Optimal shrinkage of eigenvalues in the spiked covariance model
David Donoho, Matan Gavish, Iain Johnstone
The Annals of Statistics · 2018 · Volume 46 · Issue 4
Abstract
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.
Keywords
62C20, 62H25, 90C22, 90C25, Bhattacharya/Matusita affinity, condition number loss, Covariance estimation, divergence loss, entropy loss, Fréchet distance, high-dimensional ssymptotics, optimal shrinkage, spiked covariance, Stein loss
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