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.

通过利用对偶/极性概念将最小体积的单纯结构矩阵因式分解问题转化为对偶空间中的最大体积问题，本研究证明了最大体积的对偶问题具有可辨识性，并基于该对偶形式提供了一种新颖的优化方法，将两个现有的SSMF算法家族——体积最小化和面确定——之间的差距进行了弥合。数值实验证明，所提议的方法与现有的SSMF算法相比表现优越。

简单结构矩阵分解的双对偶体积最大化