Maximum Volume Inscribed Ellipsoid: A New Simplex-Structured Matrix Factorization Framework via Facet Enumeration and Convex Optimization
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Consider a structured matrix factorization scenario where one factor is modeled to have columns lying in the unit simplex. Such a simplex-structured matrix factorization (SSMF) problem has spurred much interest in key topics such as hyperspectral unmixing in remote sensing and topic discovery in machine learning. In this paper we develop a new theoretical framework for SSMF. The idea is to study a maximum volume ellipsoid inscribed in the convex hull of the data points, which has not been attempted in prior literature. We show a sufficient condition under which this maximum volume inscribed ellipsoid (MVIE) framework can guarantee exact recovery of the factors. The condition derived is much better than that of separable non-negative matrix factorization (or pure-pixel search) and is comparable to that of another powerful framework called minimum volume enclosing simplex. From the MVIE framework we also develop an algorithm that uses facet enumeration and convex optimization to achieve the aforementioned recovery result. Numerical results are presented to demonstrate the potential of this new theoretical SSMF framework.
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