Maximum volume inscribed ellipsoid: A new simplex-structured matrix factorization framework via facet enumeration and convex optimization

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 and 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 nonnegative 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 nonseparable 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.