## Ellipsoidal Rounding for Nonnegative Matrix Factorization Under Noisy Separability

*Tomohiko Mizutani*; 15(Mar):1011−1039, 2014.

### Abstract

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.

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