Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization
This work addresses identifiability issues in interpretable data analysis models like hyperspectral unmixing and topic modeling, but it is incremental as it bridges existing algorithm families without a major breakthrough.
The paper tackles the problem of obtaining identifiable solutions in simplex-structured matrix factorization by converting minimum-volume optimization in the primal space to a maximum-volume problem in the dual space, proving identifiability and showing favorable performance compared to state-of-the-art algorithms in numerical experiments.
Simplex-structured matrix factorization (SSMF) is a generalization of nonnegative matrix factorization, a fundamental interpretable data analysis model, and has applications in hyperspectral unmixing and topic modeling. To obtain identifiable solutions, a standard approach is to find minimum-volume solutions. By taking advantage of the duality/polarity concept for polytopes, we convert minimum-volume SSMF in the primal space to a maximum-volume problem in the dual space. We first prove the identifiability of this maximum-volume dual problem. Then, we use this dual formulation to provide a novel optimization approach which bridges the gap between two existing families of algorithms for SSMF, namely volume minimization and facet identification. Numerical experiments show that the proposed approach performs favorably compared to the state-of-the-art SSMF algorithms.