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In the second part of this two-part work, another accelerator method based on the tensor format is established for solving the Sylvester tensor equations. This acceleration approach, which is called SGCRO−BTF, is based on the idea of inner-outer iteration used in the generalized conjugate residual with inner orthogonalization (GCRO) method. In SGCRO−BTF, the Simpler GMRES method based on the tensor format (SGMRES−BTF) is applied to the inner iteration, and the generalized conjugate residual based on the tensor format (GCR−BTF) method is used in the outer iteration. Furthermore, SGCRO−BTF seeks an approximate solution over a tensor subspace spanned by the approximation error tensors produced during the previous outer iterations of SGCRO−BTF and a tensor Krylov subspace constructed by the inner iteration. In order to reduce the computational storage in the outer iteration, the truncated version of SGCRO−BTF is presented, in which only some of the last approximation error tensors are kept and will be then added to the tensor Krylov subspace in order to obtain a new search subspace. Finally, the proposed methods are tested on a set of experiments and compared to some conventional and state-of-the-art Krylov subspace methods based on the tensor format. Experimental results indicate that the truncated version of SGCRO−BTF is particularly effective for solving the Sylvester tensor equations.

In recent years, Krylov subspace methods based on the tensor format have demonstrated their superiority over classical ones for handling various kinds of tensor equations. In this paper, according to the efficient computational cost of the classical Simpler GMRES method (SGMRES), we adopt the tensor format of this method, which is called SGMRES−BTF, for solving the Sylvester tensor equations. This paper is divided into two parts describing the tensor format of two types of acceleration approaches to tackle some serious problems of the restarted methods GMRES−BTF and SGMRES−BTF, such as the “stalling” and “alternating phenomenon”. In the first part of this two-part work, the truncated Loose Simpler GMRES based on the tensor format (LSGMRES−BTF) is introduced, which is an acceleration approach that aims to construct an augmented tensor Krylov subspace by means of approximation error tensors. Moreover, a detailed study is carried out on the connection between the values of sequential and skip angles and the convergence behavior of both SGMRES−BTF and truncated LSGMRES−BTF. In the second part of this paper, an acceleration approach based on the idea of inner-outer iteration in the truncated version of the generalized conjugate residual with inner orthogonalization (GCRO) method is developed. In this method, SGMRES−BTF is applied in the inner iteration, and the generalized conjugate residual based on the tensor format (GCR−BTF) method is used in the outer iteration. Numerical experiments show that SGMRES−BTF achieves an appropriate performance compared with GMRES−BTF. In addition, the numerical results reveal the high potential of the presented accelerating strategies to deal with Sylvester tensor equations.

The problem of shifted linear systems is an important and challenging issue in a number of research applications. Krylov subspace methods are effective techniques for different kinds of this problem due to their advantages in large and sparse matrix problems. In this paper, two new block projection methods based on respectively block FOM and block GMRES are introduced for solving sequences of shifted linear systems. We first express the original problem explicitly by a sequence of Sylvester matrix equations whose coefficient matrices are obtained from the shifted linear systems. Then, we show the restarted shifted block FOM (rsh-BFOM) method and derive some of its properties. We also present a framework for the restarted shifted block GMRES (rsh-BGMRES) method. In this regard, we describe two variants of rsh-BGMRES, including (1) rsh-BGMRES with an unshifted base system that applies a fixed unshifted base system and (2) rsh-BGMRES with a variable shifted base system in which the base block system can change after restart. Furthermore, we consider the use of deflation techniques for improving the performance of the rsh-BFOM and rsh-BGMRES methods. Finally, some numerical experiments are conducted to demonstrate the effectiveness of the proposed methods.

