Maximum Volume Inscribed Ellipsoid: A New Simplex-Structured Matrix Factorization Framework via Facet Enumeration and Convex Optimization

TL;DR

This paper introduces a novel matrix factorization framework based on inscribing a maximum volume ellipsoid within data convex hulls, providing relaxed conditions for exact factor recovery in structured matrix factorization tasks.

Contribution

The paper proposes the first use of maximum volume inscribed ellipsoids for simplex-structured matrix factorization, offering a new theoretical approach with practical implementation methods.

Findings

MVIE guarantees exact factor recovery under relaxed conditions.

The MVIE framework is practically implementable via facet enumeration and convex optimization.

Numerical results demonstrate the effectiveness of the MVIE approach.

Abstract

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.