  Nonnegative matrix factorization (NMF) under the separability assumption can
provably be solved efficiently, even in the presence of noise, and has been
shown to be a powerful technique in document classification and hyperspectral
unmixing. This problem is referred to as near-separable NMF and requires that
there exists a cone spanned by a small subset of the columns of the input
nonnegative matrix approximately containing all columns. In this paper, we
propose a preconditioning based on semidefinite programming making the input
matrix well-conditioned. This in turn can improve significantly the performance
of near-separable NMF algorithms which is illustrated on the popular successive
projection algorithm (SPA). The new preconditioned SPA is provably more robust
to noise, and outperforms SPA on several synthetic data sets. We also show how
an active-set method allow us to apply the preconditioning on large-scale
real-world hyperspectral images.
