrix by bulk eigenvalue matching analysis](https://mlsanity.com/paper/2006.00436/) (2020)
- [Optimal hypothesis testing for stochastic block models with growing degrees](https://mlsanity.com/paper/1705.05305/) (2017)
- [Permutation methods for factor analysis and PCA](https://mlsanity.com/paper/1710.00479/) (2017)
- [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)
- [Weak Detection in the Spiked Wigner Model with General Rank](https://mlsanity.com/paper/2001.05676/) (2020)
- [Asymptotic Normality of Log Likelihood Ratio and Fundamental Limit of the Weak Detection for Spiked Wigner Matrices](https://mlsanity.com/paper/2203.00821/) (2022)

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

- [Signal-Plus-Noise Decomposition of Nonlinear Spiked Random Matrix Models](https://mlsanity.com/paper/2405.18274/) (2024)

## Abstract (arXiv)

We study the statistical decision process of detecting the low-rank signal from various signal-plus-noise type data matrices, known as the spiked random matrix models. We first show that the principal component analysis can be improved by entrywise pre-transforming the data matrix if the noise is non-Gaussian, generalizing the known results for the spiked random matrix models with rank-1 signals. As an intermediate step, we find out sharp phase transition thresholds for the extreme eigenvalues of spiked random matrices, which generalize the Baik-Ben Arous-P'ech'e (BBP) transition. We also prove the central limit theorem for the linear spectral statistics for the spiked random matrices and propose a hypothesis test based on it, which does not depend on the distribution of the signal or the noise. When the noise is non-Gaussian noise, the test can be improved with an entrywise transformation to the data matrix with additive noise. We also introduce an algorithm that estimates the rank of the signal when it is not known a priori.