SISAL Revisited

Abstract

Simplex identification via split augmented Lagrangian (SISAL) is a popularly-used algorithm in 4 blind unmixing of hyperspectral images. Developed by José M. Bioucas-Dias in 2009, the algorithm 5 is fundamentally relevant to tackling simplex-structured matrix factorization, and by extension, non6 negative matrix factorization, which have many applications under their umbrellas. In this article, we 7 revisit SISAL and provide new meanings to this quintessential algorithm. The formulation of SISAL 8 was motivated from a geometric perspective, with no noise. We show that SISAL can be explained 9 as an approximation scheme from a probabilistic simplex component analysis framework, which is 10 statistical and is principally more powerful in accommodating the presence of noise. The algorithm 11 for SISAL was designed based on a successive convex approximation method, with a focus on practical 12 utility. It was not known, by analyses, whether the SISAL algorithm has any kind of guarantee 13 of convergence to a stationary point. By establishing associations between the SISAL algorithm 14 and a line-search-based proximal gradient method, we confirm that SISAL can indeed guarantee 15 convergence to a stationary point. Our re-explanation of SISAL also reveals new formulations and 16 algorithms. The performance of these new possibilities is demonstrated by numerical experiments. 17