Sparsity-constrained Nonnegative Matrix Factorization Research Articles

Graph-Regularized, Sparsity-Constrained Non-Negative Matrix Factorization with Earth Mover’s Distance Metric

Non-negative matrix factorization (NMF) is widely used as a powerful matrix factorization tool in data representation. However, the traditional NMF, measured by Euclidean distance or Kullback–Leibler distance, does not take into account the internal implied geometric information of the dataset and cannot measure the distance between samples as well as possible. To remedy the defects, in this paper, we propose the NMF method with Earth mover’s distance as a metric, for short GSNMF-EMD. It combines graph regularization and L1/2 smooth constraints. The GSNMF-EMD method takes into account the intrinsic implied geometric information of the dataset and can produce more sparse and stable local solutions. Experiments on two specific image datasets showed that the proposed method outperforms related state-of-the-art methods.

A Discrete-Time Neurodynamic Approach to Sparsity-Constrained Nonnegative Matrix Factorization.

Sparsity is a desirable property in many nonnegative matrix factorization (NMF) applications. Although some level of sparseness of NMF solutions can be achieved by using regularization, the resulting sparsity depends highly on the regularization parameter to be valued in an ad hoc way. In this letter we formulate sparse NMF as a mixed-integer optimization problem with sparsity as binary constraints. A discrete-time projection neural network is developed for solving the formulated problem. Sufficient conditions for its stability and convergence are analytically characterized by using Lyapunov's method. Experimental results on sparse feature extraction are discussed to substantiate the superiority of this approach to extracting highly sparse features.

Sparsity-Constrained NMF Algorithm Based on Evolution Strategy for Hyperspectral Unmixing

As a powerful and explainable blind separation tool, non-negative matrix factorization (NMF) is attracting increasing attention in Hyperspectral Unmixing(HU). By effectively utilizing the sparsity priori of data, sparsity-constrained NMF has become a representative method to improve the precision of unmixing. However, the optimization technique based on simple multiplicative update rules makes its unmixing results easy to fall into local minimum and lack of robustness. To solve these problems, this paper proposes a new hybrid algorithm for sparsity constrained NMF by intergrating evolutionary computing and multiplicative update rules (MURs). To find the superior solution in each iteration,the proposed algorithm effectively combines the MURs based on alternate optimization technique, the coefficient matrix selection strategy with sparsity measure, as well as the global optimization technique for basis matrix via the differential evolution algorithm .The effectiveness of the proposed method is demonstrated via the experimental results on real data and comparison with representative algorithms.

Quantitative Detection of Settled Dust Over Green Canopy Using Sparse Unmixing of Airborne Hyperspectral Data

The main task of environmental and geoscience applications is efficient and accurate quantitative classification of earth surfaces and spatial phenomena. In the past decade, there has been a significant interest in employing hyperspectral unmixing (HU) to retrieve accurate quantitative information latent in hyperspectral imagery data. Recently, the ground-truth and laboratory measured spectral signatures promoted by advanced algorithms are proposed as a new path toward solving the unmixing problem of hyperspectral imagery in semisupervised fashion. This paper suggests that the sensitivity of sparse unmixing techniques provides an ideal approach to extract and identify dust settled over/upon green vegetation canopy using hyperspectral airborne data. Among the available techniques, this study presents the results of seven selected algorithms: 1) non-negative matrix factorization (NMF); 2) L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> sparsity-constrained NMF (L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1_</sub> NMF); 3) L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> sparsity-constrained NMF (L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> _NMF); 4) graph regularized NMF (G_NMF); 5) structured sparse NMF (SS_NMF); 6) alternating least-square (ALS); and 7) Lin's projected gradient (LPG). The performance is evaluated on real hyperspectral imagery data via detailed experimental assessment. The results compared with performances of selected conventional unmixing techniques.

Hypergraph-Regularized Sparse NMF for Hyperspectral Unmixing

Hyperspectral image (HSI) unmixing has attracted increasing research interests in recent decades. The major difficulty of it lies in that the endmembers and the associated abundances need to be separated from highly mixed observation data with few a priori information. Recently, sparsity-constrained nonnegative matrix factorization (NMF) algorithms have been proved effective for hyperspectral unmixing (HU) since they can sufficiently utilize the sparsity property of HSIs. In order to improve the performance of NMF-based unmixing approaches, spectral and spatial constrains have been added into the unmixing model, but spectral–spatial joint structure is required to be more accurately estimated. To exploit the property that similar pixels within a small spatial neighborhood have higher possibility to share similar abundances, hypergraph structure is employed to capture the similarity relationship among the spatial nearby pixels. In the construction of a hypergraph, each pixel is taken as a vertex of the hypergraph, and each vertex with its $k$ nearest spatial neighboring pixels form a hyperedge. Using the hypergraph, the pixels with similar abundances can be accurately found, which enables the unmixing algorithm to obtain promising results. Experiments on synthetic data and real HSIs are conducted to investigate the performance of the proposed algorithm. The superiority of the proposed algorithm is demonstrated by comparing it with some state-of-the-art methods.

