(Georgia Institute of Technology, 2008)
Kim, Jingu; Park, Haesun
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.
(Georgia Institute of Technology, 2008)
Kim, Jingu; Park, Haesun
Nonnegative Matrix Factorization (NMF) is a dimension reduction method that has been widely used for
various tasks including text mining, pattern analysis, clustering, and cancer class discovery. The mathematical
formulation for NMF appears as a non-convex optimization problem, and various types of algorithms have been
devised to solve the problem. The alternating nonnegative least squares (ANLS) framework is a block coordinate
descent approach for solving NMF, which was recently shown to be theoretically sound and empirically efficient.
In this paper, we present a novel algorithm for NMF based on the ANLS framework. Our new algorithm builds
upon the block principal pivoting method for the nonnegativity constrained least squares problem that overcomes
some limitations of active set methods. We introduce ideas to efficiently extend the block principal pivoting
method within the context of NMF computation. Our algorithm inherits the convergence theory of the ANLS
framework and can easily be extended to other constrained NMF formulations. Comparisons of algorithms using
datasets that are from real life applications as well as those artificially generated show that the proposed new
algorithm outperforms existing ones in computational speed.