Blind Matrix Decomposition Via Genetic Optimization of Sparseness and Nonnegativity Constraints

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.

Spiny Lobster Sound Identification Based on Blind Source Separation (BSS) for Passive Acoustic Monitoring (PAM)

To solve the separation of multi-source signals and detect their features from a single channel, a signal separation method using multi-constraint non-negative matrix factorization (NMF) is proposed. In view of the existing NMF algorithm not performing well in the underdetermined blind source separation, the β-divergence constraints and determinant constraints are introduced in the NMF algorithm, which can enhance local feature information and reduce redundant components by constraining the objective function. In addition, the Sine-bell window function is selected as the processing method for short-time Fourier transform (STFT), and it can preserve the overall feature distribution of the original signal. The original vibration signal is first transformed into time-frequency domain with the STFT, which describes the local characteristic of the signal from the time-frequency distribution. Then, the multi-constraint NMF is applied to reduce the dimensionality of the data and separate feature components in the low dimensional space. Meanwhile, the parameter WK is constructed to filter the reconstructed signal that recombined with the feature component in the time domain. Ultimately, the separated signals will be subjected to envelope spectrum analysis to detect fault features. The simulated and experimental results indicate the effectiveness of the proposed approach, which can realize the separation of multi-source signals and their fault diagnosis of bearings. In addition, it is also confirmed that the proposed method, juxtaposed with the NMF algorithm of the traditional objective function, is more applicable for compound fault diagnosis of the rotating machinery.

Underdetermined blind separation of nonnegative dependent sources consists in decomposing a set of observed mixed signals into greater number of original nonnegative and dependent component (source) signals. That is an important problem for which very few algorithms exist. It is also practically relevant for contemporary metabolic profiling of biological samples, such as biomarker identification studies, where sources (a.k.a. pure components or analytes) are aimed to be extracted from mass spectra of complex multicomponent mixtures. This paper presents a method for underdetermined blind separation of nonnegative dependent sources. The method performs nonlinear mixture‐wise mapping of observed data in high‐dimensional reproducible kernel Hilbert space (RKHS) of functions and sparseness‐constrained nonnegative matrix factorization (NMF) therein. Thus, the original problem is converted into new one with increased number of mixtures, increased number of dependent sources, and higher‐order (error) terms generated by nonlinear mapping. Provided that amplitudes of original components are sparsely distributed, which is the case for mass spectra of analytes, sparseness‐constrained NMF in RKHS yields, with significant probability, improved accuracy relative to the case when the same NMF algorithm is performed on the original problem. The method is exemplified on numerical and experimental examples related respectively to extraction of 10 dependent components from five mixtures and to extraction of 10 dependent analytes from mass spectra of two to five mixtures. Thereby, analytes mimic complexity of components expected to be found in biological samples. Copyright © 2013 John Wiley & Sons, Ltd.

Nonnegative matrix factorization with quadratic programming

Nonnegative matrix factorization (NMF) is known as a parts-based linear representation for nonnegative data. This method has been applied for blind source separation (BSS) when the sources are nonnegative. This paper presents a new NMF method for independent component analysis (ICA), which is useful for BSS without the nonnegativity constraint. Using this method, we transform the sources by their cumulative distribution functions (CDFs) and perform the nonparametric quantization to construct a nonnegative matrix where each entry represents the joint probability density of two transformed signals. The NMF procedure is accordingly realized to find the ICA demixing matrix. The independence between sources is maximized towards attaining the uniformity in the joint probability density. In the experiments on the separation of signal and music signals, we show the effectiveness of the proposed NMF-ICA compared to the infomax ICA and FastICA algorithms.

