🕷️ Crawler Inspector

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Direct Parameter Lookup

Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 184 (from laksa110)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa184.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html\')), getAhrefsUnparsedNoserviceFromURL(\'https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html","crawl_time":1776105608,"first_indexed_time":1411351680,"http_code":200,"src_unparsed":"com,wolfram!reference,\/language\/ref\/KarhunenLoeveDecomposition.html s443","src_root_hash":"3744487911316863784","history_drop_reason":null,"meta_title":"KarhunenLoeveDecomposition: Karhunen-Loeve transform—Wolfram Documentation","meta_descriptions":["KarhunenLoeveDecomposition[{a1, a2, ...}] gives the Karhunen\\[Dash]Loeve transform {{b1, b2, ...}, m} of the numerical arrays {a1, a2, ...}, where m . ai == bi. KarhunenLoeveDecomposition[{b1, b2, ...}, m] uses the inverse of the matrix m for transforming bi to ai."],"attrs_boilerpipe_text":"See Also\nSingularValueDecomposition\nPrincipalComponents\nDimensionReduce\nRelated Guides\nMatrix Decompositions\nImage Representation\nMatrices and Linear Algebra\nUnsupervised Machine Learning\nSee Also\nSingularValueDecomposition\nPrincipalComponents\nDimensionReduce\nRelated Guides\nMatrix Decompositions\nImage Representation\nMatrices and Linear Algebra\nUnsupervised Machine Learning\nExamples\n \n \nBasic Examples\n \n \nScope\n \n \nOptions\n \n \nStandardized\n \n \nApplications\n \n \nProperties & Relations\n \n \nKarhunenLoeveDecomposition\n[\n{\na\n1\n,\na\n2\n,\n…\n}\n]\nCopy to clipboard.\nKarhunenLoeveDecomposition[{a1,a2,…}]\ngives the Karhunen\n–\nLoeve transform\n{\n{\nb\n1\n,\nb\n2\n,\n…\n}\n,\nm\n}\nof the numerical arrays\n{\na\n1\n,\na\n2\n,\n…\n}\n, where\nm\n.\na\ni\n\nb\ni\n.\nKarhunenLoeveDecomposition\n[\n{\nb\n1\n,\nb\n2\n,\n…\n}\n,\nm\n]\nCopy to clipboard.\nKarhunenLoeveDecomposition[{b1,b2,…},m]\nuses the inverse of the matrix\nm\nfor transforming\nb\ni\nto\na\ni\n.\nDetails and Options\nKarhunen\n–\nLoeve decomposition is typically used to reduce the dimensionality of data and capture the most important variation in the first few components.\nThe\na\ni\ncan be arbitrary rank arrays or images of the same dimensions.\nThe inner product of\nm\nand\n{\na\n1\n,\na\n2\n,\n…\n}\ngives\n{\nb\n1\n,\nb\n2\n,\n…\n}\n.\nIn\nKarhunenLoeveDecomposition\n[\n{\na\n1\n,\n…\n}\n]\n, rows of the transformation matrix\nm\nare the eigenvectors of the covariance matrix formed from the arrays\na\ni\n.\nThe matrix\nm\nis a linear transformation of\na\ni\n. The transformed arrays\nb\ni\nare uncorrelated, are given in order of decreasing variance, and have the same total variance as\na\ni\n.\nKarhunenLoeveDecomposition\n[\n{\nb\n1\n,\nb\n2\n,\n…\n}\n,\nm\n]\neffectively computes the inverse Karhunen\n–\nLoeve transformation. If the length of\n{\nb\n1\n,\nb\n2\n,\n…\n}\nis less than the size of\nm\n, missing components are assumed to be zero.\nWith an option setting\nStandardized\n\nTrue\n, datasets\na\ni\nare shifted so that their means are zero.\nExamples\nopen all\nclose all\nBasic Examples\n \n \n(2)\nSummary of the most common use cases\nKarhunen\n–\nLoeve decomposition of two datasets:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-knb8up\nDirect link to example\nOut[1]=\n1\nPrincipal component decomposition of RGB color channels:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-cistzr\nDirect link to example\nOut[1]=\n1\nScope\n \n \n(5)\nSurvey of the scope of standard use cases\nOptions\n \n \n(1)\nCommon values & functionality for each option\nStandardized\n \n \n(1)\nKarhunen\n–\nLoeve decomposition with datasets shifted to mean zero:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-9aifu2\nDirect link to example\nOut[1]=\n1\nApplications\n \n \n(3)\nSample problems that can be solved with this function\nEnhance the color contrast of an RGB image:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-c1h72f\nDirect link to example\nOut[1]=\n1\nReconstruct a multichannel image from 1, 2, or 3 components:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-iqpvla\nDirect link to example\nOut[1]=\n1\nTransform a list of pictorial faces:\nCopy to clipboard.\nIn[1]:=\n1\nhttps:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-exvhvo\nDirect link to example\nOut[1]=\n1\nShow the residual images when using only the first three components:\nCopy to clipboard.\nIn[2]:=\n2\nhttps:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-9lk0x\nDirect link to example\nOut[2]=\n2\nProperties & Relations\n \n \n(7)\nProperties of the function, and connections to other functions\nWolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).\nCopy to clipboard.\nWolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).\nText\nWolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).\nCopy to clipboard.\nWolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).\nCMS\nWolfram Language. 2010. \"KarhunenLoeveDecomposition.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html.\nCopy to clipboard.\nWolfram Language. 2010. \"KarhunenLoeveDecomposition.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html.\nAPA\nWolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html\nCopy to clipboard.\nWolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html\nBibTeX\n@misc{reference.wolfram_2025_karhunenloevedecomposition, author=\"Wolfram Research\", title=\"{KarhunenLoeveDecomposition}\", year=\"2015\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}\", note=[Accessed: 13-April-2026]}\nCopy to clipboard.\n@misc{reference.wolfram_2025_karhunenloevedecomposition, author=\"Wolfram Research\", title=\"{KarhunenLoeveDecomposition}\", year=\"2015\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}\", note=[Accessed: 13-April-2026]}\nBibLaTeX\n@online{reference.wolfram_2025_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}, note=[Accessed: 13-April-2026]}\nCopy to clipboard.