A convex model for non-negative matrix factorization and dimensionality reduction on physical space
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A collaborative convex framework for factoring a data matrix X into a non-negative 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 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 abundances identification and also show an application to blind source separation of NMR data.
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