  In this work, we consider nonnegative matrix factorization (NMF) with a
regularization that promotes small volume of the convex hull spanned by the
basis matrix. We present highly efficient algorithms for three different volume
regularizers, and compare them on endmember recovery in hyperspectral unmixing.
The NMF algorithms developed in this work are shown to outperform the
state-of-the-art volume-regularized NMF methods, and produce meaningful
decompositions on real-world hyperspectral images in situations where
endmembers are highly mixed (no pure pixels). Furthermore, our extensive
numerical experiments show that when the data is highly separable, meaning that
there are data points close to the true endmembers, and there are a few
endmembers, the regularizer based on the determinant of the Gramian produces
the best results in most cases. For data that is less separable and/or contains
more endmembers, the regularizer based on the logarithm of the determinant of
the Gramian performs best in general.
