The Annals of Statistics

Optimal shrinkage of eigenvalues in the spiked covariance model

David Donoho, Matan Gavish, and Iain Johnstone

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We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation.

In an asymptotic framework based on the spiked covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker $\eta$ that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker $\eta^{*}$ dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of loss function and, crucially, by inconsistency of both eigenvalues and eigenvectors of the sample covariance matrix.

Details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature, including the Stein, Entropy, Divergence, Fréchet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.

Article information

Ann. Statist., Volume 46, Number 4 (2018), 1742-1778.

Received: March 2014
Revised: May 2017
First available in Project Euclid: 27 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C20: Minimax procedures 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 90C25: Convex programming 90C22: Semidefinite programming

Covariance estimation optimal shrinkage Stein loss entropy loss divergence loss Fréchet distance Bhattacharya/Matusita affinity condition number loss high-dimensional ssymptotics spiked covariance


Donoho, David; Gavish, Matan; Johnstone, Iain. Optimal shrinkage of eigenvalues in the spiked covariance model. Ann. Statist. 46 (2018), no. 4, 1742--1778. doi:10.1214/17-AOS1601.

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Supplemental materials

  • Proofs and additional results. In the supplementary material, we provide proofs omitted from the main text for space considerations and auxiliary lemmas used in various proofs. Notably, we prove Lemma 4, and provide detailed derivations of the 17 explicit formulas for optimal shrinkers, as summarized in Table 2. In addition, in the supplementary material we offer a detailed study of the large-$\lambda$ asymptotics (asymptotic slope and asymptotic shift) of the optimal shrinkers discovered in this paper, and tabulate the asymptotic behavior of each optimal shrinker. We also study the asymptotic percent improvement of the optimal shrinkers over naive hard thresholding of the sample covariance eigenvalues.