\n@online{reference.wolfram_2025_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}, note=[Accessed: 13-April-2026]}","attrs_markdown":"#\n[Products](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n\n- [Wolfram\\|One The definitive Wolfram Language and notebook experience](https:\/\/www.wolfram.com\/wolfram-one\/?source=nav)\n- [Mathematica The original technical computing environment](https:\/\/www.wolfram.com\/mathematica\/?source=nav)\n- [Notebook Assistant + LLM Kit All-in-one AI assistance for your Wolfram experience](https:\/\/www.wolfram.com\/notebook-assistant-llm-kit\/?source=nav)\n\n- [Compute Services](https:\/\/www.wolfram.com\/compute-services\/?source=nav)\n- [System Modeler](https:\/\/www.wolfram.com\/system-modeler\/?source=nav)\n- [Finance 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Guides](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n  - [Matrix Decompositions](https:\/\/reference.wolfram.com\/language\/guide\/MatrixDecompositions.html)\n  - [Image Representation](https:\/\/reference.wolfram.com\/language\/guide\/ImageRepresentation.html)\n  - [Matrices and Linear Algebra](https:\/\/reference.wolfram.com\/language\/guide\/MatricesAndLinearAlgebra.html)\n  - [Unsupervised Machine Learning](https:\/\/reference.wolfram.com\/language\/guide\/UnsupervisedMachineLearning.html)\n- - [See Also](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n    - [SingularValueDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/SingularValueDecomposition.html)\n    - [PrincipalComponents](https:\/\/reference.wolfram.com\/language\/ref\/PrincipalComponents.html)\n    - [DimensionReduce](https:\/\/reference.wolfram.com\/language\/ref\/DimensionReduce.html)\n  - [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n    - [Matrix Decompositions](https:\/\/reference.wolfram.com\/language\/guide\/MatrixDecompositions.html)\n    - [Image Representation](https:\/\/reference.wolfram.com\/language\/guide\/ImageRepresentation.html)\n    - [Matrices and Linear Algebra](https:\/\/reference.wolfram.com\/language\/guide\/MatricesAndLinearAlgebra.html)\n    - [Unsupervised Machine Learning](https:\/\/reference.wolfram.com\/language\/guide\/UnsupervisedMachineLearning.html)\n\n[KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\\[{a1,a2,…}\\]\n\ngives the Karhunen–Loeve transform {{b1,b2,…},m} of the numerical arrays {a1,a2,…}, where m.aibi.\n\n[KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\\[{b1,b2,…},m\\]\n\nuses the inverse of the matrix m for transforming bi to ai.\n\nDetails and Options\n\n![Details and Options](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/DetailsOptions-gray.png) ![Details and Options](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/DetailsOptions-hover.png)\n\nExamples\n\nBasic Examples\n\nScope\n\nOptions\n\nStandardized\n\nApplications\n\nProperties & Relations\n\nSee Also\n\nRelated Guides\n\nHistory\n\nCite this Page\n\nBUILT-IN SYMBOL\n\n- [See Also](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n  - [SingularValueDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/SingularValueDecomposition.html)\n  - [PrincipalComponents](https:\/\/reference.wolfram.com\/language\/ref\/PrincipalComponents.html)\n  - [DimensionReduce](https:\/\/reference.wolfram.com\/language\/ref\/DimensionReduce.html)\n- [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n  - [Matrix Decompositions](https:\/\/reference.wolfram.com\/language\/guide\/MatrixDecompositions.html)\n  - [Image Representation](https:\/\/reference.wolfram.com\/language\/guide\/ImageRepresentation.html)\n  - [Matrices and Linear Algebra](https:\/\/reference.wolfram.com\/language\/guide\/MatricesAndLinearAlgebra.html)\n  - [Unsupervised Machine Learning](https:\/\/reference.wolfram.com\/language\/guide\/UnsupervisedMachineLearning.html)\n- - [See Also](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n    - [SingularValueDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/SingularValueDecomposition.html)\n    - [PrincipalComponents](https:\/\/reference.wolfram.com\/language\/ref\/PrincipalComponents.html)\n    - [DimensionReduce](https:\/\/reference.wolfram.com\/language\/ref\/DimensionReduce.html)\n  - [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n    - [Matrix Decompositions](https:\/\/reference.wolfram.com\/language\/guide\/MatrixDecompositions.html)\n    - [Image Representation](https:\/\/reference.wolfram.com\/language\/guide\/ImageRepresentation.html)\n    - [Matrices and Linear Algebra](https:\/\/reference.wolfram.com\/language\/guide\/MatricesAndLinearAlgebra.html)\n    - [Unsupervised Machine Learning](https:\/\/reference.wolfram.com\/language\/guide\/UnsupervisedMachineLearning.html)\n# KarhunenLoeveDecompositionCopy to clipboard. ✖ `KarhunenLoeveDecomposition`\n\n[KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\\[{a1,a2,…}\\]\n\nCopy to clipboard.\n\n✖\n\n`KarhunenLoeveDecomposition[{a1,a2,…}]`\n\ngives the Karhunen–Loeve transform {{b1,b2,…},m} of the numerical arrays {a1,a2,…}, where m.aibi.\n\n[KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\\[{b1,b2,…},m\\]\n\nCopy to clipboard.\n\n✖\n\n`KarhunenLoeveDecomposition[{b1,b2,…},m]`\n\nuses the inverse of the matrix m for transforming bi to ai.\n\n# Details and Options\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/details_1.png)\n\n- Karhunen–Loeve decomposition is typically used to reduce the dimensionality of data and capture the most important variation in the first few components.\n- The ai can be arbitrary rank arrays or images of the same dimensions.\n- The inner product of m and {a1,a2,…} gives {b1,b2,…}.\n- In [KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\\[{a1,…}\\], rows of the transformation matrix m are the eigenvectors of the covariance matrix formed from the arrays ai.