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• A fast weighted CP decomposition model for context-aware recommendation. • Unifying explicit and implicit feedback by assigning non-uniform weights. • Reducing computation time by element-wise alternating least squares. • A simple and effective multi-threaded computing scheme for above model. As an effective method for high-dimensional information processing, tensor factorization has been widely used in context-aware recommendation. However, the multi-dimensional processing of tensor factorization usually leads to more serious data sparsity. Making full use of the data in recommendation systems, including a small amount of explicit rating data and a large amount of implicit feedback data such as clicks, plays, and purchases, can effectively alleviate this problem. Meanwhile, the processing of sparse explicit data and huge implicit feedback data makes tensor factorization face a severe challenge of computational efficiency. Therefore, this study proposes a fast tensor factorization model named FWCP that unifies explicit and implicit feedback information, which mainly studies as follows: (1) Assigning non-uniform weights to observed and unobserved values to unify explicit and implicit feedback information; (2) Memorizing intermediate data which generated by element-wise alternating least squares(ALS), and implementing parallel optimization of FWCP, which can reduce the training time of tensor factorization based on whole-data. The evaluation results on two real datasets showed that FWCP proposed in this study is superior to other context-aware recommendation algorithms. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we give some closed-form formulas for calculating maximal and minimal ranks and inertias of with respect to , where is given, is common Hermitian least squares solutions to matrix equations and . As application, we derive necessary and sufficient conditions for in the Löwner partial ordering. In addition, we give identifying conditions for the existence of definite common Hermitian least squares solutions to matrix equations and . [Copyright &y& Elsevier]

• We introduce and compare six typically fusion strategies for saliency detection. • Overall, the non-linear fusion strategies outperform the linear-based strategies. • The devised non-linear fusion strategy based on least square method is optimum. Saliency detection is a hot topic in the field of computer vision and pattern recognition, thus plenty of saliency models have been exploited and extended to various visual correspondence applications. Currently, it's still confronted with a variety of obstacles and challenges, although it has been studied for decades. With the progress of saliency detection, different computational models and salient features have been proposed and some of them improve and compensate the deficiencies of the others. In this paper, we focus on investigating the salient feature fusion strategies in human visual attention mechanism for saliency detection (e.g., linear and non-linear), in order to efficiently incorporate various salient cues for achieving a better result. Based on the complementary principle, we firstly construct a saliency map based on the information of the image background. Then, we generate a supplemental saliency map from the compactness saliency features. Finally, we evaluate the performance of six individual fusion strategies including both linear and non-linear models in terms of three publicly available image datasets. Experimental results show that our designed non-linear fusion strategy based on least-square method outperforms the other fusion strategies in saliency detection. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose an efficient method for special least squares solution of the complex matrix equation (A X B , C X D) = (E , F). By using the real representation matrices of complex matrices, the particular structure of the real representation matrices, the Moore–Penrose generalized inverse and the Kronecker product, we obtain the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation (A X B , C X D) = (E , F) , which was studied by a product of matrices and vectors in Wang et al. (2016). Our resulting formulas only involve real matrices, and the corresponding algorithm only performs real arithmetic. Therefore our proposed method is more effective and portable. Finally, we give three numerical examples to illustrate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, by using the real representations of quaternion matrices, the particular structure of the real representations of quaternion matrices, the Kronecker product of matrices and the Moore–Penrose generalized inverse, we obtain the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation A X B + C X D = E , respectively. Our resulting formulas only involve real matrices, and therefore are simpler than those reported in Yuan (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of quaternion matrices, and therefore are very efficient and portable. Numerical examples are provided to illustrate the efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper, by applying particular structure of the real representations of quaternion matrices and the Moore–Penrose generalized inverse, we derive the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation A X = B . The resulting formulas only involve real matrices, which are simpler than those reported in (Yuan et al., 2013). The corresponding algorithms only perform real arithmetic which also consider particular structure of the real representations of quaternion matrices, therefore are very efficient and easily understood. Numerical examples are provided to illustrate the efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

Abstract: In this article we give some formulas for the maximal and minimal ranks of the submatrices in a least squares solution X to AXB = C. From these formulas, we derive necessary and sufficient conditions for the submatrices to be zero and other special forms, respectively. Finally, some Hermitian properties for least squares solution to matrix equation AXB = C are derived. [Copyright &y& Elsevier]