Optimality and sub-optimality of PCA I: Spiked random matrix models

Amelia Perry, Alexander S. Wein, Afonso S. Bandeira, Ankur Moitra

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)2416-2451
Number of pages36
JournalAnnals of Statistics
Volume46
Issue number5
DOIs
StatePublished - Oct 2018

Keywords

  • Contiguity
  • Deformed Wigner
  • Hypothesis testing
  • Phase transition
  • Power envelope
  • Principal component analysis
  • Random matrix
  • Spiked covariance

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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    Perry, A., Wein, A. S., Bandeira, A. S., & Moitra, A. (2018). Optimality and sub-optimality of PCA I: Spiked random matrix models. Annals of Statistics, 46(5), 2416-2451. https://doi.org/10.1214/17-AOS1625