The growing availability of hyperspectral data from planetary missions ranging from CubeSats to flagship programs has led to many studies aiming to infer surface mineralogy through remote sensing techniques. The Moon Mineralogical Mapper (M³) aboard Chandrayaan-1 provides visible and near-infrared data (430–3000nm) at 140 m/pixel resolution and has been widely used to infer lunar surface composition [1]. A common strategy involves applying unmixing models, often Non-Negative Matrix Factorization (NNMF), to M³ images with varying reflectance and compositional properties, using laboratory reference spectra from the RELAB database for comparison [2,3]. These spectra from standard materials represent typical lunar minerals such as olivine, pyroxenes, plagioclase, iron-titanium oxides, and volcanic glasses.This study first focuses on reproducing previous efforts to estimate the lunar mineralogy, applied to the Apollo 17 landing site (Figure 1). Then it shifts toward highlighting the methodological uncertainties involved in interpreting hyperspectral imagery using unmixing models in remote sensing applications.Figure. 1 (Top)Taurus-Littrow Valley. (Bottom) Mean (dark) reflectance spectrum and standard deviation (red) of the Taurus-Littrow Valley hyperspectral image.In the initial stage, a spatial binning with 8X8 window was applied to reduce the signal-to-noise ratio of the image. Principal Component Analysis (PCA) was then used to reduce data dimensionality and highlight surface variations. Initially applied on the binned image, PCA revealed that the first component explained 98% of the variance, mainly reflecting topography and illumination differences. To minimize these effects and enhance chemical information, the first derivative of the spectral data was applied (same preprocessing was applied to reference spectra, Figure.2). A second PCA was then performed to determine the chemical rank of the matrix, which suggested that NNMF should be carried out with 3–4 principal components.Figure. 2 ​First derivative of reference spectra from RELAB.NNMF was performed to unmix and extract surface components, possibly revealing features that may correspond to geochemical variations. In this final stage, it was assumed that three principal components could be present within the image. Figure. 3 shows the spatial distribution of the combination of these three components (top), along with the corresponding “pure” spectra (bottom) extracted by the model after applying NNMF using endmember initialization and non-negativity constraints [4] applied to the contributions. Commonly, when these results were compared with preprocessed spectra from RELAB, compounds such as pyroxenes, olivine, volcanic-glass, may be identified (Figure. 3), with acceptable lack of fit and a total variance explanation exceeding 99%.Figure. 3 False RGB showing the concentrations of the three main components after applying NNMF model using endmember initialization and non-negativity constraints.However, while spectral unmixing models have been widely applied in remote sensing for decades, limited attention has been paid to the intrinsic ambiguities they involve. Although simpler ambiguities (such as those related to scale or sign) can often be addressed using basic constraints like non-negativity, rotational ambiguity remains a significant and underexplored source of uncertainty. For these reasons, it is important to highlight that solutions derived from NNMF (commonly referred to as Multivariate Curve Resolution with non-negativity constraints) depend considerably on the degrees of freedom allowed by the optimization algorithm (e.g., Alternating Least Squares, ALS). To visualize and quantify this issue, feasibility plots were employed through MCRbands and FACpack (Figure. 4), which map the set of all admissible solutions within the NNMF framework [5-7].Figure. 4 Feasibility plots for the three components calculated. Right, the contributions and left, the spectral profiles obtained. Contributions: Blue and red represents the solution obtained and the plausible solution that could also be considered as optimal.Figure 4 shows that any space covered between those solutions could be considered feasible. On the left, the spectra calculated and the feasible spectra both indicate a strong lack of specificity and uniqueness in the solutions, clearly stating that the model is not stable enough to be deemed an optimal solution. The extent of solution variability reveals that differences between the acceptable solution and real solutions for the considered species are considerable due to rotational freedom. In conclusion, we emphasize the importance of a more rigorous consideration of ambiguity in remote sensing applications. Additionally, the presence of regolith and volcanic rocks on the lunar surface makes it unlikely that a 140 m/pixel represents a pure mineral spectrum [8].Keywords: Remote sensing, Chemometrics, Unmixing models, Near Infrared. Acknowledgements: Work supported through the PAMMAT project: “Alteration processes in Mars and Moon Meteorites, and Terrestrial Analogues at different environments: Mars2020, Rosalind Franklin and Returned Samples from Mars and Moon” (Grant No. PID2022-142750OB-I00), funded by the Spanish Agency for Research (through the Spanish Ministry of Science and Innovation, MCIN, and the European Regional Development Fund, FEDER).

In this paper we propose a new method for removing specular reflection from multispectral dermatological images which is based on blind source separation using Non-negative Matrix Factorization. The key idea of our method is based on a first step which consists to estimate the number of sources involved, by applying a Principal Component Analysis, then a first version of these sources by applying an Independent Component Analysis to our images. We exploit this first version of the sources in a second step to initialize our Non-negative Matrix Factorization algorithm instead of the random initialization that is used by most of the existing methods and that considerably affects their performances. In order to quantify numerically the performance of our method, we also propose a new protocol to artificially mix a specular reflection image with a diffuse reflection image. The tests effected on real and artificial multispectral dermatological images have shown the good performance of our method compared to two of the most used methods.KeywordsMultispectral dermatological imagesRGB imagesSpecular reflectionDiffuse reflectionBlind Source Separation (BSS)Principal Component Analysis (PCA)Independent Component Analysis (ICA)Non-negative Matrix Factorization (NMF)

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.

The problem of blind source separation (BSS) has been intensively studied by the signal processing community. The first solutions to deal with BSS were proposed in the 1980's and are founded on the concept of independent component analysis (ICA). More recently, aiming at tackling some limitations of ICA-based methods, much attention has been paid to alternative BSS approaches. In this tutorial, in addition to providing a brief review of the classical BSS framework, we present two research trends in this area, namely source separation over Galois fields and sparse component analysis. For both subjects, we provide an overview of the main criteria, highlighting scenarios that can benefit from these more recent BSS paradigms.

This paper addresses the determined convolutive blind source separation (BSS) problem. The state-of-the-art independent low-rank matrix analysis (ILRMA), unifying independent component analysis (ICA) and nonnegative matrix factorization, has the disadvantage of ignoring inter-frame and inter-frequency spectral correlation of source signals. We here propose a new BSS method that estimates a linear transformation for spectral decorrelation and performs ILRMA in the transformed domain. A newly introduced optimization problem is an extension of that for ICA based on maximum likelihood. For this problem, we provide a necessary and sufficient condition for the existence of optimal solutions, and develop algorithms based on block coordinate descent methods with closed-form solutions. Experimental results show the improved separation performance of the proposed method compared to ILRMA.

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.

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.

This paper deals with the blind source separation (BSS) problem with fewer sensors than sources. We propose a simple procedure to transform the problem with N sources and N–1 observations in a classical BSS problem with N observations which can be solved using many well-known algorithms. We will also show how to apply this idea to digital communications for separating BPSK and QPSK signals.

Characterizing sediment sources by non-negative matrix factorization of detrital geochronological data