\n- 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.\n- [KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\\[{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.\n- With an option setting [Standardized](https:\/\/reference.wolfram.com\/language\/ref\/Standardized.html)[True](https:\/\/reference.wolfram.com\/language\/ref\/True.html), datasets ai are shifted so that their means are zero.\n\n# Examples\nopen all close all\n\n## Basic Examples  (2)Summary of the most common use cases\nKarhunen–Loeve decomposition of two datasets:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_1.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-knb8up`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_1.png)\n\nPrincipal component decomposition of RGB color channels:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_2.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-cistzr`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_2.png)\n\n## Scope  (5)Survey of the scope of standard use cases\nPrincipal components of two grayscale images:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_3.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-j1ll2v`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_3.png)\n\nKarhunen–Loeve decomposition of three matrix-valued datasets:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_4.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-pyroz`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_4.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_5.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-32j1v7`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_5.png)\n\nPrincipal components of a list of color images:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_6.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-ouqstd`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_6.png)\n\nSpecify the transformation matrix:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_7.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-zsnmgd`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_7.png)\n\nUse a transformation matrix and lesser datasets:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_8.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-prlgd3`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_8.png)\n\n## Options  (1)Common values & functionality for each option\n### Standardized  (1)\nKarhunen–Loeve decomposition with datasets shifted to mean zero:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_9.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-9aifu2`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_9.png)\n\n## Applications  (3)Sample problems that can be solved with this function\nEnhance the color contrast of an RGB image:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_10.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-c1h72f`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_10.png)\n\nReconstruct a multichannel image from 1, 2, or 3 components:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_11.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-iqpvla`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_11.png)\n\nTransform a list of pictorial faces:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_12.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-exvhvo`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_12.png)\n\nShow the residual images when using only the first three components:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_13.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-9lk0x`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_13.png)\n\n## Properties & Relations  (7)Properties of the function, and connections to other functions\nThe Karhunen–Loeve decomposition preserves the total variance:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_14.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-xfttmi`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_14.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_15.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-m3ehi9`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_15.png)\n\nThe Karhunen–Loeve decomposition yields uncorrelated sets:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_16.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-f0wukw`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_17.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-f89dvf`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_16.png)\n\nThe Karhunen–Loeve decomposition yields an orthogonal transformation matrix:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_18.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-ng385z`\n\nDirect link to example\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_19.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-t4u4e8`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_17.png)\n\nRelation to [PrincipalComponents](https:\/\/reference.wolfram.com\/language\/ref\/PrincipalComponents.html):\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_20.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-dnw4l2`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_18.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_21.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-bav5vr`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_19.png)\n\nA setting [Standardized](https:\/\/reference.wolfram.com\/language\/ref\/Standardized.html)\\-\\>[True](https:\/\/reference.wolfram.com\/language\/ref\/True.html) is equivalent to subtracting the mean from the input data:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_22.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-7ff1rn`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_20.png)\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_23.