Adaptive ${L}_{\bf 1/2}$ Sparsity-Constrained NMF With Half-Thresholding Algorithm for Hyperspectral Unmixing

Hyperspectral unmixing is an important step for material classification and recognition. Recently, nonnegative matrix factorization (NMF) has been utilized to unmix the hyperspectal imagery due to the advantage that it needs no assumption for the presence of pure pixels and can determine the endmembers and abundances simultaneously. In order to improve its unmixing performance, sparsity-constrained NMF has been demonstrated to be an efficient approach. The very recent study on L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> regularization theory in compressive sensing (CS) and sparsity-constrained NMF show that the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> regularizer can yield stronger sparsity-promoting solutions than L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> or L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> regularizer. However, the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> regularization can result in a complex nonconvex optimization problem that is hard to solve efficiently. In this paper, we propose a fast and efficient adaptive half-thresholding algorithm for hyperspectral unmixing based on L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> sparsity-constrained NMF. In the proposed algorithm, iterative half-thresholding procedure that has been proved to be an efficient method for solving L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> regularization problem in CS is hybridized with the multiplicative update rule of standard NMF to deal with the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sub> sparsity-constrained NMF, which can give sparser and better unmixing results than the alternative algorithms. Furthermore, the data sparsity information can be incorporated into the algorithm to adaptively adjust the regularization parameter of the model to improve algorithm performance and usability. The effectiveness of proposed algorithm was demonstrated by comparing with the representative algorithms on synthetic and real hyperspectral data.

Development of a Real Time Sparse Non-Negative Matrix Factorization Module for Cochlear Implants by Using xPC Target

Cochlear implants (CIS) require efficient speech processing to maximize information transmission to the brain, especially in noise. A novel CI processing strategy was proposed in our previous studies, in which sparsity-constrained non-negative matrix factorization (NMF) was applied to the envelope matrix in order to improve the CI performance in noisy environments. It showed that the algorithm needs to be adaptive, rather than fixed, in order to adjust to acoustical conditions and individual characteristics. Here, we explore the benefit of a system that allows the user to adjust the signal processing in real time according to their individual listening needs and their individual hearing capabilities. In this system, which is based on MATLAB®, SIMULINK® and the xPC Target™ environment, the input/outupt (I/O) boards are interfaced between the SIMULINK blocks and the CI stimulation system, such that the output can be controlled successfully in the manner of a hardware-in-the-loop (HIL) simulation, hence offering a convenient way to implement a real time signal processing module that does not require any low level language. The sparsity constrained parameter of the algorithm was adapted online subjectively during an experiment with normal-hearing subjects and noise vocoded speech simulation. Results show that subjects chose different parameter values according to their own intelligibility preferences, indicating that adaptive real time algorithms are beneficial to fully explore subjective preferences. We conclude that the adaptive real time systems are beneficial for the experimental design, and such systems allow one to conduct psychophysical experiments with high ecological validity.

Manifold Regularized Sparse NMF for Hyperspectral Unmixing

Hyperspectral unmixing is one of the most important techniques in analyzing hyperspectral images, which decomposes a mixed pixel into a collection of constituent materials weighted by their proportions. Recently, many sparse nonnegative matrix factorization (NMF) algorithms have achieved advanced performance for hyperspectral unmixing because they overcome the difficulty of absence of pure pixels and sufficiently utilize the sparse characteristic of the data. However, most existing sparse NMF algorithms for hyperspectral unmixing only consider the Euclidean structure of the hyperspectral data space. In fact, hyperspectral data are more likely to lie on a low-dimensional submanifold embedded in the high-dimensional ambient space. Thus, it is necessary to consider the intrinsic manifold structure for hyperspectral unmixing. In order to exploit the latent manifold structure of the data during the decomposition, manifold regularization is incorporated into sparsity-constrained NMF for unmixing in this paper. Since the additional manifold regularization term can keep the close link between the original image and the material abundance maps, the proposed approach leads to a more desired unmixing performance. The experimental results on synthetic and real hyperspectral data both illustrate the superiority of the proposed method compared with other state-of-the-art approaches.

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 L1 regularizer. Unfortunately, the L1 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 L1/2 sparsity constraint, which we name L1/2 -NMF. The L1/2 regularizer not only induces sparsity but is also a better choice among Lq(0 <; q <; 1) regularizers. We propose an iterative estimation algorithm for L1/2-NMF, which provides sparser and more accurate results than those delivered using the L1 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.