whitening concept

Definition	Rescaling centred data by the inverse square root of a positive-definite covariance, or a pseudoinverse on its support, so that the covariance becomes the identity. Euclidean distance after whitening is Mahalanobis distance before it. Equation 0.6. Also whiten.
Example	With covariance diag(1, 4), the row (3, 4) whitens to (3, 2).
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `whitening`, kind concept.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 0.6.
Assumptions and scope	Rescaling centred data by the inverse square root of its covariance, so that the covariance becomes the identity. It needs a positive-definite covariance, or a pseudoinverse restricted to the covariance’s support, and after it every direction has variance one. Euclidean distance after whitening equals Mahalanobis distance before it, so whitening changes which reader the Euclidean reader is and not what the data holds. The whitened code won at every budget on the whale corpus, 0.83, 0.85, 0.97 against 0.41, 0.80, 0.89.
Prior art	none recorded
Evidence	`geometric-observation/chapters/ch08_value.md:84-97`, `lean/DataMiningAsObservation/Whitening.lean`
Reviewed	semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page.

Rescaling so that the covariance becomes the identity.

Equation

Book equation 0.6.

\[x_{\mathrm w}=\Sigma^{-1/2}(x-\mu),\qquad \Sigma^{-1/2}=\sum_i\lambda_i^{-1/2}\,v_i v_i^{\top}.\]

Conditions

Rescaling centred data by the inverse square root of its covariance, so that the covariance becomes the identity. It needs a positive-definite covariance, or a pseudoinverse restricted to the covariance’s support, and after it every direction has variance one.
Euclidean distance after whitening equals Mahalanobis distance before it, so whitening changes which reader the Euclidean reader is and not what the data holds. The whitened code won at every budget on the whale corpus, 0.83, 0.85, 0.97 against 0.41, 0.80, 0.89.

Conditions are curated in entries.toml rather than read from a record.

Ledger

none

First stated

Chapter 0 section 0.2 of Data Mining as Observation, with the program’s whitened code in Volume 14 chapter 8, geometric-observation/chapters/ch08_value.md:84-97.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 2 section 2.5	whitened code wins at every budget	`geometric-observation/chapters/ch08_value.md:84-97`
chapter 4 section 4.3	whitened code on whale 0.83, 0.85, 0.97 vs 0.41, 0.80, 0.89	`geometric-observation/chapters/ch08_value.md:84-97`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/Whitening.lean, theorems whiten_unit_variance, whiten_sq_sum, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primer L, chapters 0, 2, 4, 10.

Mahalanobis distance; identity reader; read operator; water-filling.

Status

Generated 2026-09-10 by encyclopedia/generate.py; book at observation-data-mining f3914f0; the commit of every record is listed in the encyclopedia’s provenance.

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