Nonnegative matrix factorization (NMF) has become a very popular technique inmachine learning because it automatically extracts meaningful features througha sparse and part-based representation. However, NMF has the drawback of beinghighly ill-posed, that is, there typically exist many different but equivalentfactorizations. In this paper, we introduce a completely new way to obtainingmore well-posed NMF problems whose solutions are sparser. Our technique isbased on the preprocessing of the nonnegative input data matrix, and relies onthe theory of M-matrices and the geometric interpretation of NMF. This approachprovably leads to optimal and sparse solutions under the separabilityassumption of Donoho and Stodden (NIPS, 2003), and, for rank-three matrices,makes the number of exact factorizations finite. We illustrate theeffectiveness of our technique on several image datasets.
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