This paper considers  volume minimization  (VolMin)-based structured matrix factorization. VolMin is a factorization criterion that decomposes a given data matrix into a basis matrix times a structured coefficient matrix via finding the minimum-volume simplex that encloses all the columns of the data matrix. Recent work showed that VolMin guarantees the identifiability of the factor matrices under mild conditions that are realistic in a wide variety of applications. This paper focuses on both theoretical and practical aspects of VolMin. On the theory side, exact equivalence of two independently developed sufficient conditions for VolMin identifiability is proven here, thereby providing a more comprehensive understanding of this aspect of VolMin. On the algorithm side, computational complexity and sensitivity to outliers are two key challenges associated with real-world applications of VolMin. These are addressed here via a new VolMin algorithm that handles volume regularization in a computationally simple way, and automatically detects and iteratively downweights outliers, simultaneously. Simulations and real-data experiments using a remotely sensed hyperspectral image and the Reuters document corpus are employed to showcase the effectiveness of the proposed algorithm.
['Xiao Fu', 'Kejun Huang', 'Bo Yang', 'Wing Kin Ma', 'Nicholas D. Sidiropoulos']
Robust Volume Minimization-Based Matrix Factorization for Remote Sensing and Document Clustering