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'''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})
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 <b>W</b> and the number of rows of <b>H</b> in NMF are selected so the product <b>WH</b> will become an approximation to <b>X</b> (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 are can either be negative and positive - at least in the typical application of NMF.
Early work research on non-negative matrrix factorizations was performed by a Finnish group of researchers in the middle of the 1990s under the name ''positive matrix factorization''.
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