Smooth nonnegative matrix and tensor factorizations for robust multi-way data analysis

In order to achieve best neutron flux-mapping within nuclear research reactors, we investigate in this paper, the application of blind source separation algorithms to extract different independent components that form the fission chamber output signals. More specifically, we simulated the fission chamber using python-based of Fission Chambers (pyFC) code suite. The resulting simulated signals are considered observation vectors. They are then, processed via nonnegative matrix and tensor factorization (NMF&NTF) methods to isolate the independent components that form a neutron signal. Thus, in order to select the most efficient one to analyze our set of data, various algorithms have been tested and classified according to their performance index of separability (PI). The extracted independent components have been used to characterize neutron signal through spectrogram time-frequency representation. Obtained results will be illustrated and discussed.

We have studied the feature extraction from sea ice SAR images based on non-negative factorization methods. The methods reported here are the sparseness-constrained non-negative matrix factorization (SC-NMF) and Non-negative tensor factorization (NTF). The studies performed show that these methods can be used to extract meaningful features from SAR images and that they can be used in sea ice SAR classification.

This paper proposes parallel method of non-negative large sparse matrix factorization - very popular technique in computational linguistics. Proposed blocks factorization approach using Latent Dirichlet allocation, for speedup of non-negative sparse extra-large linguistic matrix and tensor factorization. The proposed method also supports the scalability of the model.

Non-negative Tensor Factorization (NTF) is a widely used technique for decomposing a non-negative value tensor into sparse and reasonably interpretable factors. However, NTF performs poorly when the tensor is extremely sparse, which is often the case with real-world data and higher-order tensors. In this paper, we propose Non-negative Multiple Tensor Factorization (NMTF), which factorizes the target tensor and auxiliary tensors simultaneously. Auxiliary data tensors compensate for the sparseness of the target data tensor. The factors of the auxiliary tensors also allow us to examine the target data from several different aspects. We experimentally confirm that NMTF performs better than NTF in terms of reconstructing the given data. Furthermore, we demonstrate that the proposed NMTF can successfully extract spatio-temporal patterns of people's daily life such as leisure, drinking, and shopping activity by analyzing several tensors extracted from online review data sets.

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.

Although nonnegative matrix factorization (NMF) favors a part-based and sparse representation of its input, there is no guarantee for this behavior. Several extensions to NMF have been proposed in order to introduce sparseness via the ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> -norm, while little work is done using the more natural sparseness measure, the ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> -pseudo-norm. In this work we propose two NMF algorithms with ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> -sparseness constraints on the bases and the coefficient matrices, respectively. We show that classic NMF is a suited tool for ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> -sparse NMF algorithms, due to a property we call sparseness maintenance. We apply our algorithms to synthetic and real-world data and compare our results to sparse NMF and nonnegative K-SVD.

The main aim of this paper is to develop a new algorithm for computing a nonnegative low multi‐rank tensor approximation for a nonnegative tensor. In the literature, there are several nonnegative tensor factorizations or decompositions, and their approaches are to enforce the nonnegativity constraints in the factors of tensor factorizations or decompositions. In this paper, we study nonnegativity constraints in tensor entries directly, and a low rank approximation for the transformed tensor by using discrete Fourier transformation matrix, discrete cosine transformation matrix, or unitary transformation matrix. This strategy is particularly useful in imaging science as nonnegative pixels appear in tensor entries and a low rank structure can be obtained in the transformation domain. We propose an alternating projections algorithm for computing such a nonnegative low multi‐rank tensor approximation. The convergence of the proposed projection method is established. Numerical examples for multidimensional images are presented to demonstrate that the performance of the proposed method is better than that of nonnegative low Tucker rank tensor approximation and the other nonnegative tensor factorizations and decompositions.

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.

Matrix factorization is a general technique that can extract latent factors from data. Recent studies applied matrix factorization to the problem of establishing which skills are required by question items, and for assessing student skills mastery from student performance data. A number of generic algorithms, such as Non-negative Matrix Factorization and Tensor factorization, are used in these studies to perform the factorization, but few have looked at optimizing these algorithms to the specific characteristics of student performance data. In this thesis, we explore how one such characteristic can lead to better factorization: the fact that items are learnt in a constrained order and allow such inferences as if a difficult item is succeeded, an easier one should also be succeeded. In particular, we want to address this question: can a partial order knowledge structure (POKS) be used to guide matrix factorization algorithms and lead to faster or better solutions to latent skills modelling?

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.

Many spectral unmixing approaches ranging from geometry, algebra to statistics have been proposed, in which nonnegative matrix factorization (NMF)-based ones form an important family. The original NMF-based unmixing algorithm loses the spectral and spatial information between mixed pixels when stacking the spectral responses of the pixels into an observed matrix. Therefore, various constrained NMF methods are developed to impose spectral structure, spatial structure, and spectral-spatial joint structure into NMF to enforce the estimated endmembers and abundances preserve these structures. Compared with matrix format, the third-order tensor is more natural to represent a hyperspectral data cube as a whole, by which the intrinsic structure of hyperspectral imagery can be losslessly retained. Extended from NMF-based methods, a matrix-vector nonnegative tensor factorization (NTF) model is proposed in this paper for spectral unmixing. Different from widely used tensor factorization models, such as canonical polyadic decomposition CPD) and Tucker decomposition, the proposed method is derived from block term decomposition, which is a combination of CPD and Tucker decomposition. This leads to a more flexible frame to model various application-dependent problems. The matrix-vector NTF decomposes a third-order tensor into the sum of several component tensors, with each component tensor being the outer product of a vector (endmember) and a matrix (corresponding abundances). From a formal perspective, this tensor decomposition is consistent with linear spectral mixture model. From an informative perspective, the structures within spatial domain, within spectral domain, and cross spectral-spatial domain are retreated interdependently. Experiments demonstrate that the proposed method has outperformed several state-of-the-art NMF-based unmixing methods.

Nonnegative Matrix Factorization (NMF) is a widely used technique in many applications such as face recognition, motion segmentation, etc. It approximates the nonnegative data in an original high dimensional space with a linear representation in a low dimensional space by using the product of two nonnegative matrices. In many applications data are often partially corrupted with large additive noise. When the positions of noise are known, some existing variants of N-MF can be applied by treating these corrupted entries as missing values. However, the positions are often unknown in many real world applications, which prevents the usage of traditional NMF or other existing variants of NMF. This paper proposes a Robust Nonnegative Matrix Factorization (RobustNMF) algorithm that explicitly models the partial corruption as large additive noise without requiring the information of positions of noise. In particular, the proposed method jointly approximates the clean data matrix with the product of two nonnegative matrices and estimates the positions and values of outliers/noise. An efficient iterative optimization algorithm with a solid theoretical justification has been proposed to learn the desired matrix factorization. Experimental results demonstrate the advantages of the proposed algorithm.

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

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

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.