Optimal Spectral Shrinkage and PCA With Heteroscedastic Noise
Optimal Spectral Shrinkage and PCA With Heteroscedastic Noise
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DOI：
10.1109/tit.2021.3055075
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发表时间：
2021-05-01
影响因子：
2.5
通讯作者：
Romanov, Elad
中科院分区：
文献类型：
--
作者：
Leeb, William;Romanov, Elad
本文研究了异方差噪声下尖峰协方差模型的预测、协方差估计和主成分分析的相关问题。我们认为白化噪声的基础上的主成分的估计，我们得到最佳的奇异值和特征值收缩与这些估计的主成分。这些方法的基础是新的异方差噪声的高维spiked模型的渐近结果，和相关的人口参数的一致估计。我们扩展了以前的分析样本外的预测与白化预测的设置。我们证明了噪声白化的某些优点。具体来说，我们表明，在一定的渐近制度，最佳奇异值收缩与白化收敛到最好的线性预测，而没有白化它收敛到一个次优的线性预测。我们证明，对于通用信号，白化提高了主成分的估计，并增加了自然的观测信噪比。我们还表明，对于秩一信号，我们估计的主成分达到渐近极小极大率。
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.