Approximate Nonnegative Matrix Factorization via Alternating Minimization

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dc.creatorFinesso, Lorenzo
dc.creatorSpreij, Peter
dc.date2004-02-13
dc.date.accessioned2026-07-07T05:05:26Z
dc.date.available2026-07-07T05:05:26Z
dc.descriptionIn this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix $V \in \R_+^{m\times n}$ find, for assigned $k$, nonnegative matrices $W\in\R_+^{m\times k}$ and $H\in\R_+^{k\times n}$ such that $V=WH$. Exact, non trivial, nonnegative factorizations do not always exist, hence it is interesting to pose the approximate NMF problem. The criterion which is commonly employed is I-divergence between nonnegative matrices. The problem becomes that of finding, for assigned $k$, the factorization $WH$ closest to $V$ in I-divergence. An iterative algorithm, EM like, for the construction of the best pair $(W, H)$ has been proposed in the literature. In this paper we interpret the algorithm as an alternating minimization procedure à la Csiszár-Tusnády and investigate some of its stability properties. NMF is widespreading as a data analysis method in applications for which the positivity constraint is relevant. There are other data analysis methods which impose some form of nonnegativity: we discuss here the connections between NMF and Archetypal Analysis. An interesting system theoretic application of NMF is to the problem of approximate realization of Hidden Markov Models.
dc.identifierhttps://arxiv.org/abs/math/0402229
dc.identifierhttp://arxiv.org/abs/math/0402229
dc.identifierProceedings of the 16th International Symposium on Mathematical Theory of Networks and Systems, Leuven, July 5-9, 2004
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/70164
dc.subjectOptimization and Control
dc.subjectProbability
dc.subject93E03
dc.titleApproximate Nonnegative Matrix Factorization via Alternating Minimization
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