Motivating Non - Negative Matrix Factorizations ∗

Given a vector space model encoding of a large data set, a usual starting point for data analysis is rank reduction [1]. However, standard rank reduction techniques such as the QR, Singular Value (SVD), and Semi-Discrete (SDD) decompositions and Principal Component Analysis (PCA) produce low rank bases which do not respect the non-negativity or structure of the original data. Non-negative Matrix Factorization (NMF) was suggested in 1997 by Lee and Seung (see [6] and [7]) to overcome this weakness without significantly increasing the error of the associated approximation. NMF has been typically applied to image and text data (see for example: house and facial images in [6], handwriting samples in [9]), but has also been used to deconstruct music tones [4]. The additive property resulting from the non-negativity constraints of NMF has been shown to result in bases that represent local components of the original data (i.e.doors for houses, eyes for faces, curves of letters and notes in a chord). In this paper, we intend to motivate the application of NMF techniques (with noted corrections) to other types of data describing physical phenomena. The contents of this paper are as follows. Section 2 details the NMF objective functions and update strategies used here and in practice. In Section 3 we illustrate both the error and resulting basis for text and image collections. We then turn to the specific example of NMF performed on remote sensing data in Section 4. We emphasize recently proposed NMF alterations and compare the output obtained with the remote sensing literature. We conclude in Section 5 by describing future work for this promising area.