The hierarchical alternating least squares (HALS) algorithms are powerful tools for nonnegative matrix factorization (NMF), among which the Fast-HALS, proposed in [A. Cichocki and A.-H. Phan, 2009], is one of the most efficient. This paper investigates the convergence of Fast-HALS. First, a more general weak convergence (converged subsequences exist and converge to the stationary point set) is established without any assumption, while most existing results assume all the columns of iterates are strictly away from the origin. Then, a simplified strong convergence (the entire sequence converges to a stationary point) proof is provided. The existing strong convergence is attributed to the block prox-linear (BPL) method, which is a more general framework including Fast-HALS as a special case. So, the convergence proof under BPL is quite complex. Our simplified proof explores the structure of Fast-HALS and can be regarded as a complement to the results under BPL. In addition, some numerical verifications are presented.

This article presents two novel continuous- and discrete-time neural networks (NNs) for solving quadratic minimax problems with linear equality constraints. These two NNs are established based on the conditions of the saddle point of the underlying function. For the two NNs, a proper Lyapunov function is constructed so that they are stable in the sense of Lyapunov, and will converge to some saddle point(s) for any starting point under some mild conditions. Compared with the existing NNs for solving quadratic minimax problems, the proposed NNs require weaker stability conditions. The validity and transient behavior of the proposed models are illustrated by some simulation results.

In this article, we study the symmetric nonnegative matrix factorization (SNMF) which is a powerful tool in data mining for dimension reduction and clustering. The main contributions of the present work include: (i) a new descent direction for the rank-one SNMF is derived and a strategy for choosing the step size along this descent direction is established; (ii) a progressive hierarchical alternating least squares (PHALS) method for SNMF is developed, which is parameter-free and updates the variables column by column. Moreover, every column is updated by solving a rank-one SNMF subproblem; and (iii) the convergence to the Karush-Kuhn-Tucker (KKT) point set (or the stationary point set) is proved for PHALS. Several synthetical and real data sets are tested to demonstrate the effectiveness and efficiency of the proposed method. Our PHALS provides better performance in terms of the computational accuracy, the optimality gap, and the CPU time, compared with a number of state-of-the-art SNMF methods.

We study a multi-instance (MI) learning dimensionality-reduction algorithm through sparsity and orthogonality, which is especially useful for high-dimensional MI data sets. We develop a novel algorithm to handle both sparsity and orthogonality constraints that existing methods do not handle well simultaneously. Our main idea is to formulate an optimization problem where the sparse term appears in the objective function and the orthogonality term is formed as a constraint. The resulting optimization problem can be solved by using approximate augmented Lagrangian iterations as the outer loop and inertial proximal alternating linearized minimization (iPALM) iterations as the inner loop. The main advantage of this method is that both sparsity and orthogonality can be satisfied in the proposed algorithm. We show the global convergence of the proposed iterative algorithm. We also demonstrate that the proposed algorithm can achieve high sparsity and orthogonality requirements, which are very important for dimensionality reduction. Experimental results on both synthetic and real data sets show that the proposed algorithm can obtain learning performance comparable to that of other tested MI learning algorithms.

This paper presents a novel neural network for solving generally constrained variational inequality problems by constructing a system of double projection equations. By defining proper convex energy functions, the proposed neural network is proved to be stable in the sense of Lyapunov and converges to an exact solution of the original problem for any starting point under the weaker cocoercivity condition or the monotonicity condition of the gradient mapping on the linear equation set. Furthermore, two sufficient conditions are provided to ensure the stability of the proposed neural network for a special case. The proposed model overcomes some shortcomings of existing continuous-time neural networks for constrained variational inequality, and its stability only requires some monotonicity conditions of the underlying mapping and the concavity of nonlinear inequality constraints on the equation set. The validity and transient behavior of the proposed neural network are demonstrated by some simulation results.

It has always been a challenging task to develop a fast and an efficient incremental linear discriminant analysis (ILDA) algorithm. For this purpose, we conduct a new study for linear discriminant analysis (LDA) in this paper and develop a new ILDA algorithm. We propose a new batch LDA algorithm called LDA/QR. LDA/QR is a simple and fast LDA algorithm, which is obtained by computing the economic QR factorization of the data matrix followed by solving a lower triangular linear system. The relationship between LDA/QR and uncorrelated LDA (ULDA) is also revealed. Based on LDA/QR, we develop a new incremental LDA algorithm called ILDA/QR. The main features of our ILDA/QR include that: 1) it can easily handle the update from one new sample or a chunk of new samples; 2) it has efficient computational complexity and space complexity; and 3) it is very fast and always achieves competitive classification accuracy compared with ULDA algorithm and existing ILDA algorithms. Numerical experiments based on some real-world data sets demonstrate that our ILDA/QR is very efficient and competitive with the state-of-the-art ILDA algorithms in terms of classification accuracy, computational complexity, and space complexity.

