A Convex Model for Nonnegative Matrix Factorization and Dimensionality Reduction on Physical Space

Abstract

A collaborative convex framework for factoring a data matrix X into a nonnegative product AS , with a sparse coefficient matrix S, is proposed. We restrict the columns of the dictionary matrix A to coincide with certain columns of the data matrix X, thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use l(1, ∞) regularization to select the dictionary from the data and show that this leads to an exact convex relaxation of l(0) in the case of distinct noise-free data. We also show how to relax the restriction-to- X constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in X. We focus on applications of the proposed framework to hyperspectral endmember and abundance identification and also show an application to blind source separation of nuclear magnetic resonance data.

A Signal Processing Perspective on Hyperspectral Unmixing: Insights from Remote Sensing

“…It is also related to NMF, to be described in the next section. In addition, there has been interest in using the measured data Y itself as the dictionary for MMV [62]. This self-dictionary MMV (SD-MMV) approach is related to pure pixel search.…”

Feature encoding in band-limited distributed surveillance systems

“…However, the lack of physical interpretation of the compact dictionary (i.e., physical meaning of each basis in the dictionary) has been a critical shortcoming of the standard dictionary learning techniques. Inspired by the method in [26] based on non-negative matrix factorization, in this paper, we address this issue by proposing a novel probabilistic approach for selecting a group of features as a compact representation of all the features in the training set. We believe that nothing is more meaningful for representing the data than the data itself.…”

“…The convex optimization problem in Eq. 2 is solved using the Alternating Direction Method of Multipliers framework in [29] (more details about the optimization steps can be found in [26]). …”

Non-Negative Sparse Mathematical Morphology

“…We note also that in hyperspectral imaging, there is a common task called linear unmixing (or factorization into physical space) which consists in detecting the spectra of the pure materials and estimating their relative abundances. Linear unmixing can be seen as an equivalent problem to NMF (Esser et al, 2012).…”