Spectral Signal Unmixing with Interior-Point Nonnegative Matrix Factorization

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.

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.

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.

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

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) approximates a given data matrix as a product of two low rank nonnegative matrices, usually by minimizing the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> or the KL distance between the data matrix and the matrix product. This factorization was shown to be useful for several important computer vision applications. We propose here a new NMF algorithm that minimizes the Earth Mover's Distance (EMD) error between the data and the matrix product. We propose an iterative NMF algorithm (EMD NMF) and prove its convergence. The algorithm is based on linear programming. We discuss the numerical difficulties of the EMD NMF and propose an efficient approximation. Naturally, the matrices obtained with EMD NMF are different from those obtained with L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> NMF. We discuss these differences in the context of two challenging computer vision tasks - texture classification and face recognition - and demonstrate the advantages of the proposed method.

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).

Non-negative matrix factorization (NMF) is an unsupervised learning method that can be exploited for parts-based image representation due to non-negativity constraints. However, singer-layer NMF cannot capture the latent hierarchical structure features from images, while deep features play more important roles in the image representation and recognition tasks. To overcome the limitation of NMF, this paper proposes a novel deep grouped NMF (DGNMF) approach to learn different level attributes of the data. It is interesting that DGNMF approach automatically makes the data from distinct classes share different basis images and the feature vectors among different classes are mutually orthogonal at the same layer. Meanwhile, to preserve the local information, our DGNMF model establishes the objective function with graph regularization, and its optimization problem is solved using gradient descent method. The developed DGNMF algorithm is proved to be convergent and is finally evaluated on face datasets for classification. Compared with some state-of-the-art deep NMF variants, the results demonstrate the proposed DGNMF algorithm achieves surpassing performances using different layer features.

Non-negative Matrix Factorisation (NMF) is a commonly used tool in many musical signal processing tasks, including Automatic Music Transcription (AMT). However unsupervised NMF is seen to be problematic in this context, and harmonically constrained variants of NMF have been proposed. While useful, the harmonic constraints may be constrictive in mixed signals. We have previously observed that recovery of overlapping signal elements using NMF is improved through introduction of a sparse coding step, and propose here the incorporation of a sparse coding step using the Hellinger distance into a NMF algorithm. Improved AMT results for unsupervised NMF are reported.

Non-negative matrix factorization (NMF) has found numerous applications, due to its ability to provide interpretable decompositions. Perhaps surprisingly, existing results regarding its uniqueness properties are rather limited, and there is much room for improvement in terms of algorithms as well. Uniqueness and computational aspects of NMF are revisited here from a geometrical point of view. Both symmetric and asymmetric NMF are considered, the former being tantamount to element-wise non-negative square-root factorization of positive semidefinite matrices. New and insightful uniqueness results are derived, e.g., it is shown that a sufficient condition for uniqueness is that the conic hull of the latent factors is a superset of a particular second-order cone. Checking this is shown to be NP-complete; yet it offers insights on latent sparsity, as is also shown in a new necessary condition, to a smaller extent. On the computational side, a new efficient algorithm for symmetric NMF is proposed which uses Procrustes rotations. Simulation results show promising performance with respect to the state-of-art. The new algorithm is also applied to a clustering problem for co-authorship data, yielding meaningful and interpretable results.

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.

ABSTRACTHyperspectral unmixing (HU) has drawn remarkable attention because it can decompose mixed pixels into a set of endmembers and abundance fractions. And the nonnegative matrix factorization (NMF) algorithm has been widely used for solving hyperspectral spectral unmixing problem. This letter proposes a Hessian graph regularized NMF (HGNMF) algorithm, which relies upon the construction of a Hessian graph representation, to solve the hyperspectral unmixing problem. According to the HGNMF algorithm, the smoothness in the estimated abundance maps is well promoted. Moreover, the optimized problem of HGNMF algorithm is also solved by employing the multiplicative updating rules. Compared to other algorithms, the proposed HGNMF algorithm demonstrates lots of advantages based on the simulated results of the synthetic data and real data sets.

Identifying glass defects through machine visual has a vital importance for the efficient production of high quality glass. In this paper, a method based on the combination of Non-negative Matrix Factorization (NMF) and Sparse Representation Classification (SRC) was proposed for the identification of glass defects. According to the properties of glass defect image, NMF algorithm is used to decompose a defect image into one base image and another weighted coefficient matrix, so the defect image is characterized by the coefficient matrix. Then, SRC algorithm is used to classify glass defects. Simulation results show that, with NMF and SRC, glass defects images could be effectively identified.

Study subnetwork developing pattern of autism children by non-negative matrix factorization

Non-negative matrix factorization (NMF) provides a lower rank approximation of a matrix. Due to nonnegativity imposed on the factors, it gives a latent structure that is often more physically meaningful than other lower rank approximations such as singular value decomposition (SVD). Most of the algorithms proposed in literature for NMF have been based on minimizing the Frobenius norm. This is partly due to the fact that the minimization problem based on the Frobenius norm provides much more flexibility in algebraic manipulation than other divergences. In this paper we propose a fast NMF algorithm that is applicable to general Bregman divergences. Through Taylor series expansion of the Bregman divergences, we reveal a relationship between Bregman divergences and Euclidean distance. This key relationship provides a new direction for NMF algorithms with general Bregman divergences when combined with the scalar block coordinate descent method. The proposed algorithm generalizes several recently proposed methods for computation of NMF with Bregman divergences and is computationally faster than existing alternatives. We demonstrate the effectiveness of our approach with experiments conducted on artificial as well as real world data.