Non-negative matrix factorization

Abstract

Non-negative matrix factorization - NMF is a Linear Dimensionality Reduction method, which approximates a high dimensional non-negative data matrix by a multiplica- tion of two low-ranked matrices that preserves the non-negativity of the data. This property has proven to be beneficial as it allows for the approximated data to be interpreted in the same way as the original data. In addition, NMF leads to a part- based representation of the data, which supports easy identification of the essential parts/features. The thesis starts with a short introduction of NMF, which includes a motivation behind the method, a detailed comparison to the well-known Principal Component Analysis and the possible generalizations of the ”standard NMF” problem. This is followed by a chapter presenting an overview of the wide range of NMF algorithms, which are separated into algorithms based on standard nonlinear optimization schemes and so called separable NMF. All algorithms of the first group are based on the two block gradient descent scheme. In contrast, the separable NMF is restricted to a subclass of matrices characterized by a practical geometrical interpretation which is exploited in many separable NMF algorithms. The last theoretical chapter focuses on the description of the key topics that should be considered when applying NMF such as initialization methods, rank estimation and quality measures to compare the performance of the algorithms. The thesis concludes with the analysis of the NMF methods for a spectrometric dataset consisting of TOF-SIMS measurements taken from meteorites. The ability of NMF to separate spectra into two dissimilar spectra with one considered as the background and one as meteorite specific has been analyzed. The obtained results are promising and give reason to believe that NMF is an adequate method for such tasks. In addition, the robustness to noise of NMF methods in the context of spectral data has been tested and finally the task of defining an appropriate factorization rank has been discussed.