Non-negative matrix factorization: Difference between revisions

Revision as of 15:41, 15 March 2007

Revision as of 15:42, 15 March 2007

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''NMF redirects here. For the [[contract bridge|bridge]] convention, see [[new minor forcing]].''

'''Non-negative matrix factorization''' (NMF) is a group of [[algorithm]]s in [[multivariate analysis]] and [[linear algebra]] where a [[matrix (mathematics)|matrix]], <math>\mathbf{X}</math>, is factorized into (usually) two matrices, <math>\mathbf{W}</math> and <math>\mathbf{H}</math> : <math>\operatorname{nmf}(\mathbf{X}) \rightarrow \mathbf{WH} </math>

'''Non-negative matrix factorization''' (NMF) is a group of

[[algorithm]]s in [[multivariate analysis]] and [[linear algebra]]

where a [[matrix (mathematics)|matrix]], <math>\mathbf{X}</math>, is

factorized into (usually) two matrices, <math>\mathbf{W}</math> and

<math>\mathbf{H}</math> : <math>\operatorname{nmf}(\mathbf{X})

\rightarrow \mathbf{WH} </math>

Factorization of matrices is generally non-unique, and a number of different methods of doing so have been developed (e.g. [[principal component analysis]] and [[singular value decomposition]]) by incorporating different constraints; non-negative matrix factorization differs from these methods in that it enforces the constraint that all three matrices must be [[non-negative matrix|non-negative]], i.e., all elements must be equal to or greater than zero.

Factorization of matrices is generally non-unique, and a number of

different methods of doing so have been developed (e.g. [[principal

component analysis]] and [[singular value decomposition]]) by

incorporating different constraints; non-negative matrix factorization

differs from these methods in that it enforces the constraint that all

three matrices must be [[non-negative matrix|non-negative]], i.e., all

elements must be equal to or greater than zero.

Usually the number of columns of '''W''' and the number of rows of '''H''' in NMF are selected so the product '''WH''' will become an approximation to '''X''' (it has been suggested that the NMF model should be called ''nonnegative matrix approximation'' instead). The full decomposition of '''X''' then amounts to the two non-negative matrices '''W''' and '''H''' as well as a residual '''U''': : <math>\mathbf{X} = \mathbf{WH + U} </math> The elements of the residual matrix can either be negative and positive - at least in the typical application of NMF.

Usually the number of columns of '''W''' and the number of rows of

'''H''' in NMF are selected so the product '''WH''' will become an

approximation to '''X''' (it has been suggested that the NMF model

should be called ''nonnegative matrix approximation'' instead). The

full decomposition of '''X''' then amounts to the two non-negative

matrices '''W''' and '''H''' as well as a residual '''U''': :

<math>\mathbf{X} = \mathbf{WH + U} </math> The elements of the

residual matrix can either be negative and positive - at least in the

typical application of NMF.

== History ==