A Fast Sparse NMF Optimization Algorithm for Hyperspectral Unmixing

Hyperspectral remote sensing images have received extensive attention because of their high spectral resolution. However, the limitation of spatial resolution of imaging spectrometers results in a large number of mixed pixels, which restricts the accuracy of data processing. Consequently, hyperspectral unmixing (HU) becomes an important step and tool in remote sensing image processing. In recent years, many approaches based on sparse nonnegative matrix factorization (NMF) have been widely applied in the field of HU due to their explicit statistical properties. In particular, methods based on
$\ell _{1/2}$
norm regularized NMF and their extended models have attracted considerable attention. However, the
$\ell _{1/2}$
sparse regularizer leads to a nonconvex and nonsmooth problem, which is a significant challenge to solve quickly and efficiently in practice. In this article, to solve the abovementioned problems, we propose a new fast
$\ell _{1/2}$
sparse NMF algorithm, called NewSpr-NMF, which combines two famous fast gradient strategies, i.e., Barzilai–Borwein gradient and Nesterov's optimal gradient, to alternately optimize two subproblems in NMF framework under the block coordinate descent scheme. Specifically, for the abundance subproblem of sparse NMF, in order to accelerate the solving process, we use Barzilai–Borwein gradient technology to develop a half-thresholding approximate iterative subalgorithm, while for another endmember subproblem, we adopt the Nestrov's optimal gradient method with second-order convergence to solve. Finally, we integrate the two aforementioned algorithms together to achieve efficient unmixing by alternately optimizing the two subproblems of the sparse NMF. The computational complexity and convergence of the proposed algorithm are carefully analyzed and briefly proved theoretically, respectively. A series of experimental results on both synthetic and real datasets, and compared with other advanced approaches, verify the effectiveness and efficiency of the proposed algorithm.