Non-Negative Matrix Factorization, Convexity and Isometry
Abstract
In this paper we explore avenues for improving the reliability of dimensionality reduction methods such as Non-Negative Matrix Factorization (NMF) as interpretive exploratory data analysis tools. We first explore the difficulties of the optimization problem underlying NMF, showing for the first time that non-trivial NMF solutions always exist and that the optimization problem is actually convex, by using the theory of Completely Positive Factorization. We subsequently explore four novel approaches to finding globally-optimal NMF solutions using various ideas from convex optimization. We then develop a new method, isometric NMF (isoNMF), which preserves non-negativity while also providing an isometric embedding, simultaneously achieving two properties which are helpful for interpretation. Though it results in a more difficult optimization problem, we show experimentally that the resulting method is scalable and even achieves more compact spectra than standard NMF.
Cite
@article{arxiv.0810.2311,
title = {Non-Negative Matrix Factorization, Convexity and Isometry},
author = {Nikolaos Vasiloglou and Alexander G. Gray and David V. Anderson},
journal= {arXiv preprint arXiv:0810.2311},
year = {2009}
}
Comments
accpepted in SIAM Data Mining 2009, 12 pages