Positive Semidefinite Matrix Factorization Based on Truncated Wirtinger Flow

Profile-Decomposing Output of Multi-Channel Odor Sensor Array

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 introduce a probabilistic extension of non-negative matrix factorization (NMF) by considering binary coded images as a probabilistic superposition of underlying continuous-valued elementary patterns. We provide an appropriate algorithm to solve the related optimization problem with non-negativity constraints which represents an extension of the well-known NMF-algorithm to binary-valued data sets. We demonstrate the performance of our method by applying it to the detection and characterization of hidden causes for failures during semi-conductor wafer processing. We decompose binary coded (pass/fail) wafer test data into underlying elementary failure patterns and study their influence on the performance of single wafers during testing.KeywordsNonnegative Matrix FactorizationNonnegative Matrix FactorizationGradient AscentSource PatternNonnegative Matrix Factorization AlgorithmThese 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.

Non-negative matrix factorization (NMF) has been one of the most popular methods for feature learning in the field of machine learning and computer vision. Most existing works directly apply NMF on high-dimensional image datasets for computing the effective representation of the raw images. However, in fact, the common essential information of a given class of images is hidden in their low rank parts. For obtaining an effective low-rank data representation, we in this paper propose a non-negative low-rank matrix factorization (NLMF) method for image clustering. For the purpose of improving its robustness for the data in a manifold structure, we further propose a graph regularized NLMF by incorporating the manifold structure information into our proposed objective function. Finally, we develop an efficient alternating iterative algorithm to learn the low-dimensional representation of low-rank parts of images for clustering. Alternatively, we also incorporate robust principal component analysis into our proposed scheme. Experimental results on four image datasets reveal that our proposed methods outperform four representative methods.

The Positive Matrix Factorisation (PMF) algorithm (Paatero and Tapper, 1994) has been the most widely used receptor model for a long time and has only recently been challenged with new methodologies. The novel Bayesian auto-correlated matrix factorisation method (BAMF, Rusanen et al. 2024) integrates an auto-correlation term emulating real-world pollutant sources time evolution has produced higher accuracy compared to PMF. However, both PMF and BAMF struggle to provide well-separated profiles manifested as mixed time series contributions. A sparsity-handling algorithm named horseshoe (HS) regularisation has beenapplied to BAMF in order to improve profile determination. The horseshoe application pushes some parameters to be close to zero and others to have large values (Piironen and Vehtari, 2017). The BAMF+HS method reduces the dimensionality of the problem by suppressing the non-significant species for each profile. The resulting profiles are expected to be less noisy and better representing the nature of the atmospheric pollution sources. Figure 1 shows the effect of BAMF+HS (in orange) compared to the regular BAMF (in blue) and the PMF (in green) on a toy dataset, consisting on an oversimplified dataset with very sparse profiles. The BAMF+HS results show contributions pushed to zero, making the profiles closer to the truth (in black) with respect to the less sparse results of BAMF and PMF. This same comparison has been carried out on realistic synthetic datasets to show the effectiveness of sparsity introduction into source apportionment.Figure 1. Comparison to truth of source apportionment profiles resulting from three different receptor models for a toy dataset.Acknowledgement: This work is supported by the European Union's Horizon Europe research and innovation programme under the Marie Skłodowska-Curie Postdoctoral Fellowship Programme, SMASH co-funded under the grant agreement No. 101081355. The SMASH project is co-funded by the Republic of Slovenia and the European Union from the European Regional Development Fund. K.R.D. acknowledges support by SNSF Ambizione grant PZPGP2_201992.ReferencesPiironen, J., & Vehtari, A. (2017). Sparsity information and regularization in the horseshoe and other shrinkage priors.Rusanen, A., Björklund, A., Manousakas, M., Jiang, J., Kulmala, M. T., Puolamäki, K., & Daellenbach, K. R. (2023). Atmospheric Measurement Techniques Discussions, 2023, 1-28.Paatero, P., & Tapper, U. (1994). Environmetrics, 5(2), 111-126.

