  Nonnegative matrix factorization (NMF) has become a very popular technique in
machine learning because it automatically extracts meaningful features through
a sparse and part-based representation. However, NMF has the drawback of being
highly ill-posed, that is, there typically exist many different but equivalent
factorizations. In this paper, we introduce a completely new way to obtaining
more well-posed NMF problems whose solutions are sparser. Our technique is
based on the preprocessing of the nonnegative input data matrix, and relies on
the theory of M-matrices and the geometric interpretation of NMF. This approach
provably leads to optimal and sparse solutions under the separability
assumption of Donoho and Stodden (NIPS, 2003), and, for rank-three matrices,
makes the number of exact factorizations finite. We illustrate the
effectiveness of our technique on several image datasets.
