  Simplex identification via split augmented Lagrangian (SISAL) is a
popularly-used algorithm in blind unmixing of hyperspectral images. Developed
by Jos\'{e} M. Bioucas-Dias in 2009, the algorithm is fundamentally relevant to
tackling simplex-structured matrix factorization, and by extension,
non-negative matrix factorization, which have many applications under their
umbrellas. In this article, we revisit SISAL and provide new meanings to this
quintessential algorithm. The formulation of SISAL was motivated from a
geometric perspective, with no noise. We show that SISAL can be explained as an
approximation scheme from a probabilistic simplex component analysis framework,
which is statistical and is principally more powerful in accommodating the
presence of noise. The algorithm for SISAL was designed based on a successive
convex approximation method, with a focus on practical utility. It was not
known, by analyses, whether the SISAL algorithm has any kind of guarantee of
convergence to a stationary point. By establishing associations between the
SISAL algorithm and a line-search-based proximal gradient method, we confirm
that SISAL can indeed guarantee convergence to a stationary point. Our
re-explanation of SISAL also reveals new formulations and algorithms. The
performance of these new possibilities is demonstrated by numerical
experiments.
