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The class of multi-relational graph convolutional networks (MRGCNs) is a recent extension of standard graph convolutional networks (GCNs) to handle heterogenous graphs with multiple types of relationships. MRGCNs have been shown to yield results superior than traditional GCNs in various machine learning tasks. The key idea is to introduce a new kind of convolution operated on tensors that can effectively exploit correlations exhibited in multiple relationships. The main objective of this paper is to analyze the algorithmic stability and generalization guarantees of MRGCNs to confirm the usefulness of MRGCNs. Our contributions are of three folds. First, we develop a matrix representation of various tensor operations underneath MRGCNs to simplify the analysis significantly. Next, we prove the uniform stability of MRGCNs and deduce the convergence of the generalization gap to support the usefulness of MRGCNs. The analysis sheds lights on the design of MRGCNs, for instance, how the data should be scaled to achieve the uniform stability of the learning process. Finally, we provide experimental results to demonstrate the stability results. [ABSTRACT FROM AUTHOR]

• Making use of the nonnegativity and the support constraints as well as the dictionary learning approach. • Establishing a blind deconvolution model to determine the inverse filter and the deblurred image. • Using sparse and redundant representations of the objective deblurred image over learned dictionaries. • The objective image can be represented with more details by the dictionary basis. In this paper, we study a novel inverse filtering method by using a dictionary approach. The main idea is to combine a learned dictionary for the representation of the deconvoluted image and an inverse filter based on nonnegativity and support constraints, to deconvolute the observed image with an unknown point spread function. The advantage of this approach is that the target image can be represented with more details by learned basis in the dictionary. We also employ the alternating direction method of multipliers to solve the resulting optimization problem. Experimental results are presented to show that the performance of the proposed methods are better than other testing methods for several testing images. [ABSTRACT FROM AUTHOR]

In this paper, we study a discretization scheme and the corresponding fast solver for multi-dimensional time–space fractional diffusion equation with variable coefficients, in which L 1 formula and shifted Grünwald formula are employed to discretize the temporal and spatial derivatives, respectively. A divide-and-conquer strategy is applied to the large linear system assembling discrete equations of all time levels, which in turn requires to solve a series of multidimensional linear systems related to the spatial discretization. Preconditioned generalized minimal residual method is employed to solve the spatial linear systems resulting from the spatial discretization. The discretization is proven to be unconditionally stable and convergent in the sense of infinity norm for general nonnegative coefficients. Numerical results are reported to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Feature extraction is a prerequisite in many machine learning and data mining applications. As the advancement of data acquisition techniques, nowadays tensor objects are accumulated with respect to frequency or time information in a great number of fields. For instance color or hyperspectral faces in multichannel information, and human gait motion in time information are obtained. In this paper, we propose and develop a block principal component analysis (BPCA) to extract features for this kind of tensor objects. Our idea is to unfold tensor objects according to their spatial information and frequency/time information, and represent them in block matrix form. The corresponding covariance matrix for frequency/time information can be captured and used. The block eigen-decomposition of such covariance matrix is employed to seek for projection solution as features. Both reconstruction and classification problems can be solved via these projected features. Extensive experiments have been conducted on various face or gait databases to demonstrate the superiority of BPCA compared with existing methods such as PCA, (2D) 2 PCA, MPCA, and UMPCA in terms of effectiveness. Moreover, the proposed BPCA is competitively efficient compared to these existing methods. [ABSTRACT FROM AUTHOR]

Fractional partial differential equations (FPDEs) provide better modeling capabilities for challenging phenomena with long-range time memory and spatial interaction than integer-order PDEs do. A conventional numerical discretization of space–time FPDEs requires O ( N 2 + M N ) memory and O ( M N 3 + M 2 N ) computational work, where N is the number of spatial freedoms per time step and M is the number of time steps. We develop a fast finite difference method (FDM) for space–time FPDE: (i) We utilize the Toeplitz-like structure of the coefficient matrix to develop a matrix-free preconditioned fast Krylov subspace iterative solver to invert the coefficient matrix at each time step. (ii) We utilize a divide-and-conquer strategy, a recursive direct solver, to handle the temporal coupling of the numerical scheme. The fast method has an optimal memory requirement of O ( M N ) and an approximately linear computational complexity of O ( N M ( log N + log 2 M ) ) , without resorting to any lossy compression. Numerical experiments show the utility of the method. [ABSTRACT FROM AUTHOR]

