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Sparse Separable Nonnegative Matrix Factorization

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Document pages: 20 pages

Abstract: We propose a new variant of nonnegative matrix factorization (NMF), combiningseparability and sparsity assumptions. Separability requires that the columnsof the first NMF factor are equal to columns of the input matrix, whilesparsity requires that the columns of the second NMF factor are sparse. We callthis variant sparse separable NMF (SSNMF), which we prove to be NP-complete, asopposed to separable NMF which can be solved in polynomial time. The mainmotivation to consider this new model is to handle underdetermined blind sourceseparation problems, such as multispectral image unmixing. We introduce analgorithm to solve SSNMF, based on the successive nonnegative projectionalgorithm (SNPA, an effective algorithm for separable NMF), and an exact sparsenonnegative least squares solver. We prove that, in noiseless settings andunder mild assumptions, our algorithm recovers the true underlying sources.This is illustrated by experiments on synthetic data sets and the unmixing of amultispectral image.

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