Whitening, or sphering, is a common preprocessing step in statistical
analysis to transform random variables to orthogonality. However, due to
rotational freedom there are infinitely many possible whitening procedures.
Consequently, there is a diverse range of sphering methods in use, for example
based on principal component analysis (PCA), Cholesky matrix decomposition and
zero-phase component analysis (ZCA), among others.
Here we provide an overview of the underlying theory and discuss five natural
whitening procedures. Subsequently, we demonstrate that investigating the
cross-covariance and the cross-correlation matrix between sphered and original
variables allows to break the rotational invariance and to identify optimal
whitening transformations. As a result we recommend two particular approaches:
ZCA-cor whitening to produce sphered variables that are maximally similar to
the original variables, and PCA-cor whitening to obtain sphered variables that
maximally compress the original variables.

本文介绍了 Whitening 或 Sphering 的概念，强调随机变量正交性转换，探讨了基于主成分分析（PCA）、Cholesky 矩阵分解和零相位分量分析（ZCA）等五种自然白化过程，指出通过探讨 Sphered 变量和原始变量之间的交叉协方差和交叉相关矩阵可以打破旋转不变性并确定最佳白化转换，提出了两种特定方法。建议使用 ZCA-Cor 白化生产最大程度类似于原变量的 Shpered 变量以及使用 PCA-Cor 白化以获得最大压缩原变量的 Shpered 变量。