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-nvp2ii`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_21.png)\n\nNormalizing by the square root of the number of datasets better preserves the input dynamics:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_24.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-gucqhn`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_22.png)\n\n[KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html) normally returns images of a real type:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_25.png)\n\n✖\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-buunxk`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_23.png)\n\n# See Also\n[SingularValueDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/SingularValueDecomposition.html)  [PrincipalComponents](https:\/\/reference.wolfram.com\/language\/ref\/PrincipalComponents.html)  [DimensionReduce](https:\/\/reference.wolfram.com\/language\/ref\/DimensionReduce.html)\n\n# Related Guides\n- [Matrix Decompositions](https:\/\/reference.wolfram.com\/language\/guide\/MatrixDecompositions.html)\n- [Image Representation](https:\/\/reference.wolfram.com\/language\/guide\/ImageRepresentation.html)\n- [Matrices and Linear Algebra](https:\/\/reference.wolfram.com\/language\/guide\/MatricesAndLinearAlgebra.html)\n- [Unsupervised Machine Learning](https:\/\/reference.wolfram.com\/language\/guide\/UnsupervisedMachineLearning.html)\n\n# History\n[Introduced in 2010 (8.0)](https:\/\/reference.wolfram.com\/language\/guide\/SummaryOfNewFeaturesIn80)  \\|  [Updated in 2014 (10.0)](https:\/\/reference.wolfram.com\/language\/guide\/SummaryOfNewFeaturesIn100)  ▪  [2015 (10.1)](https:\/\/reference.wolfram.com\/language\/guide\/SummaryOfNewFeaturesIn101)\n\nCite this as:\n\nWolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).`\n\n![](https:\/\/reference.wolfram.com\/language-assets-v2\/img\/CitationCloseButton.svg)\n\n#### Text\nWolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).`\n\n#### CMS\nWolfram Language. 2010. \"KarhunenLoeveDecomposition.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html.\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Language. 2010. \"KarhunenLoeveDecomposition.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html.`\n\n#### APA\nWolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html\n\nCopy to clipboard.\n\n✖\n\n`Wolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html`\n\n#### BibTeX\n@misc{reference.wolfram\\_2025\\_karhunenloevedecomposition, author=\"Wolfram Research\", title=\"{KarhunenLoeveDecomposition}\", year=\"2015\", howpublished=\"\\\\url{https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}\", note=\\[Accessed: 13-April-2026\\]}\n\nCopy to clipboard.\n\n✖\n\n`@misc{reference.wolfram_2025_karhunenloevedecomposition, author=\"Wolfram Research\", title=\"{KarhunenLoeveDecomposition}\", year=\"2015\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}\", note=[Accessed: 13-April-2026]}`\n\n#### BibLaTeX\n@online{reference.wolfram\\_2025\\_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}, note=\\[Accessed: 13-April-2026\\]}\n\nCopy to clipboard.\n\n✖\n\n`@online{reference.wolfram_2025_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}, note=[Accessed: 13-April-2026]}`\n\nGive Feedback\n\n[Top](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html#top)\n\n[Introduction for Programmers](https:\/\/www.wolfram.com\/language\/fast-introduction-for-programmers\/en\/)\n\n[Introductory Book](https:\/\/www.wolfram.com\/language\/elementary-introduction\/)\n\n[Wolfram Function Repository](http:\/\/resources.wolframcloud.com\/FunctionRepository\/)  \\|  [Wolfram Data Repository](https:\/\/datarepository.wolframcloud.com\/)  \\|  [Wolfram Data Drop](https:\/\/datadrop.wolframcloud.com\/)  \\|  [Wolfram Language 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Wolfram tech through your organization](https:\/\/www.wolfram.com\/siteinfo\/)\n\n×","attrs_readable_markdown":"- [See Also](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n  - [SingularValueDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/SingularValueDecomposition.html)\n  - [PrincipalComponents](https:\/\/reference.wolfram.com\/language\/ref\/PrincipalComponents.html)\n  - [DimensionReduce](https:\/\/reference.wolfram.com\/language\/ref\/DimensionReduce.html)\n- [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n  - [Matrix Decompositions](https:\/\/reference.wolfram.com\/language\/guide\/MatrixDecompositions.html)\n  - [Image Representation](https:\/\/reference.wolfram.com\/language\/guide\/ImageRepresentation.html)\n  - [Matrices and Linear Algebra](https:\/\/reference.wolfram.com\/language\/guide\/MatricesAndLinearAlgebra.html)\n  - [Unsupervised Machine Learning](https:\/\/reference.wolfram.com\/language\/guide\/UnsupervisedMachineLearning.html)\n- - [See Also](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n    - [SingularValueDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/SingularValueDecomposition.html)\n    - [PrincipalComponents](https:\/\/reference.wolfram.com\/language\/ref\/PrincipalComponents.html)\n    - [DimensionReduce](https:\/\/reference.wolfram.com\/language\/ref\/DimensionReduce.html)\n  - [Related Guides](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\n    - [Matrix Decompositions](https:\/\/reference.wolfram.com\/language\/guide\/MatrixDecompositions.html)\n    - [Image Representation](https:\/\/reference.wolfram.com\/language\/guide\/ImageRepresentation.html)\n    - [Matrices and Linear Algebra](https:\/\/reference.wolfram.com\/language\/guide\/MatricesAndLinearAlgebra.html)\n    - [Unsupervised Machine Learning](https:\/\/reference.wolfram.com\/language\/guide\/UnsupervisedMachineLearning.html)\n\nExamples\n\nBasic Examples\n\nScope\n\nOptions\n\nStandardized\n\nApplications\n\nProperties & Relations\n\n[KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\\[{a1,a2,…}\\]\n\nCopy to clipboard.