# Estimation of the number of spiked eigenvalues in a covariance matrix by bulk eigenvalue matching analysis

arXiv: https://arxiv.org/abs/2006.00436 · HTML: https://mlsanity.com/paper/2006.00436/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2020.

## Built on by (most cited first, 15 of 15)

- [Optimal Differentially Private PCA and Estimation for Spiked Covariance Matrices](https://mlsanity.com/paper/2401.03820/) (2024)
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- [Enhancing Accuracy in Deep Learning Using Random Matrix Theory](https://mlsanity.com/paper/2310.03165/) (2023)
- [Deep Learning Weight Pruning with RMT-SVD: Increasing Accuracy and Reducing Overfitting](https://mlsanity.com/paper/2303.08986/) (2023)
- [Spectrally-Corrected and Regularized Linear Discriminant Analysis for Spiked Covariance Model](https://mlsanity.com/paper/2210.03859/) (2022)
- [Detection problems in the spiked matrix models](https://mlsanity.com/paper/2301.05331/) (2023)
- [Subject clustering by IF-PCA and several recent methods](https://mlsanity.com/paper/2306.05363/) (2023)
- [Joint estimation of high-dimensional spiked covariance matrices via a partially shared subspace](https://mlsanity.com/paper/2607.08123/) (2026)
- [A statistical framework for GWAS of high dimensional phenotypes using summary statistics, with application to metabolite GWAS](https://mlsanity.com/paper/2303.10221/) (2023)
- [On the Noise Sensitivity of the Randomized SVD](https://mlsanity.com/paper/2305.17435/) (2023)
- [Federated One-Shot Ensemble Clustering](https://mlsanity.com/paper/2409.08396/) (2024)
- [Spectrally-Corrected and Regularized QDA Classifier for Spiked Covariance Model](https://mlsanity.com/paper/2503.13582/) (2025)
- [Evaluating Singular Value Thresholds for DNN Weight Matrices based on Random Matrix Theory](https://mlsanity.com/paper/2512.12911/) (2025)
- [Enhancing Spectral Embedding through Robust and Flexible Knowledge Transfer in Electronic Health Records](https://mlsanity.com/paper/2606.11570/) (2026)
- [FiGuRO: Intrinsic Dimension Estimation for Multi-Modal Data](https://mlsanity.com/paper/2608.10857/) (2026)

## Abstract (arXiv)

The spiked covariance model has gained increasing popularity in high-dimensional data analysis. A fundamental problem is determination of the number of spiked eigenvalues, K. For estimation of K, most attention has focused on the use of top eigenvalues of sample covariance matrix, and there is little investigation into proper ways of utilizing bulk eigenvalues to estimate K. We propose a principled approach to incorporating bulk eigenvalues in the estimation of K. Our method imposes a working model on the residual covariance matrix, which is assumed to be a diagonal matrix whose entries are drawn from a gamma distribution. Under this model, the bulk eigenvalues are asymptotically close to the quantiles of a fixed parametric distribution. This motivates us to propose a two-step method: the first step uses bulk eigenvalues to estimate parameters of this distribution, and the second step leverages these parameters to assist the estimation of K. The resulting estimator hatK aggregates information in a large number of bulk eigenvalues. We show the consistency of hatK under a standard spiked covariance model. We also propose a confidence interval estimate for K. Our extensive simulation studies show that the proposed method is robust and outperforms the existing methods in a range of scenarios. We apply the proposed method to analysis of a lung cancer microarray data set and the 1000 Genomes data set.