15647217
Common Nature of Learning Exemplified by BP and Hopfield Neural Networks for Solving Online a System of Linear Equations
Many computational problems widely encountered in scientific and engineering applications could finally be transformed to the online linear-equations solving. Classic numerical methods for solving linear equations include Gaussian elimination and matrix factorization methods, which are usually of O(n3) operations. Being important parallel-computational models, both BP (back propagation) and Hopfield neural networks could be exploited for solving such linear equations. BP neural network is evidently different from Hopfield neural network in terms of network definition, architecture and learning pattern. However, both of these two neural networks could have a common nature of learning (i.e., governed by the same mathematical iteration formula) during the online solution of linear equations. In addition, computer-simulation results substantiate the theoretical analysis of both BP and Hopfield neural networks for solving online such a set of linear equations.
3106142
MATLAB Simulation and Comparison of Zhang Neural Network and Gradient Neural Network for Online Solution of Linear Time-Varying Matrix Equation AXB-C=0
Different from gradient neural networks (GNN), a special kind ofrecurrent neural networks has been proposed recently by Zhanget alfor solving online linear matrix equations withtime-varying coefficients. Such recurrent neural networks, designedbased on a matrix-valued error-function, could achieve globalexponential convergence when solving online time-varying problemsin comparison with gradient neural networks. This paperinvestigates the MATLAB simulation of Zhang neural networks (ZNN)for real-time solution of linear time-varying matrix equationAXB- C= 0. Gradient neural networks aresimulated and compared as well. