@article{kessy_optimal_2018,
  title = {Optimal {Whitening} and {Decorrelation}},
  volume = {72},
  issn = {0003-1305},
  url = {https://doi.org/10.1080/00031305.2016.1277159},
  doi = {10.1080/00031305.2016.1277159},
  number = {4},
  urldate = {2023-09-08},
  journal = {The American Statistician},
  author = {Kessy, Agnan and Lewin, Alex and Strimmer, Korbinian},
  month = oct,
  year = {2018},
  note = {Publisher: Taylor \& Francis
  \_eprint: https://doi.org/10.1080/00031305.2016.1277159},
  keywords = {CAR score, CAT score, Cholesky decomposition, Decorrelation, Principal components analysis, Whitening, ZCA-Mahalanobis transformation},
  pages = {309--314},
  arxiv = {arXiv:1512.00809 [stat]},
  arxivdoi = {https://doi.org/10.48550/arXiv.1512.00809},
  tldr = {Covers `whitening', linear transforms that convert random vectors to another random vector, where the new random vector has covariance equal to the identity matrix. Five types discussed: zero-phase components analysis (ZCA) or Mahalanobis whitening, PCA whitening, Cholesky whitening, ZCA-cor, and PCA-cor. ZCA whitening is used in paper `CamP: Camera Preconditioning for Neural Radiance Fields', Park et al. 2023. `Whitening' is equivalent to the term `sphering'.}
}

Whitening, or sphering, is a common preprocessing step in statistical analysis to transform random variables to orthogonality. However, due to rotational freedom there are infinitely many possible whitening procedures. Consequently, there is a diverse range of sphering methods in use, for example, based on principal component analysis (PCA), Cholesky matrix decomposition, and zero-phase component analysis (ZCA), among others. Here, we provide an overview of the underlying theory and discuss five natural whitening procedures. Subsequently, we demonstrate that investigating the cross-covariance and the cross-correlation matrix between sphered and original variables allows to break the rotational invariance and to identify optimal whitening transformations. As a result we recommend two particular approaches: ZCA-cor whitening to produce sphered variables that are maximally similar to the original variables, and PCA-cor whitening to obtain sphered variables that maximally compress the original variables.

arXiv: http://doi.org/https://doi.org/10.48550/arXiv.1512.00809
Publisher: http://doi.org/10.1080/00031305.2016.1277159