In the framework of the FD (frequent directions) algorithm, we first develop two efficient algorithms for low-rank matrix approximations under the embedding matrices composed of the product of any SpEmb (sparse embedding) matrix and any standard Gaussian matrix, or any SpEmb matrix and any SRHT (subsampled randomized Hadamard transform) matrix. The theoretical results are also achieved based on the bounds of singular values of standard Gaussian matrices and the theoretical results for SpEmb and SRHT matrices. With a given Tucker-rank, we then obtain several efficient FD-based randomized variants of T-HOSVD (the truncated high-order singular value decomposition) and ST-HOSVD (sequentially T-HOSVD), which are two common algorithms for computing the approximate Tucker decomposition of any tensor with a given Tucker-rank. We also consider efficient FD-based randomized algorithms for computing the approximate TT (tensor-train) decomposition of any tensor with a given TT-rank. Finally, we illustrate the efficiency and accuracy of these algorithms using synthetic and real-world matrix (and tensor) data.

This research is the first study focused on studying the total least squares (TLS) for time-varying (TV) tensor multilinear equations (TE). Obtained results extend existing methods for solving the TLS problem in both constant tensor settings and constant and TV matrix settings. Two models are suggested to solve simultaneous perturbations in both tensor coefficient and right-hand vector inside TV TEs. The first model establishes a continuous-time neural network designed for solving the TLS of TV tensor equations (termed as NNTVTETLS), while the second (termed as discrete neural network for total-least-squares solution of time-varying tensor equations (DNNTVTETLS)) is developed as its discrete counterpart. Unlike existing gradient-based neural networks and least-squares methods, the proposed models address TLS perturbation in a high-order TV multilinear settings. A rigorous theoretical convergence analysis is provided, establishing local convergence for both continuous and discrete models. An efficient iterative scheme based on the DNNTVTETLS scheme is also developed. Extensive numerical experiments demonstrate the efficacy of both continuous and discrete-time models. Numerical tests under various TV, constant, bounded vanishing, and bounded nonvanishing noise, confirm the rapid convergence, stability, and robustness of the proposed methods. The suggested framework is applied to solve the Bellman equation, which emerges in dynamic programming, economics, reinforcement learning, and optimal decision-making problems.

Completing multidimensional color images is a fundamental challenge in image processing and computer vision. However, some tensor-based methods often treat RGB channels as independent modes, thereby neglecting their intrinsic correlations. To address this limitation, we represent RGB values as pure quaternions and organize them into a quaternion tensor for holistic modeling that preserves chromatic relationships. To better capture the nonlinear characteristics inherent in visual data and to improve the compactness of low-rank representations, we propose a nonlinear transformation within the quaternion domain. This design enables more expressive modeling compared to conventional linear approaches. In addition, we introduce two novel regularization terms that jointly encode global low-rankness and local smoothness, with the nonlinear transformation further enhancing the exploitation of structural priors. The overall model is optimized via a nonlinear alternating direction method of multipliers (ADMM), with theoretical guarantees of convergence. Extensive experiments on several datasets demonstrate that the proposed method significantly outperforms state-of-the-art low-rank tensor and quaternion tensor recovery techniques in multidimensional color image completion tasks.

This paper develops several new dynamical designs, based on the gradient neural network (GNN), from the perspective of control theory to solve the time-varying Sylvester equation (TVSE). We start with an adaptive gradient neural network called AGNN-S. Then, based on Lyapunov theory, we propose three improved models: ACGNN, AIGNN, and ABGNN. Among them, the ABGNN model stands out for its fastest convergence speed and strongest robustness to noise. Theoretical analysis confirms that all proposed models solve the TVSE effectively, with ACGNN, AIGNN, and ABGNN converging faster than AGNN-S. Theoretical analysis shows that using certain nonlinear activation functions can further boost convergence speed. A robustness analysis indicates that the AGNN-S, ACGNN, and ABGNN models maintain stable convergence even in the presence of differentiation or model-implementation errors. Comparisons with the classical zeroing neural network (ZNN) and other six state-of-the-art GNN- and ZNN-type models demonstrate the superior accuracy and efficiency of our approaches, especially the ABGNN model. Numerical experiments notably highlight ABGNN’s advantages over other models under certain parameter setting. Finally, we showcase applications in time-varying quadratic programming and robotic arm trajectory tracking to verify the practical value of the models.

The primary focus of this study is to estimate least squares solutions for time-varying multi-linear systems (TVMLS) that involve time-varying (TV) perturbations on the right-hand side. To address this issue, we introduce two neural network models: the hybrid neural network (HNN) and a nonlinear enhancement of the HNN design that uses a modified weighted sign-bi-power (Mwsbp) activation function, referred to as MwsbpHNN. Theoretical findings demonstrate the convergence of both the HNN and MwsbpHNN dynamic models to the exact solutions of underlying TVMLS. Furthermore, we rigorously prove a fixed-time convergence of MwsbpHNN and provide upper bounds for the convergence time. Computational complexity of our theoretical framework is compared with current state-of-the-art methods for solving TVMLS, highlighting its efficiency. Numerical simulation further confirms the robustness of the proposed HNN and MwsbpHNN models. A model arising from discretized high-order Bellman equation is considered.

