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October 2018 Optimality and sub-optimality of PCA I: Spiked random matrix models
Amelia Perry, Alexander S. Wein, Afonso S. Bandeira, Ankur Moitra
Ann. Statist. 46(5): 2416-2451 (October 2018). DOI: 10.1214/17-AOS1625

Abstract

A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, introduced by Johnstone, in which a prominent eigenvector (or “spike”) is planted into a random matrix. These distributions form natural statistical models for principal component analysis (PCA) problems throughout the sciences. Baik, Ben Arous and Péché showed that the spiked Wishart ensemble exhibits a sharp phase transition asymptotically: when the spike strength is above a critical threshold, it is possible to detect the presence of a spike based on the top eigenvalue, and below the threshold the top eigenvalue provides no information. Such results form the basis of our understanding of when PCA can detect a low-rank signal in the presence of noise. However, under structural assumptions on the spike, not all information is necessarily contained in the spectrum. We study the statistical limits of tests for the presence of a spike, including nonspectral tests. Our results leverage Le Cam’s notion of contiguity and include:

(i) For the Gaussian Wigner ensemble, we show that PCA achieves the optimal detection threshold for certain natural priors for the spike.

(ii) For any non-Gaussian Wigner ensemble, PCA is sub-optimal for detection. However, an efficient variant of PCA achieves the optimal threshold (for natural priors) by pre-transforming the matrix entries.

(iii) For the Gaussian Wishart ensemble, the PCA threshold is optimal for positive spikes (for natural priors) but this is not always the case for negative spikes.

Citation

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Amelia Perry. Alexander S. Wein. Afonso S. Bandeira. Ankur Moitra. "Optimality and sub-optimality of PCA I: Spiked random matrix models." Ann. Statist. 46 (5) 2416 - 2451, October 2018. https://doi.org/10.1214/17-AOS1625

Information

Received: 1 April 2017; Revised: 1 July 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964337
MathSciNet: MR3845022
Digital Object Identifier: 10.1214/17-AOS1625

Subjects:
Primary: 62B15, 62H15

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 5 • October 2018
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