Multi-instance multi-label (MIML) learning is one of challenging research problems in machine learning. In the literature, there are several methods for solving MIML problems. However, they may take a long computational time and have a huge storage cost for large MIML data sets. The main aim of this paper is to propose and develop an efficient Markov Chain learning algorithm for MIML problems, especially for data represented by non-negative features. Our idea is to perform labels classification iteratively through two Markov chains constructed by using objects and features respectively. The classification of objects can be obtained by using labels propagation via training data in the iterative method. Moreover, we demonstrate that the proposed method can be formulated by considering normalized linear kernel. Because linear kernel function is explicit and separable, it is not necessary to compute and store a huge affinity matrix among objects/instances compared with the use of other kernel functions. Therefore, both the storage and computational time of the proposed algorithm are very efficient. Experimental results are presented to show that the classification performance of the proposed method using normalized linear kernel function is about the same as those using the other kernel functions, while the required computational time is much less, which together suggest that the linear kernel can be good enough for MIML problem. Also experimental results on some benchmark data sets are reported to illustrate the effectiveness of the proposed method in one-error, ranking loss, coverage and average precision, and show that it is competitive with the other MIML methods. [ABSTRACT FROM AUTHOR]

Deep neural networks have achieved great success in solving many machine learning and computer vision problems. In this paper, we propose a deep neural network called the Tucker network derived from the Tucker format and analyze its expressive power. The results demonstrate that the Tucker network has exponentially higher expressive power than the shallow network. In other words, a shallow network with an exponential width is required to realize the same score function as that computed by the Tucker network. Moreover, we discuss the expressive power between the hierarchical Tucker tensor network (HT network) and the proposed Tucker network. To generalize the Tucker network into a deep version, we combine the hierarchical Tucker format and Tucker format to propose a deep Tucker tensor decomposition. Its corresponding deep Tucker network is presented. Experiments are conducted on three datasets: MNIST, CIFAR-10 and CIFAR-100. The results experimentally validate the theoretical results and show that the Tucker network and deep Tucker network have better performance than the shallow network and HT network. [ABSTRACT FROM AUTHOR]

Abstract: In a radar detecting system, the radar cross section (RCS) of a target is an important application in electromagnetics computation. Cavity prediction is usually required in the computation process. When wavelengths of electromagnetic field are large, cavity calculation is a challenging problem. The main aim of this paper is to employ the preconditioned GMRES method to solve linear systems arising in cavity calculation of the transverse magnetic and transverse electric problems. We develop new preconditioners based on fast transforms and Toeplitz solvers. Our numerical results show that the proposed preconditioners are quite efficient and effective, especially when the wavelength is large. [Copyright &y& Elsevier]

Objectives: To reduce the complications associated with placenta percreta (PP) by adequate preoperative planning with a multidisciplinary team. PP is a rare and potentially morbid condition of pregnancy, particularly if the urinary tract is involved. Cesarean delivery and hysterectomy are typically required to reduce the pelvic hemorrhage, placing the urinary tract at risk. Methods: We reviewed our urologic consultations and experience with PP in the past year. Maternal factors, surgical timing, blood loss, surgical complications, and outcomes were recorded. The timing of the urologic assessment was divided into preoperative and perioperative. Results: Five cases of PP were available. Of the 5 cases, 4 had been successfully diagnosed by prenatal ultrasonography, with 2 also requiring magnetic resonance imaging. All patients underwent cesarean delivery and hysterectomy, with significant blood loss (median 12 U transfused). A preoperative urologic assessment was done in 2 of the 5 patients, with no urinary complications found in this group. Both patients had undergone cystoscopy with placement of temporary ureteral catheters, even though the cases were emergent. In contrast, 3 patients underwent urologic consultation during or immediately after surgery. All 3 had bladder injuries, with 1 ureteral injury and delayed convalescence in this group of patients. Conclusions: PP remains a technically challenging and high-risk obstetric condition. In the setting of urinary tract involvement, adequate imaging, surgical planning, and preoperative urologic assessment with placement of temporary ureteral catheters were associated with a lower incidence of urologic complications in our series. Adequate preoperative planning with a multidisciplinary team is recommended to reduce the complications associated with PP. [Copyright &y& Elsevier]

