A manifold Hessian-regularized NMF for hyperspectral data unmixing

Hyperspectral unmixing is a powerful method of the remote sensing image mining that identifies the constituent materials and estimates the corresponding fractions from the mixture. We consider the application of nonnegative matrix factorization (NMF) for the mining and analysis of spectral data. In this paper, we develop two effective active set type NMF algorithms for hyperspectral unmixing. Because the factor matrices used in unmixing have sparse features, the active set strategy helps reduce the computational cost. These active set type algorithms for NMF is based on an alternating nonnegative constrained least squares (ANLS) and achieve a quadratic convergence rate under the reasonable assumptions. Finally, numerical tests demonstrate that these algorithms work well and that the function values decrease faster than those obtained with other algorithms.

Nonnegative matrix factorization (NMF) based hyperspectral unmixing aims at estimating pure spectral signatures and their fractional abundances at each pixel. During the past several years, manifold structures have been introduced as regularization constraints into NMF. However, most methods only consider the constraints on abundance matrix while ignoring the geometric relationship of endmembers. Although such relationship can be described by traditional graph construction approaches based on k-nearest neighbors, its accuracy is questionable. In this paper, we propose a novel hyperspectral unmixing method, namely NMF with endmember sparse graph learning, to tackle the above drawbacks. This method first integrates endmember sparse graph structure into NMF, then simultaneously performs unmixing and graph learning. It is further extended by incorporating abundance smoothness constraint to improve the unmixing performance. Experimental results on both synthetic and real datasets have validated the effectiveness of the proposed method.

Lee and Seung proposed nonnegative matrix factorization (NMF) algorithms to decompose patterns and images for structure retrieving. The NMF algorithms have been applied to various optimization problems. However, it is difficult to prove the convergence of this class of learning algorithms. This paper presents the global minima analysis of the NMF algorithms. In the analysis, invariant set is constructed so that the non-divergence of the algorithms can be guaranteed in the set. Using the features of linear equation systems and their solutions, the fixed points and convergence properties of the update algorithms are discussed in detail. The analysis shows that, although the cost function is not convex in both A and X together, it is possible to obtain the global minima from the particular learning algorithms. For different initializations, simulations are presented to confirm the analysis results.

Hyperspectral imaging captures detailed spectral data for remote sensing. However, due to the limited spatial resolution of hyperspectral sensors, each pixel of a hyperspectral image (HSI) may contain information from multiple materials. Although the hyperspectral unmixing (HU) process involves estimating endmembers, identifying pure spectral components, and estimating pixel abundances, existing algorithms mostly focus on just one or two tasks. Blind source separation (BSS) based on nonnegative matrix factorization (NMF) algorithms identify endmembers and their abundances at each pixel of HSI simultaneously. Although they perform well, the factorization results are unstable, require high computational costs, and are difficult to interpret from the original HSI. CUR matrix decomposition selects specific columns and rows from a dataset to represent it as a product of three small submatrices, resulting in interpretable low-rank factorization. In this paper, we propose a new blind HU framework based on CUR factorization called CUR-HU that performs the entire HU process by exploiting the low-rank structure of given HSIs. CUR-HU incorporates several techniques to perform the HU process with a performance comparable to state-of-the-art methods but with higher computational efficiency. We adopt a deterministic sampling method to select the most informative pixels and spectrum components in HSIs. We use an incremental QR decomposition method to reduce computation complexity and estimate the number of endmembers. Various experiments on synthetic and real HSIs are conducted to evaluate the performance of CUR-HU. CUR-HU performs comparably to state-of-the-art methods for estimating the number of endmembers and abundance maps, but it outperforms other methods for estimating the endmembers and the computational efficiency. It has a 9.4 to 249.5 times speedup over different methods for different real HSIs.

Nonnegative matrix factorization(NMF) has been applied to hyperspectral unmixing in recent years. Different constraints based on geometrical or statistical properties of endmember and abundance are incorporated into NMF model to improve unmixing result. In this paper, a new regularizer based on spectral cluster information is proposed to strengthen the constrained relationship between original image and abundance maps. The new algorithm makes abundances of similar pixels close and abundances of dissimilar pixels be separated completely. Additionally, L 1/2 sparsity constraint is adopted to make the solutions sparse. Comparative results on real and synthetic hyperspectral datasets prove our proposed method could improve the hyperspectral unmixing accuracy.

