Non-negative matrix factorization: Difference between revisions

Revision as of 08:46, 6 March 2007

Revision as of 09:59, 6 March 2007

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The elements of the residual matrix can either be negative and positive - at least in the typical application of NMF.

== History ==

Early work research on non-negative matrix factorizations was performed by a Finnish group of researchers in the middle of the 1990s under the name ''positive matrix factorization''.<ref>{{Cite journal

| author = P. Paatero, U. Tapper

* | P.title Paatero,= U. Tapper, "Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values", ''[[Environmetrics]]'', 5:111-126, [[1994]].

| journal = [[Environmetrics]]

| volume = 5

| pages = 111-126

| year = [[1994]]

| doi = 10.1002/env.3170050203

}}</ref><ref>{{Cite journal

| author = [[Pia Anttila]], [[Pentti Paatero]], Unto Tapper, Olli Järvinen

* | Piatitle Anttila,= Pentti Paatero, Unto Tapper, Olli Järvinen. "Source identification of bulk wet deposition in Finland by positive matrix factorization", ''[[Atmospheric Environment]]'', 29(14):1705-1718, 1995

| journal = [[Atmospheric Environment]]

| volume = 29

| issue = 14

| pages = 1705&ndash;1718

| year = 1995

| doi = 10.1016/1352-2310(94)00367-T

}}</ref>.

It became more widely known after Lee and Seung's investigations of the properties of the algorithm, and after they published a simple useful algorithm.

== Types ==

There are different types of non-negative matrix factorizations and one of these is related to [[probabilistic latent semantic analysis]] and the [[latent class model]].

The different types arise from using different [[cost function]]s (divergence functions) and/or by [[regularization (mathematics)|regularization]] of the '''W''' and/or '''H''' matrices<ref>[[Inderjit S. Dhillon]], [[Suvrit Sra]], "[http://books.nips.cc/papers/files/nips18/NIPS2005_0203.pdf Generalized Nonnegative Matrix Approximations with Bregman Divergences]", [[NIPS]], 2005.</ref>.

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== Uniqueness ==

The factorization is not unique: A matrix and its [[inverse matrix|inverse]] can be used to transform the two factorization matrices by, e.g.,

: <math>\mathbf{WH} = \mathbf{WBB}^{-1}\mathbf{H}</math>

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== Sources and external links ==

* J. Shen, G. W. Israël, "A receptor model using a specific non-negative transformation technique for ambient aerosol", ''[[Atmospheric Environment]]'', 23(10):2289-2298, [[1989]].

* P. Paatero, U. Tapper, "Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values", ''[[Environmetrics]]'', 5:111-126, [[1994]].

* Pia Anttila, Pentti Paatero, Unto Tapper, Olli Järvinen. "Source identification of bulk wet deposition in Finland by positive matrix factorization", ''[[Atmospheric Environment]]'', 29(14):1705-1718, 1995

* [[Pentti Paatero]], "Least squares formulation of robust non-negative factor analysis", ''[[Chemometrics and Intelligent Laboratory Systems]]'', 37(1):23-35, 1997 May.

* [[Daniel D. Lee]] and [[H. Sebastian Seung]], "Learning the parts of objects by non-negative matrix factorization", ''[[Nature (journal)|Nature]]'', 401(6755):788-791, 1999 October.

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=== References ===

<div class="references-small"><references/></div>