Longitudinal dispersion coefficient (LDC) is known as the most remarkable environmental variables which plays a key role in evaluation of pollution profiles in water pipelines. Even though, there is a wide range of numerical models to estimate coefficient of longitudinal dispersion, these mathematical techniques may often come in quite few inaccuracies due to complex mechanism of convection-diffusion processes in pollutant transition in water pipelines. In this research work, to obtain more accurate prediction of LDC, general structure of group method of data handling (GMDH) is modified by means of extreme learning machine (ELM) conceptions. In fact, with getting inspiration from ELM, a novel GMDH method, called GMDH network based on using extreme learning machine (GMDH-ELM), is proposed in which weighting coefficients of quadratic polynomials applied in conventional GMDH are no longer required to be updated either using back propagation technique or other evolutionary algorithms through training stage. In fact, an intermediate parameter is employed to establish a relationship between the input and output in each neuron of the GMDH model. In this way, a well-known and reliable dataset (233 experimental data) related to LDC in water network pipelines, as output vector, is applied to conduct training and testing phases. Through datasets, the Re number, the average longitudinal flow velocity, the friction factor of pipeline and the diameter of pipe are considered as inputs of the proposed approach. The results of GMDH-ELM model indicate a highly satisfying level of precision in both training and testing phases. Furthermore, feed forward structure of GMDH model was improved by particle swarm optimization (PSO) and gravitational search algorithm (GSA) to predict LDC. Through a sound judgment, a comparison is drawn between the performance of GMDH-ELM and other developed GMDH models. Moreover, several empirical equations existing in literature have been applied for comparisons. Overall, results of GMDH-ELM have permissible superiority over the other soft computing tools and conventional predictive models.

One of the most critical challenges in managing complex diseases like COVID-19 is to establish an intelligent triage system that can optimize the clinical decision-making at the time of a global pandemic. The clinical presentation and patients’ characteristics are usually utilized to identify those patients who need more critical care. However, the clinical evidence shows an unmet need to determine more accurate and optimal clinical biomarkers to triage patients under a condition like the COVID-19 crisis. Here we have presented a machine learning approach to find a group of clinical indicators from the blood tests of a set of COVID-19 patients that are predictive of poor prognosis and morbidity. Our approach consists of two interconnected schemes: Feature Selection and Prognosis Classification. The former is based on different Matrix Factorization (MF)-based methods, and the latter is performed using Random Forest algorithm. Our model reveals that Arterial Blood Gas (ABG) O₂ Saturation and C-Reactive Protein (CRP) are the most important clinical biomarkers determining the poor prognosis in these patients. Our approach paves the path of building quantitative and optimized clinical management systems for COVID-19 and similar diseases.

Most of existing research works in the field of feature selection via matrix factorization techniques have been employed for unsupervised learning problems. This paper introduces a new framework for the supervised feature selection, called supervised feature selection by constituting a basis for the original space of features and matrix factorization (SFS-BMF). To this end, SFS-BMF is a guided search to find a basis for the original space of features that inherently contains linearly independent features and can be replaced with the original space. For finding the best subset of features regarding the class attribute, information gain is utilized for the process of constructing a basis. In fact, a basis for the original features is constructed according to the most informative features in terms of the information gain. Then, this basis is decomposed through a matrix factorization form in order to select a subset of features. Our proposed method guarantees the maximum relevancy of selected features to the output by using the information gain while simultaneously secures the minimum redundancy among them based on the linear independence property. Several experiments on high-dimensional microarray datasets are conducted for illustrating the efficiency of SFS-BMF. The experimental results show that the proposed SFS-BMF method outperforms some state-of-the-art feature selection methods with respect to classification performance and also according to the computational complexity.

The extraction of predictive features from the complex high-dimensional multi-omic data is necessary for decoding and overcoming the therapeutic responses in systems pharmacology. Developing computational methods to reduce high-dimensional space of features inin vitro, in vivoand clinical data is essential to discover the evolution and mechanisms of the drug responses and drug resistance. In this paper, we have utilized the Matrix Factorization (MF) as a modality for high dimensionality reduction in systems pharmacology. In this respect, we have proposed three novel feature selection methods using the mathematical conception of a basis for features. We have applied these techniques as well as three other matrix factorization methods to analyze eight different gene expression datasets to investigate and compare their performance for feature selection. Our results show that these methods are capable of reducing the feature spaces and find predictive features in terms of phenotype determination. The three proposed techniques outperform the other methods used and can extract a 2-gene signature predictive of a Tyrosine Kinase Inhibitor (TKI) treatment response in the Cancer Cell Line Encyclopedia (CCLE).Key PointsMatrix Factorization (MF) is a useful framework for high dimensionality reduction in systems pharmacology.Novel feature selection methods using the incorporation of the mathematical conception of a basis for features into MF increases the performance of feature selection process.Feature selection based on the basis-concept in MF can provide predictive gene signatures for therapeutic responses in systems pharmacology.

