  This paper studies the related problems of prediction, covariance estimation,
and principal component analysis for the spiked covariance 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 shrinkers for use with these estimated principal components.
Underlying these methods are 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 certain
advantages of noise whitening. Specifically, we show that in a certain
asymptotic regime, optimal singular value shrinkage with whitening converges to
the best linear predictor, whereas without whitening it converges to a
suboptimal linear predictor. We prove 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 for rank one
signals, our estimated principal components achieve the asymptotic minimax
rate.
