Approximate Nonnegative Matrix Factorization via Alternating Minimization

TL;DR

This paper analyzes an EM-like iterative algorithm for approximate nonnegative matrix factorization, exploring its stability, connections to archetypal analysis, and applications to Hidden Markov Model realization.

Contribution

It interprets the NMF algorithm as an alternating minimization method and investigates its stability and connections to other data analysis techniques.

Findings

The algorithm exhibits certain stability properties.

Connections between NMF and Archetypal Analysis are discussed.

Applications to Hidden Markov Model approximation are explored.

Abstract

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. An interesting system theoretic application of NMF is to the problem of approximate realization of Hidden Markov Models.