%0 Journal Article
%T Nonnegative Matrix Factorization and I-Divergence Alternating Minimization
%A Lorenzo Finesso
%A Peter Spreij
%J Mathematics
%D 2004
%I arXiv
%R 10.1016/j.laa.2005.11.012
%X In 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 \`a la Csisz\'ar-Tusn\'ady 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.
%U http://arxiv.org/abs/math/0412070v2