SISAL Revisited

TL;DR

This paper revisits the SISAL algorithm for hyperspectral unmixing, providing a probabilistic interpretation, convergence guarantees, and new formulations, enhancing its robustness and practical utility in noisy environments.

Contribution

It offers a probabilistic framework for SISAL, proves convergence guarantees, and introduces new formulations and algorithms for improved performance.

Findings

SISAL can be interpreted as an approximation in a probabilistic framework.

The algorithm is proven to converge to a stationary point.

New formulations and algorithms outperform previous methods in experiments.

Abstract

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.