Non-negative matrix factorization: Ill-posedness and a geometric algorithm

Non-negative matrix factorization (NMF) has been proposed as a mathematical tool for identifying the components of a dataset. However, popular NMF algorithms tend to operate slowly and do not always identify the components which are most representative of the data. In this paper, an alternative algorithm for performing NMF is developed using the geometry of the problem. The computational costs of the algorithm are explored, and it is shown to successfully identify the components of a simulated dataset. The development of the geometric algorithm framework illustrates the ill-posedness of the NMF problem and suggests that NMF is not sufficiently constrained to be applied successfully outside of a particular class of problems.