Code supplement to "Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model"

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Abstract/Contents

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, Frechet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.

Type of resource	software, multimedia
Date created	October 10, 2015

Subject	Covariance Estimation
Subject	Precision Estimation
Subject	Optimal Nonlinearity
Subject	Stein Loss
Subject	Entropy Loss
Subject	Divergence Loss
Subject	Fr\'{e}chet Distance
Subject	Bhattacharya/Matusita Affinity
Subject	Quadratic Loss
Subject	Condition Number Loss
Subject	High-Dimensional Asymptotics
Subject	Spiked Covariance
Subject	Principal Component Shrinkage

Related Publication	Donoho, D, Gavish, M and Johnstone, I. (2018). Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model. Annals of Statistics. https://doi.org/10.1214%2F17-AOS1601
Related item	Title Arxiv paper
Location	https://purl.stanford.edu/xy031gt1574

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License: This work is licensed under a Creative Commons Attribution 3.0 Unported license (CC BY).

Preferred Citation: Donoho, D, Gavish, M and Johnstone, I, Code supplement to "Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model",
https://purl.stanford.edu/xy031gt1574

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