  Consider a structured matrix factorization model where one factor is
restricted to have its columns lying in the unit simplex. This
simplex-structured matrix factorization (SSMF) model and the associated
factorization techniques have spurred much interest in research topics over
different areas, such as hyperspectral unmixing in remote sensing, topic
discovery in machine learning, to name a few. In this paper we develop a new
theoretical SSMF framework whose idea is to study a maximum volume ellipsoid
inscribed in the convex hull of the data points. This maximum volume inscribed
ellipsoid (MVIE) idea has not been attempted in prior literature, and we show a
sufficient condition under which the MVIE framework guarantees exact recovery
of the factors. The sufficient recovery condition we show for MVIE is much more
relaxed than that of separable non-negative matrix factorization (or pure-pixel
search); coincidentally it is also identical to that of minimum volume
enclosing simplex, which is known to be a powerful SSMF framework for
non-separable problem instances. We also show that MVIE can be practically
implemented by performing facet enumeration and then by solving a convex
optimization problem. The potential of the MVIE framework is illustrated by
numerical results.
