  We present a numerical algorithm for nonnegative matrix factorization (NMF)
problems under noisy separability. An NMF problem under separability can be
stated as one of finding all vertices of the convex hull of data points. The
research interest of this paper is to find the vectors as close to the vertices
as possible in a situation in which noise is added to the data points. Our
algorithm is designed to capture the shape of the convex hull of data points by
using its enclosing ellipsoid. We show that the algorithm has correctness and
robustness properties from theoretical and practical perspectives; correctness
here means that if the data points do not contain any noise, the algorithm can
find the vertices of their convex hull; robustness means that if the data
points contain noise, the algorithm can find the near-vertices. Finally, we
apply the algorithm to document clustering, and report the experimental
results.