\n\n`KarhunenLoeveDecomposition[{a1,a2,…}]`\n\ngives the Karhunen–Loeve transform {{b1,b2,…},m} of the numerical arrays {a1,a2,…}, where m.aibi.\n\n[KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\\[{b1,b2,…},m\\]\n\nCopy to clipboard.\n\n`KarhunenLoeveDecomposition[{b1,b2,…},m]`\n\nuses the inverse of the matrix m for transforming bi to ai.\n\n## Details and Options\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/details_1.png)\n\n- Karhunen–Loeve decomposition is typically used to reduce the dimensionality of data and capture the most important variation in the first few components.\n- The ai can be arbitrary rank arrays or images of the same dimensions.\n- The inner product of m and {a1,a2,…} gives {b1,b2,…}.\n- In [KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\\[{a1,…}\\], rows of the transformation matrix m are the eigenvectors of the covariance matrix formed from the arrays ai.\n- 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.\n- [KarhunenLoeveDecomposition](https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html)\\[{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.\n- With an option setting [Standardized](https:\/\/reference.wolfram.com\/language\/ref\/Standardized.html)[True](https:\/\/reference.wolfram.com\/language\/ref\/True.html), datasets ai are shifted so that their means are zero.\n\n## Examples\nopen all close all\n\n## Basic Examples  (2)Summary of the most common use cases\nKarhunen–Loeve decomposition of two datasets:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_1.png)\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-knb8up`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_1.png)\n\nPrincipal component decomposition of RGB color channels:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_2.png)\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-cistzr`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_2.png)\n## Scope  (5)Survey of the scope of standard use cases\n## Options  (1)Common values & functionality for each option\n### Standardized  (1)\nKarhunen–Loeve decomposition with datasets shifted to mean zero:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_9.png)\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-9aifu2`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_9.png)\n\n## Applications  (3)Sample problems that can be solved with this function\nEnhance the color contrast of an RGB image:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_10.png)\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-c1h72f`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_10.png)\n\nReconstruct a multichannel image from 1, 2, or 3 components:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_11.png)\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-iqpvla`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_11.png)\n\nTransform a list of pictorial faces:\n\nCopy to clipboard.\n\nIn\\[1\\]:=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_12.png)\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-exvhvo`\n\nDirect link to example\n\nOut\\[1\\]=1\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_12.png)\n\nShow the residual images when using only the first three components:\n\nCopy to clipboard.\n\nIn\\[2\\]:=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/I_13.png)\n\n`https:\/\/wolfram.com\/xid\/0b6jc6p63p4r5i51zoyjrm-9lk0x`\n\nDirect link to example\n\nOut\\[2\\]=2\n\n![](https:\/\/reference.wolfram.com\/language\/ref\/Files\/KarhunenLoeveDecomposition.en\/O_13.png)\n## Properties & Relations  (7)Properties of the function, and connections to other functions\n\nWolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).\n\nCopy to clipboard.\n\n`Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).`\n\n#### Text\nWolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).\n\nCopy to clipboard.\n\n`Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html (updated 2015).`\n\n#### CMS\nWolfram Language. 2010. \"KarhunenLoeveDecomposition.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html.\n\nCopy to clipboard.\n\n`Wolfram Language. 2010. \"KarhunenLoeveDecomposition.\" Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html.`\n\n#### APA\nWolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html\n\nCopy to clipboard.\n\n`Wolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html`\n\n#### BibTeX\n@misc{reference.wolfram\\_2025\\_karhunenloevedecomposition, author=\"Wolfram Research\", title=\"{KarhunenLoeveDecomposition}\", year=\"2015\", howpublished=\"\\\\url{https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}\", note=\\[Accessed: 13-April-2026\\]}\n\nCopy to clipboard.\n\n`@misc{reference.wolfram_2025_karhunenloevedecomposition, author=\"Wolfram Research\", title=\"{KarhunenLoeveDecomposition}\", year=\"2015\", howpublished=\"\\url{https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}\", note=[Accessed: 13-April-2026]}`\n\n#### BibLaTeX\n@online{reference.wolfram\\_2025\\_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}, note=\\[Accessed: 13-April-2026\\]}\n\nCopy to clipboard.\n\n`@online{reference.wolfram_2025_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https:\/\/reference.wolfram.com\/language\/ref\/KarhunenLoeveDecomposition.html}, note=[Accessed: 13-April-2026]}`","meta_canonical":null}

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Meta Description	KarhunenLoeveDecomposition[{a1, a2, ...}] gives the Karhunen\[Dash]Loeve transform {{b1, b2, ...}, m} of the numerical arrays {a1, a2, ...}, where m . ai == bi. KarhunenLoeveDecomposition[{b1, b2, ...}, m] uses the inverse of the matrix m for transforming bi to ai.
