Chapter 13 - Non-negative mixtures

Abstract

This chapter discusses some algorithms for the use of non-negativity constraints in unmixing problems, including positive matrix factorization, nonnegative matrix factorization (NMF), and their combination with other unmixing methods such as non-negative independent component analysis and sparse non-negative matrix factorization. The 2D models can be naturally extended to multiway array (tensor) decompositions, especially non-negative tensor factorization (NTF) and non-negative tucker decomposition (NTD). The standard NMF model has been extended in various ways, including semi-NMF, multilayer NMF, tri-NMF, orthogonal NMF, nonsmooth NMF, and convolutive NMF. When gradient descent is a simple procedure, convergence can be slow, and the convergence can be sensitive to the step size. This can be overcome by applying multiplicative update rules, which have proved particularly popular in NMF. These multiplicative update rules have proved to be attractive since they are simple, do not need the selection of an update parameter, and their multiplicative nature, and non-negative terms on the RHS ensure that the elements cannot become negative.