ZCA Whitening

Whitening is a transformation of data in such a way that its covariance matrix $\Sigma$ is the identity matrix. Hence whitening decorrelates features. It is used as a preprocessing method.

When you have $N$ data points in $\mathbb{R}^n$, then the covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ is estimated to be:

$$\hat{\Sigma}_{jk} = \frac{1}{N-1} \sum_{i=1}^N (x_{ij} - \bar{x}_j) \cdot (x_{ik} - \bar{x}_k)$$

where $\bar{x}_j$ denotes the $j$-th component of the estimated mean of the samples $x$.

Any matrix $W \in \mathbb{R}^{n \times n}$ which satisfies the condition

$$W^T W = C^{-1}$$

whitens the data. ZCA whitening is the choice $W = M^{- \frac{1}{2}}$. PCA is another choice. According to "Neural Networks: Tricks of the Trade" PCA and ZCA whitening differ only by a rotation.

How to do it

When you look at the Keras code, you can see the following:

# Calculate principal components
sigma = np.dot(flat_x.T, flat_x) / flat_x.shape[0]
u, s, _ = linalg.svd(sigma)
principal_components = np.dot(np.dot(u, np.diag(1.0 / np.sqrt(s + 10e-7))), u.T)

# Apply ZCA whitening
whitex = np.dot(flat_x, principal_components)

So, at first you compute the covariance matrix $\Sigma$. I'm not quite sure, but I think they should divide by flat_x.shape[0] - 1 for the unbiased estimator.

Then you apply singular value decomposition to the estimated covariance matrix. The matrix $u \in \mathbb{R}^{n \times n}$ is unitary and $s \in \mathbb{R}^{n \times n}$ is a diagonal matrix with non-negative real numbers on the diagonal. Those numbers are the singular values of $\Sigma$.

Next, the principal components are calculated:

$$u \cdot \frac{1}{\sqrt{s + 10^{-7}}} I \cdot u^T$$

By adding 10e-7 one prevents division by zero.

Whitening is then simply the multiplication with the principal components.

How to do it

See also