In this paper, we study canonical correlation analysis (CCA), which is a powerful tool in multivariate data analysis for finding the correlation between two sets of multidimensional variables. The main contributions of the paper are: 1) to reveal the equivalent relationship between a recursive formula and a trace formula for the multiple CCA problem, 2) to obtain the explicit characterization for all solutions of the multiple CCA problem even when the corresponding covariance matrices are singular, 3) to develop a new sparse CCA algorithm, and 4) to establish the equivalent relationship between the uncorrelated linear discriminant analysis and the CCA problem. We test several simulated and real-world datasets in gene classification and cross-language document retrieval to demonstrate the effectiveness of the proposed algorithm. The performance of the proposed method is competitive with the state-of-the-art sparse CCA algorithms.

In this paper, we analyze and establish the stability and convergence of the dynamical system proposed by Xia and Feng, whose equilibria solve variational inequality and related problems. Under the pseudo-monotonicity and other conditions, this system is proved to be stable in the sense of Lyapunov and converges to one of its equilibrium points for any starting point. Meanwhile, the global exponential stability of this system is also shown under some mild conditions without the strong monotonicity of the mapping. The obtained results improve and correct some existing ones. The validity and performance of this system are demonstrated by some numerical examples.

In this paper, the interior point based continuous method is proposed for linear programming. The continuous method model can be viewed as the continuous realization of the existing interior point method for linear programming. Our study will be under the framework of the continuous method for optimization. As a result, we are able to study the behaviors of these continuous method models in a unified format. A key component of the continuous method is an ordinary differential equation (ODE), which is established based on the interior point method for linear programming. The properties of the ODE along with the convergence issues will be addressed.

In this paper, we present a new neural network for solving linear and quadratic programming problems in real time by introducing some new vectors. The proposed neural network is stable in the sense of Lyapunov and can converge to an exact optimal solution of the original problem when the objective function is convex on the set defined by equality constraints. Compared with existing one-layer neural networks for quadratic programming problems, the proposed neural network has the least neurons and requires weak stability conditions. The validity and transient behavior of the proposed neural network are demonstrated by some simulation results.

This paper presents a new neural network model for solving constrained variational inequality problems by converting the necessary and sufficient conditions for the solution into a system of nonlinear projection equations. Five sufficient conditions are provided to ensure that the proposed neural network is stable in the sense of Lyapunov and converges to an exact solution of the original problem by defining a proper convex energy function. The proposed neural network includes an existing model, and can be applied to solve some nonmonotone and nonsmooth problems. The validity and transient behavior of the proposed neural network are demonstrated by some numerical examples.

Based on the inherent properties of convex quadratic minimax problems, this article presents a new neural network model for a class of convex quadratic minimax problems. We show that the new model is stable in the sense of Lyapunov and will converge to an exact saddle point in finite time by defining a proper convex energy function. Furthermore, global exponential stability of the new model is shown under mild conditions. Compared with the existing neural networks for the convex quadratic minimax problem, the proposed neural network has finite-time convergence, a simpler structure, and lower complexity. Thus, the proposed neural network is more suitable for parallel implementation by using simple hardware units. The validity and transient behavior of the proposed neural network are illustrated by some simulation results.

Variational inequality is a uniform approach for many important optimization and equilibrium problems. Based on the sufficient and necessary conditions of the solution, this paper presents a novel neural network model for solving variational inequalities with linear and nonlinear constraints. Three sufficient conditions are provided to ensure that the proposed network with an asymmetric mapping is stable in the sense of Lyapunov and converges to an exact solution of the original problem. Meanwhile, the proposed network with a gradient mapping is also proved to be stable in the sense of Lyapunov and to have a finite-time convergence under some mild condition by using a new energy function. Compared with the existing neural networks, the new model can be applied to solve some nonmonotone problems, has no adjustable parameter, and has lower complexity. Thus, the structure of the proposed network is very simple. Since the proposed network can be used to solve a broad class of optimization problems, it has great application potential. The validity and transient behavior of the proposed neural network are demonstrated by several numerical examples.

In this paper, we propose a neural network for solving a class of convex quadratic minimax problems with constraints. Four sufficient conditions are provided to ensure the asymptotic stability of the proposed network. Furthermore, the exponential stability of the proposing network is also proved under certain conditions. The results obtained here can be further extended to the globally projected dynamical system. In addition, some new stability conditions for the system are also obtained. Since our stability conditions can be easily checked in practice, these results becomes more attractive in real applications.