NMF (Non-negative Matrix Factorization)
Non-negative matrix factorization (NMF) decomposes a non-negative data matrix into the product of two smaller, also non-negative matrices. We explain why that constraint yields interpretable parts-based representations, its uses—from topic modeling to audio separation—and how it is computed.
Non-negative matrix factorization (NMF) approximately decomposes a data matrix with non-negative values into the product of two smaller, also non-negative matrices: V ≈ W·H. The matrix W holds some “components” or latent patterns, and H indicates how to combine them to reconstruct each sample.
Why non-negativity matters
The constraint that all values be non-negative only allows additive combinations of the components, never subtractive ones. That produces a “parts-based,” interpretable representation: the parts add up. Daniel Lee and Sebastian Seung popularized it in Nature in 1999, showing that NMF learns “the parts of objects”—for example, facial features—unlike principal component analysis (PCA), which allows negative values and gives less interpretable representations.
What it is used for
NMF is used for dimensionality reduction when the data have no negative values, for topic modeling in text (each document as a non-negative mixture of topics), for audio source separation, in recommender systems, and in image analysis or bioinformatics.
How it is computed
W and H are fit by minimizing the reconstruction error between the original matrix and their product, usually measured with the Frobenius norm or with divergences such as Kullback-Leibler; the classic algorithms use multiplicative update rules. The number of components is a hyperparameter set beforehand, and the problem is not convex, so different initializations can give different results.
This article was produced with artificial intelligence under human editorial oversight.