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Boilerpipe Text	See Also SingularValueDecomposition PrincipalComponents DimensionReduce Related Guides Matrix Decompositions Image Representation Matrices and Linear Algebra Unsupervised Machine Learning See Also SingularValueDecomposition PrincipalComponents DimensionReduce Related Guides Matrix Decompositions Image Representation Matrices and Linear Algebra Unsupervised Machine Learning Examples Basic Examples Scope Options Standardized Applications Properties & Relations KarhunenLoeveDecomposition [ { a 1 , a 2 , … } ] Copy to clipboard. KarhunenLoeveDecomposition[{a1,a2,…}] gives the Karhunen – Loeve transform { { b 1 , b 2 , … } , m } of the numerical arrays { a 1 , a 2 , … } , where m . a i  b i . KarhunenLoeveDecomposition [ { b 1 , b 2 , … } , m ] Copy to clipboard. KarhunenLoeveDecomposition[{b1,b2,…},m] uses the inverse of the matrix m for transforming b i to a i . 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 a i can be arbitrary rank arrays or images of the same dimensions. The inner product of m and { a 1 , a 2 , … } gives { b 1 , b 2 , … } . In KarhunenLoeveDecomposition [ { a 1 , … } ] , rows of the transformation matrix m are the eigenvectors of the covariance matrix formed from the arrays a i . The matrix m is a linear transformation of a i . The transformed arrays b i are uncorrelated, are given in order of decreasing variance, and have the same total variance as a i . KarhunenLoeveDecomposition [ { b 1 , b 2 , … } , m ] effectively computes the inverse Karhunen – Loeve transformation. If the length of { b 1 , b 2 , … } is less than the size of m , missing components are assumed to be zero. With an option setting Standardized  True , datasets a i are shifted so that their means are zero. Examples open all close all Basic Examples (2) Summary of the most common use cases Karhunen – Loeve decomposition of two datasets: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-knb8up Direct link to example Out[1]= 1 Principal component decomposition of RGB color channels: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-cistzr Direct link to example Out[1]= 1 Scope (5) Survey of the scope of standard use cases Options (1) Common values & functionality for each option Standardized (1) Karhunen – Loeve decomposition with datasets shifted to mean zero: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-9aifu2 Direct link to example Out[1]= 1 Applications (3) Sample problems that can be solved with this function Enhance the color contrast of an RGB image: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-c1h72f Direct link to example Out[1]= 1 Reconstruct a multichannel image from 1, 2, or 3 components: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-iqpvla Direct link to example Out[1]= 1 Transform a list of pictorial faces: Copy to clipboard. In[1]:= 1 https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-exvhvo Direct link to example Out[1]= 1 Show the residual images when using only the first three components: Copy to clipboard. In[2]:= 2 https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-9lk0x Direct link to example Out[2]= 2 Properties & Relations (7) Properties of the function, and connections to other functions Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015). Copy to clipboard. 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). Copy to clipboard. 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. Copy to clipboard. 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 Copy to clipboard. Wolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html BibTeX @misc{reference.wolfram_2025_karhunenloevedecomposition, author="Wolfram Research", title="{KarhunenLoeveDecomposition}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}", note=[Accessed: 13-April-2026]} Copy to clipboard. @misc{reference.wolfram_2025_karhunenloevedecomposition, author="Wolfram Research", title="{KarhunenLoeveDecomposition}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}", note=[Accessed: 13-April-2026]} BibLaTeX @online{reference.wolfram_2025_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}, note=[Accessed: 13-April-2026]} Copy to clipboard. @online{reference.wolfram_2025_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}, note=[Accessed: 13-April-2026]}
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[Connect & Follow](https://www.wolfram.com/connect/?source=nav) - [Wolfram\\|Alpha](https://www.wolframalpha.com/?source=nav) - [Wolfram Cloud](https://www.wolframcloud.com/?source=nav) - [Your Account](https://account.wolfram.com/?source=nav) - [User Portal](https://user.wolfram.com/?source=nav) [Wolfram Language & System Documentation Center](https://reference.wolfram.com/language/) KarhunenLoeveDecomposition - [See Also](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [SingularValueDecomposition](https://reference.wolfram.com/language/ref/SingularValueDecomposition.html) - [PrincipalComponents](https://reference.wolfram.com/language/ref/PrincipalComponents.html) - [DimensionReduce](https://reference.wolfram.com/language/ref/DimensionReduce.html) - [Related Guides](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [Matrix Decompositions](https://reference.wolfram.com/language/guide/MatrixDecompositions.html) - [Image Representation](https://reference.wolfram.com/language/guide/ImageRepresentation.html) - [Matrices and Linear Algebra](https://reference.wolfram.com/language/guide/MatricesAndLinearAlgebra.html) - [Unsupervised Machine Learning](https://reference.wolfram.com/language/guide/UnsupervisedMachineLearning.html) - - [See Also](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [SingularValueDecomposition](https://reference.wolfram.com/language/ref/SingularValueDecomposition.html) - [PrincipalComponents](https://reference.wolfram.com/language/ref/PrincipalComponents.html) - [DimensionReduce](https://reference.wolfram.com/language/ref/DimensionReduce.html) - [Related Guides](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [Matrix Decompositions](https://reference.wolfram.com/language/guide/MatrixDecompositions.html) - [Image Representation](https://reference.wolfram.com/language/guide/ImageRepresentation.html) - [Matrices and Linear Algebra](https://reference.wolfram.com/language/guide/MatricesAndLinearAlgebra.html) - [Unsupervised Machine Learning](https://reference.wolfram.com/language/guide/UnsupervisedMachineLearning.html) [KarhunenLoeveDecomposition](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html)\[{a1,a2,…}\] gives the Karhunen–Loeve transform {{b1,b2,…},m} of the numerical arrays {a1,a2,…}, where m.aibi. [KarhunenLoeveDecomposition](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html)\[{b1,b2,…},m\] uses the inverse of the matrix m for transforming bi to ai. Details and Options ![Details and Options](https://reference.wolfram.com/language-assets-v2/img/DetailsOptions-gray.png) ![Details and Options](https://reference.wolfram.com/language-assets-v2/img/DetailsOptions-hover.png) Examples Basic Examples Scope Options Standardized Applications Properties & Relations See Also Related Guides History Cite this Page BUILT-IN SYMBOL - [See Also](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [SingularValueDecomposition](https://reference.wolfram.com/language/ref/SingularValueDecomposition.html) - [PrincipalComponents](https://reference.wolfram.com/language/ref/PrincipalComponents.html) - [DimensionReduce](https://reference.wolfram.com/language/ref/DimensionReduce.html) - [Related Guides](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [Matrix Decompositions](https://reference.wolfram.com/language/guide/MatrixDecompositions.html) - [Image Representation](https://reference.wolfram.com/language/guide/ImageRepresentation.html) - [Matrices and Linear Algebra](https://reference.wolfram.com/language/guide/MatricesAndLinearAlgebra.html) - [Unsupervised Machine Learning](https://reference.wolfram.com/language/guide/UnsupervisedMachineLearning.html) - - [See