Non-negative matrix factorization: Difference between revisions

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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>\mathbf{WHX} \leftarrow= \operatorname{nmf}(\mathbf{XWH}) </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.

ItEarly work research on non-negative matrrix factorizations was usedperformed by a Finnish group of researchers in the middle of the 1990s under the name ''positive matrix factorization''.

Usually all three matrices must be [[non-negative matrix|non-negative]], i.e., all elements must be equal to or greater than zero.

It might be said to be a sort of non-negative version of [[singular value decomposition]].

Usually the numbers of columns of <b>W</b> and the numbers of rows of <b>H</b> is selected so the product <b>WH</b> will become an approximation to <b>X</b>, and it has been suggested that the NMF model should be called ''nonnegative matrix approximation'' instead.

It was used by a Finnish group of researchers in the middle of the 1990s under the name ''positive matrix factorization''.

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.