Sparsity-constrained Nonnegative Matrix Factorization

Hyperspectral unmixing is a crucial preprocessing step for material classification and recognition. In the last decade, nonnegative matrix factorization (NMF) and its extensions have been intensively studied to unmix hyperspectral imagery and recover the material end-members. As an important constraint for NMF, sparsity has been modeled making use of the L1 regularizer. Nonetheless, recent studies show that the solutions with L1-norm are less sparse than those yielded by its L0 counterpart, while solving the L0-norm is an NP hard problem. Furthermore, the L1 regularizer conflicts with the full additivity constraint of material abundances, hence, limiting the practical efficacy of NMF methods in hyperspectral unmixing. In this paper, we extend the NMF method by incorporating the L1/2 sparsity constraint, which we name L1/2-NMF. The L1/2 regularizer not only induces sparsity, but is also an unbiased estimator. We provide an iterative estimation algorithm for L1/2-NMF, which provides more sparse and accurate results than those delivered making use of the L1 norm. We do this by considering the end-member additivity constraint explicitly in the optimization process. We illustrate the utility of our method on synthetic and real hyperspectral data. Index Terms Hyperspectral unmixing; Nonnegative matrix factorization; Sparse coding, L1/2 regularizer Y. Qian is with the Institute of Artificial Intelligence, College of Computer Science, Zhejiang University, Hangzhou, P.R. China. He is supported by Australia China Special Fund for Science and Technology Cooperation No.61011120054 S. Jia is with the College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, P.R. China. He is supported by National Natural Science Foundation of China No.60902070 and the Doctor Starting Project of Natural Science Foundation of Guangdong Province No.9451806001002287. Corresponding author. J. Zhou and A. Robles-Kelly are with Canberra Research Laboratory, NICTA, PO BOX 8001, Canberra, ACT 2601, Australia. They are also with College of Engineering and Computer Science, The Australian National University, Canberra, ACT 0200, Australia. NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program. February 2, 2011 DRAFT