Power amplifiers (PAs) are essential components in wireless communication systems, and the design of their behavioral models has been an important research topic for many years. The widely used generalized memory polynomial (GMP) model suffers from rapid growth in the number of parameters with increasing memory depths and nonlinearity order, which leads to a significant increase in model complexity and the risk of overfitting. In this study, we introduce tensor networks to compress the unknown coefficient tensor of the GMP model, resulting in three novel tensor-based GMP models. These models can achieve comparable performance to the GMP model, but with far fewer parameters and lower complexity. For the identification of these models, we derive the alternating least-squares (ALS) method to ensure the rapid updates and convergence of model parameters in an iterative manner. In addition, we notice that the horizontal slices of the third-order data tensor constructed from the input signals are Vandermonde matrices, which have a numerically low-rank structure. Hence, we further propose the RP-ALS algorithm, which first performs a truncated higher-order singular value decomposition on the data tensor to generate random projections, then conducts the ALS algorithm for the identification of projected models with downscaled dimensions, thus reducing the computational effort of the iterative process. The experimental results show that the proposed models outperform the full GMP model and sparse GMP model via LASSO regression in terms of the reduction in the number of parameters and running complexity. Note to Practitioners—This paper is motivated by the demand for low-complexity and accurate PA behavioral models, offering a viable option for modeling wideband PAs in modern wireless communication systems. A good model should provide satisfactory performance with as low complexity as possible and allow for quick identification of model parameters. Based on the advantages of tensor networks, this paper proposes three tensor-based GMP models that can utilize the potential low-rank tensor structure of the GMP model, and derives the alternating least-squares method to rapidly identify the model parameters. These models can efficiently reduce the number of parameters and running complexity while maintaining comparable accuracy to the GMP model.

Two complex Zhang neural network (ZNN) models for computing the Drazin inverse of arbitrary time-varying complex square matrix are presented. The design of these neural networks is based on corresponding matrix-valued error functions arising from the limit representations of the Drazin inverse. Two types of activation functions, appropriate for handling complex matrices, are exploited to develop each of these networks. Theoretical results of convergence analysis are presented to show the desirable properties of the proposed complex-valued ZNN models. Numerical results further demonstrate the effectiveness of the proposed models.

Two linear recurrent neural networks for generating outer inverses with prescribed range and null space are defined. Each of the proposed recurrent neural networks is based on the matrix-valued differential equation, a generalization of dynamic equations proposed earlier for the nonsingular matrix inversion, the Moore-Penrose inversion, as well as the Drazin inversion, under the condition of zero initial state. The application of the first approach is conditioned by the properties of the spectrum of a certain matrix; the second approach eliminates this drawback, though at the cost of increasing the number of matrix operations. The cases corresponding to the most common generalized inverses are defined. The conditions that ensure stability of the proposed neural network are presented. Illustrative examples present the results of numerical simulations.

In this letter, we present the dynamical equation and corresponding artificial recurrent neural network for computing the Drazin inverse for arbitrary square real matrix, without any restriction on its eigenvalues. Conditions that ensure the stability of the defined recurrent neural network as well as its convergence toward the Drazin inverse are considered. Several illustrative examples present the results of computer simulations.

This paper presents a recurrent neural network (RNN) for computing the Drazin inverse of a real matrix in real time. This recurrent neural network (RNN) is composed of n independent parts (subnetworks), where n is the order of the input matrix. These subnetworks can operate concurrently, so parallel and distributed processing can be achieved. In this way, the computational advantages over the existing sequential algorithms can be attained in real-time applications. The RNN defined in this paper is convenient for an implementation in an electronic circuit. The number of neurons in the neural network is the same as the number of elements in the output matrix, which represents the Drazin inverse. The difference between the proposed RNN and the existing ones for the Drazin inverse computation lies in their network architecture and dynamics. The conditions that ensure the stability of the defined RNN as well as its convergence toward the Drazin inverse are considered. In addition, illustrative examples and examples of application to the practical engineering problems are discussed to show the efficacy of the proposed neural network.

Recently, an efficient lattice reduction method, called the effective LLL (ELLL) algorithm, was presented for the detection of multiinput multioutput (MIMO) systems. In this letter, a novel lattice reduction criterion, called diagonal reduction, is proposed. The diagonal reduction is weaker than the ELLL reduction, however, like the ELLL reduction, it has identical performance with the LLL reduction when applied for the sphere decoding and successive interference cancelation (SIC) decoding. It improves the efficiency of the ELLL algorithm by significantly reducing the size-reduction operations. Furthermore, we present a greedy column traverse strategy, which reduces the column swap operations in addition to the size-reduction operations.

PageRank is one of the most important ranking techniques used in today's search engines. A recent very interesting research track focuses on exploiting efficient numerical methods to speed up the computation of PageRank, among which the Arnoldi-type algorithm and the GMRES algorithm are competitive candidates. In essence, the former deals with the PageRank problem from an eigenproblem, while the latter from a linear system, point of view. However, there is little known about the relations between the two approaches for PageRank. In this article, we focus on a theoretical and numerical comparison of the two approaches. Numerical experiments illustrate the effectiveness of our theoretical results.