ZCA

PCA is a technique that decorrelates the components, but this does not prevent the eigenvalues from being different. If all eigenvalues are forced to be equal (see also 2.4.1), effectively changing the unit of measurement so that all principal components are equal (that is, their variances are equal), the distribution is said to be spherized, and the procedure is called data whitening.

$\mathbf{W}$ is called the whitening matrix (whitening matrix) and is identified as the Zero Components Analysis (ZCA) solution of the equation

$$\mathbf{Y}^{\top} \mathbf{Y} = \mathbf{I}$$ (2.75)

After whitening, the data have zero mean, are decorrelated, and have identity covariance.

The PCA-whitened matrix is obtained as

$$\mathbf{X}_{PCA} = \mathbf{V}^{\top} \mathbf{X}^{\top} = \mathbf{S} \mathbf{U}^{\top}$$ (2.76)

that is, $\mathbf{W}_{PCA} = \mathbf{V}^{\top}$, whereas the ZCA whitening matrix can be obtained from

$$\mathbf{X}_{ZCA} = \boldsymbol\Delta^{-1} \mathbf{X}_{PCA}=... ...}^{-1} \mathbf{V}^{\top} \mathbf{X}^{\top} = \mathbf{U}^{\top}$$ (2.77)

that is, $\mathbf{W}_{ZCA} = \mathbf{S}^{-1} \mathbf{V}^{\top}$, and, most importantly, the notable result $\mathbf{X}_{ZCA} = \mathbf{U}^{\top}$.

It should be noted that the matrix after the PCA transformation may have fewer components than the input data, whereas ZCA always has the same number of components.