Non-Negative Matrix Factorization, Convexity and Isometry

Details Non-Negative Matrix Factorization, Convexity and Isometry Non-Negative Matrix Factorization, Convexity and Isometry

2008-10-13

arxiv.org/abs/0810.2311v2

Computer Science

Artificial Intelligence

accpepted in SIAM Data Mining 2009, 12 pages

Scientific paper

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.

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