I. Introduction
Over the past few years, the non-negative matrix factorization (NMF) algorithm and its alternatives have proven to be very useful for several problems, especially in facial image characterization and representation problems [1]–[9]. NMF, similar to the principal component analysis (PCA) algorithm [10], represents a facial image as a linear combination of basis images and does not allow negative elements in either the basis images or the representation coefficients used in the linear combination of the basis images. Thus, it represents a facial image only by the additions of weighted basis images. The non-negativity constraints correspond better to the intuitive notion of combining facial parts to create a complete facial image. The bases of PCA are the Eigenfaces, resembling distorted versions of the entire face, while the bases of NMF are localized features that correspond better to the intuitive notion of facial parts [1]. The original NMF algorithm does not incorporate any sparseness constraints in the decomposition, even though in many cases, it has been experimentally verified that it produces sparse bases (i.e., bases with components that are spatially distributed without any connectivity).