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KarhunenLoeveDecomposition[{a1,a2,…}]

gives the Karhunen–Loeve transform {{b1,b2,…},m} of the numerical arrays {a1,a2,…}, where m.aibi.

KarhunenLoeveDecomposition[{b1,b2,…},m]

uses the inverse of the matrix m for transforming bi to ai.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Standardized  
Applications  
Properties & Relations  
See Also
Related Guides
History
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KarhunenLoeveDecomposition

KarhunenLoeveDecomposition[{a1,a2,…}]

gives the Karhunen–Loeve transform {{b1,b2,…},m} of the numerical arrays {a1,a2,…}, where m.aibi.

KarhunenLoeveDecomposition[{b1,b2,…},m]

uses the inverse of the matrix m for transforming bi to ai.

Details and Options

  • Karhunen–Loeve decomposition is typically used to reduce the dimensionality of data and capture the most important variation in the first few components.
  • The ai can be arbitrary rank arrays or images of the same dimensions.
  • The inner product of m and {a1,a2,…} gives {b1,b2,…}.
  • In KarhunenLoeveDecomposition[{a1,…}], rows of the transformation matrix m are the eigenvectors of the covariance matrix formed from the arrays ai.
  • The matrix m is a linear transformation of ai. The transformed arrays bi are uncorrelated, are given in order of decreasing variance, and have the same total variance as ai.
  • KarhunenLoeveDecomposition[{b1,b2,…},m] effectively computes the inverse Karhunen–Loeve transformation. If the length of {b1,b2,…} is less than the size of m, missing components are assumed to be zero.
  • With an option setting StandardizedTrue, datasets ai are shifted so that their means are zero.

Examples

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Basic Examples  (2)

Karhunen–Loeve decomposition of two datasets:

Wolfram Language code: KarhunenLoeveDecomposition[{{1, 2, 4}, {2, 3, 10}}]

Principal component decomposition of RGB color channels:

Wolfram Language code: KarhunenLoeveDecomposition[ColorSeparate[[image]]][[1]]

Scope  (5)

Principal components of two grayscale images:

Wolfram Language code: KarhunenLoeveDecomposition[ {[image], [image]}]

Karhunen–Loeve decomposition of three matrix-valued datasets:

Wolfram Language code: a = {{{7, 3}, {0, 1}}, {{6, 8}, {10, 6}}, {{3, 8}, {3, 1}}}; {b, m} = KarhunenLoeveDecomposition[a]
Wolfram Language code: m.a == b

Principal components of a list of color images:

Wolfram Language code: First@KarhunenLoeveDecomposition[{[image], [image], [image]}]

Specify the transformation matrix:

Wolfram Language code: KarhunenLoeveDecomposition[{{0, 0, 1}, {0, 1, 1}}, RotationMatrix[45 Degree]]

Use a transformation matrix and lesser datasets:

Wolfram Language code: KarhunenLoeveDecomposition[{{0, 0, 1}}, RotationMatrix[45 Degree]]

Options  (1)

Standardized  (1)

Karhunen–Loeve decomposition with datasets shifted to mean zero:

Wolfram Language code: First@KarhunenLoeveDecomposition[{{1, 2, 3, 4}, {0.8, 2.1, 3.2, 3.9}}, Standardized -> True]

Applications  (3)

Enhance the color contrast of an RGB image:

Wolfram Language code: {b, m} = KarhunenLoeveDecomposition[ColorSeparate@[image]]; ColorCombine[First@KarhunenLoeveDecomposition[ImageAdjust /@ b, m]]//ImageAdjust

Reconstruct a multichannel image from 1, 2, or 3 components:

Wolfram Language code: {b, m} = KarhunenLoeveDecomposition[ColorSeparate@[image]]; Table[ColorCombine[First@KarhunenLoeveDecomposition[Take[b, i], m]], {i, 3}]

Transform a list of pictorial faces:

Wolfram Language code: {b, m} = KarhunenLoeveDecomposition[faces = {[image], [image], [image], [image], [image], [image], [image], [image], [image], [image]}]; ImageAdjust /@ b

Show the residual images when using only the first three components:

Wolfram Language code: MapThread[ImageDifference, {faces, First@KarhunenLoeveDecomposition[b[[ ;; 3]], m]}]

Properties & Relations  (7)

The Karhunen–Loeve decomposition preserves the total variance:

Wolfram Language code: Variance /@ {{1, 2, 3}, {4, 6, 8}}
Wolfram Language code: Variance /@ First@KarhunenLoeveDecomposition[{{1, 2, 3}, {4, 6, 8}}]

The Karhunen–Loeve decomposition yields uncorrelated sets:

Wolfram Language code: a = RandomReal[1, {2, 5}]; b = First@KarhunenLoeveDecomposition[a];
Wolfram Language code: Covariance[Transpose[b]]//Chop

The Karhunen–Loeve decomposition yields an orthogonal transformation matrix:

Wolfram Language code: a = RandomReal[1, {2, 5}]; m = Last@KarhunenLoeveDecomposition[a];
Wolfram Language code: OrthogonalMatrixQ[m]

Relation to PrincipalComponents:

Wolfram Language code: KarhunenLoeveDecomposition[ Transpose[{{1, 2}, {2, 3}, {4, 10}}], Standardized -> True ] [[1]] // Transpose
Wolfram Language code: PrincipalComponents[{{1, 2}, {2, 3}, {4, 10}}] // N

A setting Standardized->True is equivalent to subtracting the mean from the input data:

Wolfram Language code: KarhunenLoeveDecomposition[Standardize[#, Mean, 1&] & /@ Transpose[{{1, 2}, {2, 3}, {4, 10}}] ]
Wolfram Language code: KarhunenLoeveDecomposition[Transpose[{{1, 2}, {2, 3}, {4, 10}}], Standardized -> True]

Normalizing by the square root of the number of datasets better preserves the input dynamics:

Wolfram Language code: KarhunenLoeveDecomposition[{{1, 2, 3, 4}, {1, 2, 3, 4}}] [[1]] / Sqrt[2]

KarhunenLoeveDecomposition normally returns images of a real type:

Wolfram Language code: Map[ImageType, {#, First@KarhunenLoeveDecomposition[#]}&@ {[image], [image]}, {2}]
Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015).

Text

Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015).

CMS

Wolfram Language. 2010. "KarhunenLoeveDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html.

APA

Wolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html

BibTeX

@misc{reference.wolfram_2026_karhunenloevedecomposition, author="Wolfram Research", title="{KarhunenLoeveDecomposition}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}, note=[Accessed: 12-June-2026]}