In many sparsity-based image processing problems, compared with the convex ℓ 1 norm approximation of the nonconvex ℓ 0 quasi-norm, one can often preserve the structures better by taking full advantage of the nonconvex ℓ p quasi-norm (0 ≤ p < 1). In this paper, we propose a nonconvex ℓ p quasi-norm approximation in the total generalized variation (TGV)-shearlet regularization for image reconstruction. By introducing some auxiliary variables, the nonconvex nonsmooth objective function can be solved by an efficient alternating direction method of multipliers with convergence analysis. Especially, we use a generalized iterated shrinkage operator to deal with the ℓ p quasi-norm subproblem, which is easy to implement. Extensive experimental results show clearly that the proposed nonconvex sparsity approximation outperforms some state-of-the-art algorithms in both the visual and quantitative measures for different sampling ratios and noise levels. [ABSTRACT FROM AUTHOR]

Abstract: Markov chains are commonly used in modeling many practical systems such as queuing systems, manufacturing systems and inventory systems. They are also effective in modeling categorical data sequences. In a conventional th order multivariate Markov chain model of chains, and each chain has the same set of states, the total number of parameters required to set up the model is . Such huge number of states discourages researchers or practitioners from using them directly. In this paper, we propose an th-order multivariate Markov chain model for modeling multiple categorical data sequences such that the total number of parameters are of . The proposed model requires significantly less parameters than the conventional one. We develop efficient estimation methods for the model parameters. An application to demand predictions in inventory control is also discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we develop a semi-supervised regression algorithm to analyze data sets which contain both categorical and numerical attributes. This algorithm partitions the data sets into several clusters and at the same time fits a multivariate regression model to each cluster. This framework allows one to incorporate both multivariate regression models for numerical variables (supervised learning methods) and k-mode clustering algorithms for categorical variables (unsupervised learning methods). The estimates of regression models and k-mode parameters can be obtained simultaneously by minimizing a function which is the weighted sum of the least-square errors in the multivariate regression models and the dissimilarity measures among the categorical variables. Both synthetic and real data sets are presented to demonstrate the effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: We consider a class of inverse positive matrices, which is called Maximum inverse positive (MIP) matrices. If is an MIP matrix, then for any matrix which has at least one entry larger than that of , then is no longer an inverse positive matrix. Some existence and nonexistence results for MIP matrices are presented. [Copyright &y& Elsevier]

Abstract: We present new preconditioners based on matrix splittings for the saddle point problems. The spectral property of one of the preconditioned matrix is studied in detail. Numerical examples are used to illustrate the efficiency of the new preconditioners. [Copyright &y& Elsevier]

Abstract: In this paper, we are interested in a nonsymmetric algebraic Riccati equation arising in transport theory. The effects of a parameter α on the minimal positive solution X*(α) of this equation are studied. We show that X*(α) decreases in α only when α is close to one and X*(α) cannot attain its maximum for α ∈[0,1). A matrix lower bound and a matrix upper bound for X*(α) are also given. [Copyright &y& Elsevier]

