World Library  
Flag as Inappropriate
Email this Article

Whitening transformation

Article Id: WHEBN0008145581
Reproduction Date:

Title: Whitening transformation  
Author: World Heritage Encyclopedia
Language: English
Subject: White noise, Xor-encrypt-xor, Chiasmus (cipher), BEAR and LION ciphers, CRYPTON
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Whitening transformation

A whitening transformation is a decorrelation transformation that transforms a set of random variables having a known covariance matrix M into a set of new random variables whose covariance is the identity matrix (meaning that they are uncorrelated and all have variance 1).

The transformation is called "whitening" because it changes the input vector into a white noise vector. It differs from a general decorrelation transformation in that the latter only makes the covariances equal to zero, so that the correlation matrix may be any diagonal matrix.

The inverse coloring transformation transforms a vector Y of uncorrelated variables (a white random vector) into a vector X with a specified covariance matrix.

Definition

Suppose X is a random (column) vector with covariance matrix M and mean 0. The matrix M can be written as the expected value of the outer product of X with itself, namely:

M = \operatorname{E}[X X^T]

Since M is symmetric and positive semidefinite, it has a square root M^{1/2}, a matrix(not necessarily symmetric) such that M^{1/2}( M^{1/2})^T = M. If M is positive definite, M^{1/2} is invertible. Then the vector Y = M^{-1/2}X has covariance matrix:

\operatorname{Cov}(Y) = \operatorname{E}[Y Y^T] = M^{-1/2} \operatorname{E}[X X^T] (M^{-1/2})^T = M^{-1/2} M (M^{-1/2})^T

= M^{-1/2} (M^{1/2}( M^{1/2})^T) (M^{-1/2})^T = (M^{-1/2} M^{1/2}) (M^{-1/2} M^{1/2})^T = I

and is therefore a white random vector.

Since the square root of a matrix is not unique, the whitening transformation is not unique either.

If M is not positive definite, then M^{1/2} is not invertible, and it is impossible to map X to a white vector with the same number of components. In that case the vector X can still be mapped to a smaller white vector Y with m elements, where m is the number of non-zero eigenvalues of M.

See also

References

External links

  • http://courses.media.mit.edu/2010fall/mas622j/whiten.pdf
  • The ZCA whitening transformation. Appendix A of Learning Multiple Layers of Features from Tiny Images by A. Krizhevsky.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.