This article uses the projected gradient method (PG) for a non-negative matrix factorization problem (NMF), where one or both matrix factors must have orthonormal columns or rows. We penalize the orthonormality constraints and apply the PG method via a block coordinate descent approach. This means that at a certain time one matrix factor is fixed and the other is updated by moving along the steepest descent direction computed from the penalized objective function and projecting onto the space of non-negative matrices. Our method is tested on two sets of synthetic data for various values of penalty parameters. The performance is compared to the well-known multiplicative update (MU) method from Ding (2006), and with a modified global convergent variant of the MU algorithm recently proposed by Mirzal (2014). We provide extensive numerical results coupled with appropriate visualizations, which demonstrate that our method is very competitive and usually outperforms the other two methods.

In this paper, we present a novel muscle synergy extraction method based on multivariate curve resolution–alternating least squares (MCR-ALS) to overcome the limitation of the nonnegative matrix factorization (NMF) method for extracting non-sparse muscle synergy, and we study its potential application for evaluating motor function of stroke survivors. Nonnegative matrix factorization (NMF) is the most widely used method for muscle synergy extraction. However, NMF is susceptible to components’ sparseness and usually provides inferior reliability, which significantly limits the promotion of muscle synergy. In this study, MCR-ALS was employed to extract muscle synergy from electromyography (EMG) data. Its performance was compared with two other matrix factorization algorithms, NMF and self-modeling mixture analysis (SMMA). Simulated data sets were utilized to explore the influences of the sparseness and noise on the extracted synergies. As a result, the synergies estimated by MCR-ALS were the most similar to true synergies as compared with SMMA and NMF. MCR-ALS was used to analyze the muscle synergy characteristics of upper limb movements performed by healthy (n = 11) and stroke (n = 5) subjects. The repeatability and intra-subject consistency were used to evaluate the performance of MCR-ALS. As a result, MCR-ALS provided much higher repeatability and intra-subject consistency as compared with NMF, which were important for the reliability of the motor function evaluation. The stroke subjects had lower intra-subject consistency and seemingly had more synergies as compared with the healthy subjects. Thus, MCR-ALS is a promising muscle synergy analysis method for motor function evaluation of stroke patients.

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.

Non-negative Matrix Factorization (NMF) has been frequently applied to image feature extraction and clustering. Especially in image clustering tasks, it can achieve the similar or better performance than most of the matrix factorization algorithms due to its parts-based representations in the brain. However, the features extracted by NMF are not sparse and localized enough and the error of factorization is not small enough. Semi-tensor product of matrices (STP) is a novel operation of matrix multiplication, it is a generalization of the conventional matrix product for allowing the dimensions of factor matrices to be unequal. STP can manage the data hierarchically and the inverse process of STP can separate the data hierarchically. Based on this character of STP, we propose the Non-Negative Matrix Semi-Tensor Factorization (NMSTF). In this algorithm, we use the inverse process of Semi-Tensor Product of matrices for non-negative matrix factorization. This algorithm effectively optimizes the above two problems in NMF. While achieving similar even better performance on image clustering tasks, the size of features extracted by STNMF is at least 50 % smaller than the ones’ extracted by NMF and the error of factorization reduces 30 % in average.

Normalized dimensionality reduction using nonnegative matrix factorization

Real world sounds often exhibit non-stationary spectral characteristics such as those produced by a harpsichord or a guitar. The classical Non-negative Matrix Factorization (NMF) needs a number of atoms to accurately decompose the spectrogram of such sounds. An extension of NMF is proposed hereafter which includes time-frequency activations based on ARMA modeling. This leads to an efficient single-atom decomposition for a single audio event. The new algorithm is tested on real audio data and shows promising results.

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

Nonnegative matrix factorization (NMF) is the problem of approximating an input nonnegative matrix, $V$, as the product of two smaller nonnegative matrices, $W$ and $H$. In this paper, we introduce a general framework to design multiplicative updates (MU) for NMF based on $\beta$-divergences ($\beta$-NMF) with disjoint equality constraints, and with penalty terms in the objective function. By disjoint, we mean that each variable appears in at most one equality constraint. Our MU satisfy the set of constraints after each update of the variables during the optimization process, while guaranteeing that the objective function decreases monotonically. We showcase this framework on three NMF models, and show that it competes favorably the state of the art: (1)~$\beta$-NMF with sum-to-one constraints on the columns of $H$, (2) minimum-volume $\beta$-NMF with sum-to-one constraints on the columns of $W$, and (3) sparse $\beta$-NMF with $\ell_2$-norm constraints on the columns of $W$.

Deep probability multi-view feature learning for data clustering

Analysis of two-mode network data using nonnegative matrix factorization