Optimal spectral shrinkage and PCA with heteroscedastic noise

This paper studies the related problems of denoising, covariance estimation, and principal component analysis for the spiked model with heteroscedastic noise. We consider an estimator of the principal components based on whitening the noise, and we derive optimal singular value and eigenvalue denoisers for use with these estimated principal components. As part of this derivation, we obtain new asymptotic results for the high-dimensional spiked model with heteroscedastic noise, and consistent estimators for the relevant population parameters. We extend previous analysis on out-of-sample prediction to the setting of predictors with whitening. We demonstrate the advantages of noise whitening theoretically and through simulations. Specifically, we prove that in a certain asymptotic regime, optimal singular value denoising with whitening converges to the best linear predictor, whereas without whitening it converges to a suboptimal linear filter. We show that for generic signals, whitening improves estimation of the principal components, and increases a natural signal-to-noise ratio of the observations. We also show that our estimated principal components achieve the optimal minimax rate for subspace estimation in the spiked model.