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Non-negative matrix factorization

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Illustration of approximate non-negative matrix factorization: the matrix V is represented by the two smaller matrices W and H, which, when multiplied, approximately reconstruct V.

Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation[1][2] is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms or muscular activity, non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically.

NMF finds applications in such fields as astronomy,[3][4] computer vision, document clustering,[1] missing data imputation,[5] chemometrics, audio signal processing, recommender systems,[6][7] and bioinformatics.[8]

  1. ^ a b Cite error: The named reference dhillon was invoked but never defined (see the help page).
  2. ^ Tandon, Rashish; Sra, Suvrit (September 13, 2010). Sparse nonnegative matrix approximation: new formulations and algorithms (PDF) (Report). Max Planck Institute for Biological Cybernetics. Technical Report No. 193.
  3. ^ Cite error: The named reference blantonRoweis07 was invoked but never defined (see the help page).
  4. ^ Cite error: The named reference ren18 was invoked but never defined (see the help page).
  5. ^ Ren, Bin; Pueyo, Laurent; Chen, Christine; Choquet, Elodie; Debes, John H; Duechene, Gaspard; Menard, Francois; Perrin, Marshall D. (2020). "Using Data Imputation for Signal Separation in High Contrast Imaging". The Astrophysical Journal. 892 (2): 74. arXiv:2001.00563. Bibcode:2020ApJ...892...74R. doi:10.3847/1538-4357/ab7024. S2CID 209531731.
  6. ^ Rainer Gemulla; Erik Nijkamp; Peter J. Haas; Yannis Sismanis (2011). Large-scale matrix factorization with distributed stochastic gradient descent. Proc. ACM SIGKDD Int'l Conf. on Knowledge discovery and data mining. pp. 69–77.
  7. ^ Yang Bao; et al. (2014). TopicMF: Simultaneously Exploiting Ratings and Reviews for Recommendation. AAAI.
  8. ^ Ben Murrell; et al. (2011). "Non-Negative Matrix Factorization for Learning Alignment-Specific Models of Protein Evolution". PLOS ONE. 6 (12): e28898. Bibcode:2011PLoSO...628898M. doi:10.1371/journal.pone.0028898. PMC 3245233. PMID 22216138.

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