Also](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [SingularValueDecomposition](https://reference.wolfram.com/language/ref/SingularValueDecomposition.html) - [PrincipalComponents](https://reference.wolfram.com/language/ref/PrincipalComponents.html) - [DimensionReduce](https://reference.wolfram.com/language/ref/DimensionReduce.html) - [Related Guides](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [Matrix Decompositions](https://reference.wolfram.com/language/guide/MatrixDecompositions.html) - [Image Representation](https://reference.wolfram.com/language/guide/ImageRepresentation.html) - [Matrices and Linear Algebra](https://reference.wolfram.com/language/guide/MatricesAndLinearAlgebra.html) - [Unsupervised Machine Learning](https://reference.wolfram.com/language/guide/UnsupervisedMachineLearning.html) # KarhunenLoeveDecompositionCopy to clipboard. ✖ `KarhunenLoeveDecomposition` [KarhunenLoeveDecomposition](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html)\[{a1,a2,…}\] Copy to clipboard. ✖ `KarhunenLoeveDecomposition[{a1,a2,…}]` gives the Karhunen–Loeve transform {{b1,b2,…},m} of the numerical arrays {a1,a2,…}, where m.aibi. [KarhunenLoeveDecomposition](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html)\[{b1,b2,…},m\] Copy to clipboard. ✖ `KarhunenLoeveDecomposition[{b1,b2,…},m]` uses the inverse of the matrix m for transforming bi to ai. # Details and Options ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/details_1.png) - 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](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html)\[{a1,…}\], 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](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html)\[{b1,b2,…},m\] 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 [Standardized](https://reference.wolfram.com/language/ref/Standardized.html)[True](https://reference.wolfram.com/language/ref/True.html), datasets ai are shifted so that their means are zero. # Examples open all close all ## Basic Examples (2)Summary of the most common use cases Karhunen–Loeve decomposition of two datasets: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_1.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-knb8up` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_1.png) Principal component decomposition of RGB color channels: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_2.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-cistzr` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_2.png) ## Scope (5)Survey of the scope of standard use cases Principal components of two grayscale images: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_3.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-j1ll2v` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_3.png) Karhunen–Loeve decomposition of three matrix-valued datasets: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_4.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-pyroz` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_4.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_5.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-32j1v7` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_5.png) Principal components of a list of color images: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_6.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-ouqstd` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_6.png) Specify the transformation matrix: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_7.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-zsnmgd` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_7.png) Use a transformation matrix and lesser datasets: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_8.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-prlgd3` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_8.png) ## Options (1)Common values & functionality for each option ### Standardized (1) Karhunen–Loeve decomposition with datasets shifted to mean zero: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_9.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-9aifu2` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_9.png) ## Applications (3)Sample problems that can be solved with this function Enhance the color contrast of an RGB image: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_10.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-c1h72f` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_10.png) Reconstruct a multichannel image from 1, 2, or 3 components: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_11.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-iqpvla` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_11.png) Transform a list of pictorial faces: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_12.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-exvhvo` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_12.png) Show the residual images when using only the first three components: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_13.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-9lk0x` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_13.png) ## Properties & Relations (7)Properties of the function, and connections to other functions The Karhunen–Loeve decomposition preserves the total variance: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_14.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-xfttmi` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_14.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_15.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-m3ehi9` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_15.png) The Karhunen–Loeve decomposition yields uncorrelated sets: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_16.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-f0wukw` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_17.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-f89dvf` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_16.png) The Karhunen–Loeve decomposition yields an orthogonal transformation matrix: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_18.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-ng385z` Direct link to example Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_19.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-t4u4e8` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_17.png) Relation to [PrincipalComponents](https://reference.wolfram.com/language/ref/PrincipalComponents.html): Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_20.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-dnw4l2` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_18.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_21.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-bav5vr` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_19.png) A setting [Standardized](https://reference.wolfram.com/language/ref/Standardized.html)\-\>[True](https://reference.wolfram.com/language/ref/True.html) is equivalent to subtracting the mean from the input data: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_22.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-7ff1rn` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_20.png) Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_23.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-nvp2ii` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_21.png) Normalizing by the square root of the number of datasets better preserves the input dynamics: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_24.