Face recognition is a challenging issue in field of multi-science, main contents of research is how to make computer have ability of face recognition face recognition technology involved in a lot, which is a key feature extraction and classification method, this paper focuses on study of related theory. Non-negative matrix factorization trapped MF) algorithm and local non-negative matrix factorization (LNMF) algorithm is a feature extraction method based on local features, has been successfully used in face recognition NMF algorithm in face recognition rate is low, although LNMF algorithm to a certain extent, improve recognition rate, but its price is to increase number of iterations. In addition, two algorithms have failed to solve good nonlinear separable problems the kernel method combined with LNMF algorithm, kernel local non-negative matrix factorization (KLNMF) algorithm, first by a nonlinear transformation of original space to high-dimensional space, making samples linearly separable, and then use LNMF algorithm to extract face features. In classification part, paper presents decision rules of classification of their own, and design based on NMF subspace classifier. DOI : http://dx.doi.org/10.11591/telkomnika.v12i3.4356 Full Text: PDF

The significance of Non-negative Matrix Factorization ((NMF) in the field of automatic text summarization is rapidly increasing due to its interpretation and storage capabilities. Interpretation defines the ease at which the structure of high dimensional data can be understood. While, storage capability relates to the extent of data reduction process achieved by NMF. The parametric values that serve as input to the NMF process include initialization method, rank of factorization, sparseness measure and maximum iteration. These inputs are vital to the output produced by NMF. These parameters of NMF were not been considered in the existing literature on Automatic Text Summarization (ATS). This paper sheds light specifically on sparseness of NMF when applied to ATS and the impact it makes to the quality of the summary generated. In this study on NMF algorithm that supports sparseness is treated with various degree of sparsity and its performance impact on text summary generated is compared with other NMF algorithms without sparseness constraints.

Nonnegative matrix factorization (NMF) is a widely-used method for multivariate analysis of nonnegative data, the goal of which is decompose a data matrix into a basis matrix and an encoding variable matrix with all of these matrices allowed to have only nonnegative elements. In this paper we present simple algorithms for orthogonal NMF, where orthogonality constraints are imposed on basis matrix or encoding matrix. We develop multiplicative updates directly from the true gradient (natural gradient) in Stiefel manifold, whereas existing algorithms consider additive orthogonality constraints. Numerical experiments on face image data for a image representation task show that our orthogonal NMF algorithm preserves the orthogonality, while the goodness-of-fit (GOF) is minimized. We also apply our orthogonal NMF to a clustering task, showing that it works better than the original NMF, which is confirmed by experiments on several UCI repository data sets.

Non-negative matrix factorization (NMF) approximates a given matrix as a product of two non-negative matrix factors. Multiplicative algorithms deliver reliable results, but they show slow convergence for high-dimensional data and may be stuck away from local minima. Gradient descent methods have better behavior, but only apply to smooth losses. For non-smooth losses such as the Kullback-Leibler (KL) loss, surprisingly, these methods are lacking. In this paper, we propose a first-order primal-dual algorithm for non-negative decomposition problems (one of the two factors is fixed) with the KL distance. All required computations may be obtained in closed form and we provide an efficient heuristic way to select step-sizes. By using alternating optimization, our algorithm readily extends to NMF and, on synthetic or real world data, it is either faster than existing algorithms, or leads to improved local optima, or both.

Hyperspectral unmixing (HU) has become an important technique in exploiting hyperspectral data since it decomposes a mixed pixel into a collection of endmembers weighted by fractional abundances. The endmembers of a hyperspectral image (HSI) are more likely to be generated by independent sources and be mixed in a macroscopic degree before arriving at the sensor element of the imaging spectrometer as mixed spectra. Over the past few decades, many attempts have focused on imposing auxiliary regularizes on the conventional nonnegative matrix factorization (NMF) framework in order to effectively unmix these mixed spectra. As a promising step toward finding an optimum regularizer to extract endmembers, this paper presents a novel blind HU algorithm, referred to as Kurtosis-based Smooth Nonnegative Matrix Factorization (KbSNMF) which incorporates a novel regularizer based on the statistical independence of the probability density functions of endmember spectra. Imposing this regularizer on the conventional NMF framework promotes the extraction of independent endmembers while further enhancing the parts-based representation of data. Experiments conducted on diverse synthetic HSI datasets (with numerous numbers of endmembers, spectral bands, pixels, and noise levels) and three standard real HSI datasets demonstrate the validity of the proposed KbSNMF algorithm compared to several stateof- the-art NMF-based HU baselines. The proposed algorithm exhibits superior performance especially in terms of extracting endmember spectra from hyperspectral data; therefore, it could uplift the performance of recent deep learning HU methods which utilize the endmember spectra as supervisory input data for abundance extraction.

