A whitening transformation or sphering transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix, meaning that they are uncorrelated and each have variance 1.
[1] The transformation is called "whitening" because it changes the input vector into a white noise vector.
Several other transformations are closely related to whitening: Suppose
is a random (column) vector with non-singular covariance matrix
yields the whitened random vector
with unit diagonal covariance.
[4] Optimal whitening transforms can be singled out by investigating the cross-covariance and cross-correlation of
[3] For example, the unique optimal whitening transformation achieving maximal component-wise correlation between original
is produced by the whitening matrix
the diagonal variance matrix.
Whitening a data matrix follows the same transformation as for random variables.
An empirical whitening transform is obtained by estimating the covariance (e.g. by maximum likelihood) and subsequently constructing a corresponding estimated whitening matrix (e.g. by Cholesky decomposition).
One of the main issues of extending whitening to infinite dimensions is that the covariance operator has an unbounded inverse in
Nevertheless, if one assumes that Picard condition holds for
in the range space of the covariance operator, whitening becomes possible.
[5] A whitening operator can be then defined from the factorization of the Moore–Penrose inverse of the covariance operator, which has effective mapping on Karhunen–Loève type expansions of
The advantage of these whitening transformations is that they can be optimized according to the underlying topological properties of the data, thus producing more robust whitening representations.
High-dimensional features of the data can be exploited through kernel regressors or basis function systems.
The R package "pfica"[8] allows the computation of high-dimensional whitening representations using basis function systems (B-splines, Fourier basis, etc.