Abstract: Preconditioned iterative methods are described for the solution of an elliptic partial differential equation over an unit square region with Robbins boundary conditions. Transform based preconditioners are constructed and analyzed. The motivation is to exploit the fast inversion of transform based systems via the fast transform. We prove that transform based preconditioners can be chosen so that the condition number of the preconditioned system can be reduced from O(n 2) to O(1). Numerical results are reported to illustrate the effectiveness of the preconditioners. [Copyright &y& Elsevier]

Abstract: For the large sparse implicit linear initial value problem, we present a block successive overrelaxation scheme for the alternating direction implicit waveform relaxation method to further accelerate its convergence speed, and discuss the convergence property of the resulting iteration method in detail. Numerical implementations about several non-Hermitian implicit linear initial value problems show that the alternating direction implicit waveform relaxation method is very effective, and the block successive overrelaxation technique really accelerates its convergence speed. [Copyright &y& Elsevier]

We consider an ill-posed deconvolution problem with a noise-contaminated observation, and a known convolution kernel. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting blurring matrices are block Toeplitz-plus-Hankel matrices with Toeplitz-plus-Hankel blocks. We study the application of the preconditioned iterative regularization scheme for solving these linear systems, where the blurring matrices are approximated by cosine transform preconditioners. We give a simple approach for finding these preconditioners and show how iterations can be effectively and efficiently regularized for solving ill-posed problems by using the spectral decomposition of the preconditioner. [Copyright &y& Elsevier]

This paper is concerned with the localization of the Perron root of a nonnegative irreducible matrix A. A new localization method that utilizes the relationship between the Perron root of a nonnegative matrix and the estimates of the row sums of its generalized Perron complement is presented. The method is efficient because it gives the bounds on ρ(A) only by computing the estimates of the row sums of the generalized Perron complement rather than the generalized Perron complement itself. Several numerical examples are given to illustrate the effectiveness of our method. [Copyright &y& Elsevier]

In this paper, we consider the stability of the algorithms emerging from Toeplitz matrix inversion formulas. We show that if the Toeplitz matrix is nonsingular and well-conditioned, then they are numerically forward stable. [Copyright &y& Elsevier]

Abstract: We modify the algorithm of [1], based on Newton''s iteration and on the concept of ε-displacement rank, to the computation of the Moore-Penrose inverse of a rank-deficient Toeplitz matrix. Numerical results are presented to illustrate the effectiveness of the method. [Copyright &y& Elsevier]

We study efficient iterative methods for Toeplitz systems based on the circulant and skew-circulant splitting (CSCS) of the Toeplitz matrix. Theoretical analysis show that if the circulant and the skew-circulant splitting matrices are positive definite, then the CSCS method converges to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the CSCS iteration which is dependent solely on the spectra of the circulant and the skew-circulant matrices involved. Numerical examples are presented to demonstrate the method. [Copyright &y& Elsevier]

Docosahexaenoic acid [22:6(n-3)] is required in large amounts for membrane lipid synthesis during brain growth. The functional importance of differences in dietary fatty acid intakes that alter brain 22:6(n-3), however, is not well understood. We used a dietary approach to manipulate 22:6(n-3) in piglet brain and assessed the effects on behavior and change in behavior on an elevated plus maze after administration of L-dihydroxyphenylalanine (L-Dopa) or sulperide, a dopamine D2 receptor blocker. Piglets were fed 1.2% energy 18:2(n-6) and 0.05% energy 18:3(n-3) (low PUFA), or 10.7% energy 18:2(n-6), 1.1% energy 18:3(n-3), 0.3% energy 20:4(n-6) and 0.3% energy 22:6(n-3) (high PUFA) from 1 d of age and behavior assessed at 18-22 d of age. At 30 d of age, frontal cortex dopamine, and phosphatidylcholine (PC), phosphatidylserine (PS), phosphatidyethanolamine (PE) and phosphatidylinositol (PI) fatty acids were quantified. Piglets fed the low PUFA diet had fewer arm entries on the maze than piglets fed the high PUFA diet, P = 0.02. L-Dopa increased the open (P = 0.005) and closed (P = 0.04) arm entries by piglets fed the low PUFA diet. Behavior did not differ between piglets fed the low and high PUFA diets when given L-Dopa. Frontal cortex PC, PS and PE 22:6(n-3) was lower and 22:5(n-6) was higher in piglets fed the low compared with the high PUFA diet, P < 0.01. Our work establishes the neonatal piglet as a model with which to study the behavioral effects of diet-induced changes in brain 22:6(n-3), and provides functional evidence that brain 22:6(n-3) is important in central dopamine metabolism. [ABSTRACT FROM AUTHOR]