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-gucqhn` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_22.png) [KarhunenLoeveDecomposition](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) normally returns images of a real type: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_25.png) ✖ `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-buunxk` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_23.png) # See Also [SingularValueDecomposition](https://reference.wolfram.com/language/ref/SingularValueDecomposition.html) [PrincipalComponents](https://reference.wolfram.com/language/ref/PrincipalComponents.html) [DimensionReduce](https://reference.wolfram.com/language/ref/DimensionReduce.html) # Related Guides - [Matrix Decompositions](https://reference.wolfram.com/language/guide/MatrixDecompositions.html) - [Image Representation](https://reference.wolfram.com/language/guide/ImageRepresentation.html) - [Matrices and Linear Algebra](https://reference.wolfram.com/language/guide/MatricesAndLinearAlgebra.html) - [Unsupervised Machine Learning](https://reference.wolfram.com/language/guide/UnsupervisedMachineLearning.html) # History [Introduced in 2010 (8.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn80) \\| [Updated in 2014 (10.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn100) ▪ [2015 (10.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn101) Cite this as: Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015). Copy to clipboard. ✖ `Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015).` ![](https://reference.wolfram.com/language-assets-v2/img/CitationCloseButton.svg) #### Text Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015). Copy to clipboard. ✖ `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. Copy to clipboard. ✖ `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 Copy to clipboard. ✖ `Wolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. 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Readable Markdown	- [See Also](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [SingularValueDecomposition](https://reference.wolfram.com/language/ref/SingularValueDecomposition.html) - [PrincipalComponents](https://reference.wolfram.com/language/ref/PrincipalComponents.html) - [DimensionReduce](https://reference.wolfram.com/language/ref/DimensionReduce.html) - [Related Guides](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [Matrix Decompositions](https://reference.wolfram.com/language/guide/MatrixDecompositions.html) - [Image Representation](https://reference.wolfram.com/language/guide/ImageRepresentation.html) - [Matrices and Linear Algebra](https://reference.wolfram.com/language/guide/MatricesAndLinearAlgebra.html) - [Unsupervised Machine Learning](https://reference.wolfram.com/language/guide/UnsupervisedMachineLearning.html) - - [See Also](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [SingularValueDecomposition](https://reference.wolfram.com/language/ref/SingularValueDecomposition.html) - [PrincipalComponents](https://reference.wolfram.com/language/ref/PrincipalComponents.html) - [DimensionReduce](https://reference.wolfram.com/language/ref/DimensionReduce.html) - [Related Guides](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html) - [Matrix Decompositions](https://reference.wolfram.com/language/guide/MatrixDecompositions.html) - [Image Representation](https://reference.wolfram.com/language/guide/ImageRepresentation.html) - [Matrices and Linear Algebra](https://reference.wolfram.com/language/guide/MatricesAndLinearAlgebra.html) - [Unsupervised Machine Learning](https://reference.wolfram.com/language/guide/UnsupervisedMachineLearning.html) Examples Basic Examples Scope Options Standardized Applications Properties & Relations [KarhunenLoeveDecomposition](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html)\[{a1,a2,…}\] Copy to clipboard. `KarhunenLoeveDecomposition[{a1,a2,…}]` gives the Karhunen–Loeve transform {{b1,b2,…},m} of the numerical arrays {a1,a2,…}, where m.aibi. [KarhunenLoeveDecomposition](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html)\[{b1,b2,…},m\] Copy to clipboard. `KarhunenLoeveDecomposition[{b1,b2,…},m]` uses the inverse of the matrix m for transforming bi to ai. ## Details and Options ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/details_1.png) - 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](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html)\[{a1,…}\], 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](https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html)\[{b1,b2,…},m\] 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 [Standardized](https://reference.wolfram.com/language/ref/Standardized.html)[True](https://reference.wolfram.com/language/ref/True.html), datasets ai are shifted so that their means are zero. ## Examples open all close all ## Basic Examples (2)Summary of the most common use cases Karhunen–Loeve decomposition of two datasets: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_1.png) `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-knb8up` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_1.png) Principal component decomposition of RGB color channels: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_2.png) `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-cistzr` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_2.png) ## Scope (5)Survey of the scope of standard use cases ## Options (1)Common values & functionality for each option ### Standardized (1) Karhunen–Loeve decomposition with datasets shifted to mean zero: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_9.png) `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-9aifu2` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_9.png) ## Applications (3)Sample problems that can be solved with this function Enhance the color contrast of an RGB image: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_10.png) `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-c1h72f` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_10.png) Reconstruct a multichannel image from 1, 2, or 3 components: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_11.png) `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-iqpvla` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_11.png) Transform a list of pictorial faces: Copy to clipboard. In\[1\]:=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_12.png) `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-exvhvo` Direct link to example Out\[1\]=1 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_12.png) Show the residual images when using only the first three components: Copy to clipboard. In\[2\]:=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/I_13.png) `https://wolfram.com/xid/0b6jc6p63p4r5i51zoyjrm-9lk0x` Direct link to example Out\[2\]=2 ![](https://reference.wolfram.com/language/ref/Files/KarhunenLoeveDecomposition.en/O_13.png) ## Properties & Relations (7)Properties of the function, and connections to other functions Wolfram Research (2010), KarhunenLoeveDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html (updated 2015). Copy to clipboard. `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). Copy to clipboard. `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. Copy to clipboard. `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 Copy to clipboard. `Wolfram Language. (2010). KarhunenLoeveDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html` #### BibTeX @misc{reference.wolfram\_2025\_karhunenloevedecomposition, author="Wolfram Research", title="{KarhunenLoeveDecomposition}", year="2015", howpublished="\\url{https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}", note=\[Accessed: 13-April-2026\]} Copy to clipboard. `@misc{reference.wolfram_2025_karhunenloevedecomposition, author="Wolfram Research", title="{KarhunenLoeveDecomposition}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}", note=[Accessed: 13-April-2026]}` #### BibLaTeX @online{reference.wolfram\_2025\_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}, note=\[Accessed: 13-April-2026\]} Copy to clipboard. `@online{reference.wolfram_2025_karhunenloevedecomposition, organization={Wolfram Research}, title={KarhunenLoeveDecomposition}, year={2015}, url={https://reference.wolfram.com/language/ref/KarhunenLoeveDecomposition.html}, note=[Accessed: 13-April-2026]}`
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