Gene expression data have become increasingly important in machine learning and computational biology over the past few years. In the field of gene expression analysis, several matrix factorization-based dimensionality reduction methods have been developed. However, such methods can still be improved in terms of efficiency and reliability. In this paper, an innovative approach to feature selection, called Dual Regularized Unsupervised Feature Selection Based on Matrix Factorization and Minimum Redundancy (DR-FS-MFMR), is introduced. The major focus of DR-FS-MFMR is to discard redundant features from the set of original features. In order to reach this target, the primary feature selection problem is defined in terms of two aspects: (1) the matrix factorization of data matrix in terms of the feature weight matrix and the representation matrix, and (2) the correlation information related to the selected features set. Then, the objective function is enriched by employing two data representation characteristics along with an inner product regularization criterion to perform both the redundancy minimization process and the sparsity task more precisely. To demonstrate the proficiency of the DR-FS-MFMR method, a large number of experimental studies are conducted on nine gene expression datasets. The obtained computational results indicate the efficiency and productivity of DR-FS-MFMR for the gene selection task.

One of the most critical challenges in managing complex diseases like COVID-19 is to establish an intelligent triage system that can optimize the clinical decision-making at the time of a global pandemic. The clinical presentation and patients’ characteristics are usually utilized to identify those patients who need more critical care. However, the clinical evidence shows an unmet need to determine more accurate and optimal clinical biomarkers to triage patients under a condition like the COVID-19 crisis. Here we have presented a machine learning approach to find a group of clinical indicators from the blood tests of a set of COVID-19 patients that are predictive of poor prognosis and morbidity. Our approach consists of two interconnected schemes: Feature Selection and Prognosis Classification. The former is based on different Matrix Factorization (MF)-based methods, and the latter is performed using Random Forest algorithm. Our model reveals that Arterial Blood Gas (ABG) O2Saturation and C-Reactive Protein (CRP) are the most important clinical biomarkers determining the poor prognosis in these patients. Our approach paves the path of building quantitative and optimized clinical management systems for COVID-19 and similar diseases.

In the first part of this paper, we investigate the use of Hessenberg-based methods for solving the Sylvester matrix equation A X + X B = C . To achieve this goal, the Sylvester form of the global generalized Hessenberg process is presented. Using this process, different methods based on a Petrov–Galerkin or on a minimal norm condition are derived. In the second part, we focus on the SGl-CMRH method which is based on the Sylvester form of the Hessenberg process with pivoting strategy combined with a minimal norm condition. In order to accelerate the SGl-CMRH method, a preconditioned framework of this method is also considered. It includes both fixed and flexible variants of the SGl-CMRH method. Moreover, the connection between the flexible preconditioned SGl-CMRH method and the fixed one is studied and some upper bounds for the residual norm are obtained. In particular, application of the obtained theoretical results is investigated for the special case of solving linear systems of equations with several right-hand sides. Finally, some numerical experiments are given in order to evaluate the effectiveness of the proposed methods.

Recently many neural network methods have been proposed for multi-label classification in the literature. One of these recent researches is the multi-layer extreme learning machines (ML-ELM) in which stack auto encoders have been used for tuning the weights. However, ML-ELM suffers from three primary drawbacks: First, input weights and biases are chosen randomly. Second, the pseudo-inverse solution for calculating output weights will cause to increase the reconstruction error. Third, memory and execution time of transformation matrices are proportional to the number of hidden layers. In this paper multi-layer kernel extreme learning machine, that uses a linear combination of base kernels in each layer, is proposed for multi-label classification. The proposed approach effectively addresses the above-mentioned drawbacks. Furthermore, multi-label classification data inherently have multi-modal aspects due to the variety of labels assigned to each instance. Applying a combination of different kernels is the added advantage of ML-CK-ELM that leads to implicitly assess the inherent multi-modal aspects of multi-label data; each kernel can be effectively used to cover one of the modals better than the other kernels. The empirical study indicates that ML-CK-ELM represents competitive performance against other state-of-the-art methods, and experimental results over multi-label datasets verify the feasibility of ML-CK-ELM.

In this research, group method of data handling (GMDH) as a one of the self-organized approaches is utilized to predict three-dimensional free span expansion rates around pipeline due to waves. The...

In this paper, three novel frameworks based on the widespread regression methods Ridge, LASSO and Elastic Net are established to perform the task of feature selection. The suggested frameworks, which benefit from the joint advantages of the dual-manifold learning and the hesitant fuzzy correlation (HFC), utilize the concept of the hesitant fuzzy correlation matrix (HFCM) of the features and samples. In order to compute the HFCM, a fuzzy clustering algorithm is applied to samples (or features), and a hesitant fuzzy set on each sample (or each feature) is obtained after projecting the cluster membership functions on different samples (or features). Then, two kinds of HFCMs are calculated for samples of each class and the whole of features, respectively. In specific, a feature manifold regularization term based on the HFCM among features is added to the objective function in order to preserve the similarity between the features and the feature weights. Furthermore, a sample manifold regularization term is also considered for the purpose of preserving the local correlation among the samples of each class. Eventually, a set of experiments are conducted on twenty-four datasets in order to validate the performance of each method. The results confirm that the proposed approaches are effective to select suitable features in terms of both the number of selected features and the classification accuracy.

