Abstract. In image compression and feature extraction, linear expansions are standardly used. It was recently pointed out by Lee and Seung that the positivity or non-negativity of a linear expansion is a very powerful constraint, that seems to lead to sparse representations for the images. Their technique, called Non-negative Matrix Factorization (NMF), was shown to be a useful technique in approximating high dimensional data where the data are comprised of non-negative components. We propose here a new variant of the NMF method for learning spatially localized, sparse, part-based subspace representations of visual patterns. The algorithm is based on positively constrained projections and is related both to NMF and to the conventional SVD or PCA decomposition. Two iterative positive projection algorithms are suggested, one based on minimizing Euclidean distance and the other on minimizing the divergence of the original data matrix and its non-negative approximation. Experimental results show that P-NMF derives bases which are somewhat better suitable for a localized representation than NMF.

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We propose a new variant of Non-negative Matrix Factorization (NMF), including its model and two optimization rules. Our method is based on positively constrained projections and is related to the conventional SVD or PCA decomposition. The new model can potentially be applied to image compression and feature extraction problems. Of the latter, we consider processing of facial images, where each image consists of several parts and for each part the observations with different lighting mainly distribute along a straight line through the origin. No regularization terms are required in the objective functions and both suggested optimization rules can easily be implemented by matrix manipulations. The experiments show that the derived base vectors are spatially more localized than those of NMF. In turn, the better part-based representations improve the recognition rate of semantic classes such as the gender or existence of mustache in the facial images.