  This paper considers \emph{volume minimization} (VolMin)-based structured
matrix factorization (SMF). 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.