The symmetric Sinc–Galerkin method applied to a sparable second-order self-adjoint elliptic boundary value problem gives rise to a system of linear equations(Ψx⊗Dy+Dx⊗Ψy)u=g,where⊗ is the Kronecker product symbol, Ψx and Ψy are Toeplitz-plus-diagonal matrices, and Dx and Dy are diagonal matrices. The main contribution of this paper is to present and analyze a two-step preconditioning strategy based on the banded matrix approximation (BMA) and the alternating direction implicit (ADI) iteration for these Sinc–Galerkin systems. In particular, we show that the two-step preconditioner is symmetric positive definite, and the condition number of the preconditioned matrix is bounded by the convergence factor of the involved ADI iteration. Numerical examples show that the new preconditioner is practical and efficient to precondition the conjugate gradient method for solving the above symmetric Sinc–Galerkin linear system. [Copyright &y& Elsevier]

Clustering methods partition a set of objects into clusters such that objects in the same cluster are more similar to each other than objects in different clusters according to some defined criteria. The fuzzy k-means-type algorithm is best suited for implementing this clustering operation because of its effectiveness in clustering data sets. However, working only on numeric values limits its use because data sets often contain categorical values. In this paper, we present a tabu search based clustering algorithm, to extend the k-means paradigm to categorical domains, and domains with both numeric and categorical values. Using tabu search based techniques, our algorithm can explore the solution space beyond local optimality in order to aim at finding a global solution of the fuzzy clustering problem. It is found that the clustering results produced by the proposed algorithm are very high in accuracy. [Copyright &y& Elsevier]

In 1992, Labahn and Shalom showed that the inverse of a Toeplitz matrix can be reconstructed by two of its columns and by some entries of the original Toeplitz matrix. We here present a modification of this result and another (shorter) proof. [Copyright &y& Elsevier]

We consider the system of equations arising from finite difference discretization of a three-dimensional convection–diffusion model problem. This system is typically nonsymmetric. The GMRES method with the Strang block-circulant preconditioner is proposed for solving this linear system. We show that our preconditioners are invertible and study the spectra of the preconditioned matrices. Numerical results are reported to illustrate the effectiveness of our methods. [Copyright &y& Elsevier]

This paper describes a new algorithm for computing Nonnegative Low Rank Matrix (NLRM) approximation for nonnegative matrices. Our approach is completely different from classical nonnegative matrix factorization (NMF) which has been studied for more than twenty five years. For a given nonnegative matrix, the usual NMF approach is to determine two nonnegative low rank matrices such that the distance between their product and the given nonnegative matrix is as small as possible. However, the proposed NLRM approach is to determine a nonnegative low rank matrix such that the distance between such matrix and the given nonnegative matrix is as small as possible. There are two advantages. (i) The minimized distance by the proposed NLRM method can be smaller than that by the NMF method, and it implies that the proposed NLRM method can obtain a better low rank matrix approximation. (ii) Our low rank matrix admits a matrix singular value decomposition automatically which provides a significant index based on singular values that can be used to identify important singular basis vectors, while this information cannot be obtained in the classical NMF. The proposed NLRM approximation algorithm was derived using the alternating projection on the low rank matrix manifold and the non-negativity property. Experimental results are presented to demonstrate the above mentioned advantages of the proposed NLRM method compared the NMF method. [ABSTRACT FROM AUTHOR]