In this paper we consider the Nonnegative Matrix Factorization (NMF) problem:
given an (elementwise) nonnegative matrix V∈R+m×n​ find, for
assigned k, nonnegative matrices W∈R+m×k​ and
H∈R+k×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. An interesting system theoretic application of NMF is to the problem
of approximate realization of Hidden Markov Models