Nonnegative matrix factorization and I-divergence alternating minimization

In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: n an (elementwise) nonnegative matrix V is an element of R-+(mxn) find, for assigned k, nonnegative matrices W is an element of R(+)(mxk)m and H is an element of R-+(kxn) such that V = WH. Exact, nontrivial, 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 Csiszar-Tusnady 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. (c) 2005 Elsevier Inc. All rights reserved.