Sparse Nonnegative Matrix Factorization for Clustering

Abstract

Properties of Nonnegative Matrix Factorization (NMF) as a clustering method are studied by relating its formulation to other methods such as K-means clustering. We show how interpreting the objective function of K-means as that of a lower rank approximation with special constraints allows comparisons between the constraints of NMF and K-means and provides the insight that some constraints can be relaxed from K-means to achieve NMF formulation. By introducing sparsity constraints on the coefficient matrix factor in NMF objective function, we in term can view NMF as a clustering method. We tested sparse NMF as a clustering method, and our experimental results with synthetic and text data shows that sparse NMF does not simply provide an alternative to K-means, but rather gives much better and consistent solutions to the clustering problem. In addition, the consistency of solutions further explains how NMF can be used to determine the unknown number of clusters from data. We also tested with a recently proposed clustering algorithm, Affinity Propagation, and achieved comparable results. A fast alternating nonnegative least squares algorithm was used to obtain NMF and sparse NMF.