Nonnegative matrix factorization (NMF) is a popular method for multivariate analysis of nonnegative data, the goal of which is decompose a data matrix into a product of two factor matrices with all entries in factor matrices restricted to be nonnegative. NMF was shown to be useful in a task of clustering (especially document clustering). In this paper we present an algorithm for orthogonal nonnegative matrix factorization, where an orthogonality constraint is imposed on the nonnegative decomposition of a term-document matrix. We develop multiplicative updates directly from true gradient on Stiefel manifold, whereas existing algorithms consider additive orthogonality constraints. Experiments on several different document data sets show our orthogonal NMF algorithms perform better in a task of clustering, compared to the standard NMF and an existing orthogonal NMF.KeywordsNonnegative Matrix FactorizationDocument ClusterOrthogonality ConstraintMultiplicative UpdateProbabilistic Latent Semantic IndexingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

For the quantitation of cerebral blood flow (CBF) using H/sub 2//sup 15/O PET, the measurement of arterial input function (AIF) is essential. In this study, for the simplified quantitation, we present a method for the blind and noninvasive extraction of carotid input function (CIF) from dynamic PET images. On 8 healthy volunteers, the PET scans of C/sup 15/O and H/sub 2//sup 15/O were sequentially performed with arterial blood sampling using detector. With the inhalation of C/sup 15/O gas, dynamic PET data was acquired. And then after the injection of H/sub 2//sup 15/O, dynamic PET scans of H/sub 2//sup 15/O was perform. For 4 subjects, PET data for both rest and Acetazolamide (ACZ)-induced stress states were acquired, respectively. In the transverse dynamic images of C/sup 15/O and H/sub 2//sup 15/O, the regions of dynamic images including carotid artery were selected by masking. Non-negative matrix factorization (NMF) algorithm was used to extract the CIF from the selected dynamic images. The partial volume correction estimated from dynamic C/sup 15/O image, was applied to the extracted CIF. Whole brain CBF was estimated by kinetic analysis based on single compartment model. The error was analyzed using the area under the curve (AUC) and estimated CBF. NMF provided the good separation of the component of CIF from others in both images of C/sup 15/O and H/sub 2//sup 15/O. The shape of estimated CIF by NMF was similar with AIF measured by blood sampling. The AUC between the measured AIF and the estimated CIF was not so different at both rest and ACZ states. The difference of CBF before and after the injection of ACZ was also same between the measured AIF and the estimated CIF. These results show that NMF technique may be a tool for the noninvasive extraction of carotid input function, and this carotid input function might be used in the noninvasive quantitation of CBF using H/sub 2//sup 15/O PET.

Non-negative matrix factorization (NMF) is widely used in feature extraction and dimension reduction fields. Essentially, it is an optimization problem to determine two non-negative low rank matrices \(W_{m \times k}\) and \(H_{k \times n}\) for a given matrix \(A_{m \times n}\), satisfying \(A_{m \times n} \approx W_{m \times k}H_{k \times n}\). In this paper, a novel approach to improve the image decomposing and reconstruction effects by introducing the Singular Value Decomposing (SVD)-based initialization scheme of factor matrices W and H, and another measure called choosing rule to determine the optimum value of factor rank k, are proposed. The input image is first decomposed using SVD to get its singular values and corresponding eigenvectors. Then, the number of main components as the rank value k is extracted. Then, the singular values and corresponding eigenvectors are used to initialize W and H based on selected rank k. Finally, convergent results are obtained using multiplicative and additive update rules. However, iterative NMF algorithms’ convergence is very slow on most platforms limiting its practicality. To this end, a parallel implementation frame of described improved NMF algorithm using CUDA, a tool for algorithms parallelization on massively parallel processors, i.e., many-core graphics processors, is presented. Experimental results show that our approach can get better decomposing effect than traditional NMF implementations and dramatic accelerate rate comparing to serial schemes as well as existing distributed-system implementations.

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