ZCA

PCA is a technique that allows for the decorrelation of components, but this does not prevent the eigenvalues from being different. If all eigenvalues are forced to be equal (see also 2.4.1), and in fact the unit of measurement is changed so that all principal components are equal (the variances are equal), the distribution is referred to as spherical, and the process is known as data whitening.

The matrix referred to as the whitening matrix is denoted as the solution to Zero Components Analysis (ZCA) of the equation

$$\mathbf{Y}^{\top} \mathbf{Y} = \mathbf{I}$$ (2.72)

. After the whitening transformation, the data will not only have zero mean and be decorrelated, but will also exhibit identity covariance.

The whitened matrix from PCA is obtained as

$$\mathbf{X}_{PCA} = \mathbf{V}^{\top} \mathbf{X}^{\top} = \mathbf{S} \mathbf{U}^{\top}$$ (2.73)

or equivalently $\mathbf{W}_{PCA} = \mathbf{V}^{\top}$, while the whitening matrix from ZCA can be derived from

$$\mathbf{X}_{ZCA} = \boldsymbol\Delta^{-1} \mathbf{X}_{PCA}=... ...}^{-1} \mathbf{V}^{\top} \mathbf{X}^{\top} = \mathbf{U}^{\top}$$ (2.74)

or, in other words, $\mathbf{W}_{ZCA} = \mathbf{S}^{-1} \mathbf{V}^{\top}$. Most importantly, the remarkable result is given by $\mathbf{X}_{ZCA} = \mathbf{U}^{\top}$.

It is noteworthy that the matrix after the PCA transformation may have a number of components less than the input data, whereas ZCA always retains the same number of components.