A new weighted ℓp-norm for sparse hyperspectral unmixing
In this paper, we propose a new approach to approximate the ℓ<inf>0</inf>-norm for the sparse hyperspectral unmixing. In our method, we first approximate the ℓ<inf>0</inf>-norm with the ℓ<inf>p</inf>-norm and reduce p iteratively. It is motivated by our theorem that the set of minima for the ℓ<inf>p</inf>-norm problem is continuous in terms of p, which implies that smooth iterative reduction of p results in an enhanced solution. We introduce a weighted ℓ<inf>1</inf>-norm approximation of the ℓ<inf>p</inf>-norm problem involving a parameter e to deal with the fact that the ℓ<inf>p</inf>-norm problem is not Lipschitz continuous for p < 1 and propose an updating rule for the pair (p, ∊). To solve this approximated problem using the previous solution, we employ the alternating direction method of multipliers (ADMM) approach to derive the remaining updating steps. Experimental results show that our proposed method outperforms several state-of-the-art methods in terms of the reconstruction errors and their probability of success.