Existence of debris structures inevitably ascends the rate of scour process around bridge piers and flow area not only lead into remarkable deviation of flow but also increase the velocity around b...

Multi-label learning has been received much attention due to its applicability in machine learning problems. In current years, quite few approaches based on either extreme learning machine (ELM) or radial basis function (RBF) neural network have been proposed with the aim of increasing the efficiency of the multi-label classification. Most existing multi-label learning algorithms focus on information about the feature space. In this paper, our major intention is to regularize the objective function of multi-label learning methods via Locally Linear Embedding (LLE). To achieve this goal, two neural network architectures namely Multi-Label RBF (ML-RBF) and Multi-Label Multi Layer ELM (ML-ELM) are utilized. Then, a regularized multi-label learning method via feature manifold learning (RMLFM) and a regularized multi-label learning method via dual-manifold learning (RMLDM) are established for training two network structures. RMLDM simultaneously exploits the geometry structure of both feature and data space. Furthermore, eight different configurations of applying training algorithms (i.e., RMLFM and RMLDM) to model architectures (i.e., ML-RBF and ML-ELM) are considered for conducting comparisons. The validity and effectiveness of these eight classifiers are indicated by a number of experimental studies on several multi-label datasets. Furthermore, the experiments indicate that the efficiency of the classification can be improved considerably against some cutting-the-edge multi-label techniques for the neural classifiers in which the dual-manifold learning is used as the training method.

Summary This paper deals with studying some of well-known iterative methods in their tensor forms to solve a Sylvester tensor equation. More precisely, the tensor form of the Arnoldi process and full orthogonalization method are derived by using a product between two tensors. Then tensor forms of the conjugate gradient and nested conjugate gradient algorithms are also presented. Rough estimation of the required number of operations for the tensor form of the Arnoldi process is obtained, which reveals the advantage of handling the algorithms based on tensor format over their classical forms in general. Some numerical experiments are examined, which confirm the feasibility and applicability of the proposed algorithms in practice. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a general framework using tensor Krylov projection techniques is proposed for solving high order Sylvester tensor equations. After describing the tensor format of the generalized Hessenberg process, we combine the obtained different processes with a Galerkin orthogonality condition or with a minimal norm condition in order to derive the Hess − BTF and CMRH − BTF methods which are based on the tensor format of the Hessenberg process. In addition, we also recover the FOM − BTF and GMRES − BTF which are known methods based on the tensor format of the Arnoldi process. To accelerate the convergence or prevent a possible stagnation of the different obtained methods, we incorporate a weighting strategy based on the use of a weighted inner product instead of the classical one when building a basis for the tensor Krylov subspace. Numerical experiments are described in order to compare the new proposed methods that are Hess − BTF and CMRH − BTF with the known methods FOM − BTF and GMRES − BTF and to show the efficiency of the weighting strategy. Moreover, we utilize a flexible preconditioning framework for the unweighted and weighted forms of the proposed methods, and the flexible version is validated by satisfactory numerical results.

In this paper, the Ridge, LASSO and Elastic Net regression methods are adapted for the task of selecting feature. In order to enhance the feature selection performance via these methods, a Hesitant Fuzzy Correlation Matrix (HFCM) is added to the objective functions of these models for addressing the minimum redundancy of features. To this end, the fuzzy C-means clustering is utilized, and the obtained fuzzy clusters are projected on the features in a way that the number of fuzzy Membership Functions (MF) for each feature is equal to the number of clusters. Then, the projected MFs on each feature are considered as a Hesitant Fuzzy Set (HFS), and thereby the hesitant fuzzy correlation between features is calculated. Afterward, the obtained HFCM is employed in the regression methods for securing the minimum redundancy of features. Eventually, the accuracies of the selected features, achieved by these methods, are determined by three different classification models such as Naive Bayes, SVM and Decision Tree. A large number of experiments are conducted over twenty-four classification datasets to demonstrate the efficiency and applicability of using HFCM in some classical regression methods.