Hyperspectral Unmixing via $L_{1/2}$ 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 <i>L</i><sub>1</sub> regularizer. Unfortunately, the <i>L</i><sub>1</sub> regularizer cannot enforce further sparsity when the full additivity constraint of material abundances is used, hence limiting the practical efficacy of NMF methods in hyperspectral unmixing. In this paper, we extend the NMF method by incorporating the <i>L</i><sub>1/2</sub> sparsity constraint, which we name <i>L</i><sub>1/2</sub> -NMF. The <i>L</i><sub>1/2</sub> regularizer not only induces sparsity but is also a better choice among <i>Lq</i>(0 <; <i>q</i> <; 1) regularizers. We propose an iterative estimation algorithm for <i>L</i><sub>1/2</sub>-NMF, which provides sparser and more accurate results than those delivered using the <i>L</i><sub>1</sub> norm. We illustrate the utility of our method on synthetic and real hyperspectral data and compare our results to those yielded by other state-of-the-art methods.
Linear and Nonlinear Projective Nonnegative Matrix Factorization
A variant of nonnegative matrix factorization (NMF) which was proposed earlier is analyzed here. It is called projective nonnegative matrix factorization (PNMF). The new method approximately factorizes a projection matrix, minimizing the reconstruction error, into a positive low-rank matrix and its transpose. The dissimilarity between the original data matrix and its approximation can be measured by the Frobenius matrix norm or the modified Kullback-Leibler divergence. Both measures are minimized by multiplicative update rules, whose convergence is proven for the first time. Enforcing orthonormality to the basic objective is shown to lead to an even more efficient update rule, which is also readily extended to nonlinear cases. The formulation of the PNMF objective is shown to be connected to a variety of existing NMF methods and clustering approaches. In addition, the derivation using Lagrangian multipliers reveals the relation between reconstruction and sparseness. For kernel principal component analysis (PCA) with the binary constraint, useful in graph partitioning problems, the nonlinear kernel PNMF provides a good approximation which outperforms an existing discretization approach. Empirical study on three real-world databases shows that PNMF can achieve the best or close to the best in clustering. The proposed algorithm runs more efficiently than the compared NMF methods, especially for high-dimensional data. Moreover, contrary to the basic NMF, the trained projection matrix can be readily used for newly coming samples and demonstrates good generalization.
neural network power system internet of things electric vehicle data analysi renewable energy smart grid learning algorithm power grid image compression hyperspectral image matrix factorization source separation cyber-physical system energy management system sparse representation deep convolutional cloud storage blind source separation demand response blind source gradient method renewable energy system grid system dictionary learning hyperspectral datum latent semantic spectral clustering nonnegative matrix nonnegative matrix factorization hyperspectral imagery low rank image representation image inpainting public cloud matrix completion spectral datum smart grid system smart grid technology remote datum smart grid communication tensor factorization data matrix latent factor future smart grid factorization method spectral unmixing grid communication hyperspectral unmixing international system future smart smart power grid nonnegative matrice power grid system dictionary learning algorithm matrix factorization method data possession projected gradient graph regularized factorization based nonnegative tensor provable data possession system of units image inpainting method smart grid security provable datum public cloud storage matrix factorization technique projected gradient method factorization technique nonnegative tensor factorization nmf algorithm low-rank matrix factorization exemplar-based image inpainting image inpainting technique emerging smart grid matrix factorization problem multiplicative update based image inpainting regularized nonnegative matrix constrained nonnegative matrix sparse nonnegative kernel k-means clustering regularized nonnegative sparse nonnegative matrix matrix and tensor sparse nmf constrained nonnegative high-dimensional vector nmf method orthogonal nonnegative matrix graph regularized nonnegative nonnegative datum multi-way datum nonnegative tucker decomposition lee and seung weighted nonnegative matrix weighted nonnegative robust nonnegative matrix projective nonnegative matrix als algorithm robust nonnegative input data matrix projective nonnegative semantic image inpainting fast nonnegative wind power