The Annals of Statistics

Sharp detection in PCA under correlations: All eigenvalues matter

Edgar Dobriban

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Principal component analysis (PCA) is a widely used method for dimension reduction. In high-dimensional data, the “signal” eigenvalues corresponding to weak principal components (PCs) do not necessarily separate from the bulk of the “noise” eigenvalues. Therefore, popular tests based on the largest eigenvalue have little power to detect weak PCs. In the special case of the spiked model, certain tests asymptotically equivalent to linear spectral statistics (LSS)—averaging effects over all eigenvalues—were recently shown to achieve some power.

We consider a “local alternatives” model for the spectrum of covariance matrices that allows a general correlation structure. We develop new tests to detect PCs in this model. While the top eigenvalue contains little information, due to the strong correlations between the eigenvalues we can detect weak PCs by averaging over all eigenvalues using LSS. We show that it is possible to find the optimal LSS, by solving a certain integral equation. To solve this equation, we develop efficient algorithms that build on our recent method for computing the limit empirical spectrum [Dobriban (2015)]. The solvability of this equation also presents a new perspective on phase transitions in spiked models.

Article information

Ann. Statist., Volume 45, Number 4 (2017), 1810-1833.

Received: February 2016
Revised: August 2016
First available in Project Euclid: 28 June 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 62H15: Hypothesis testing 45B05: Fredholm integral equations

Principal component analysis linear spectral statistic random matrix theory linear integral equation optimal testing


Dobriban, Edgar. Sharp detection in PCA under correlations: All eigenvalues matter. Ann. Statist. 45 (2017), no. 4, 1810--1833. doi:10.1214/16-AOS1514.

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Supplemental materials

  • Supplement to “Sharp detection in PCA under correlations: All eigenvalues matter”. In the supplementary material, we give the remaining details of the proofs, algorithms implementing our method and further simulations.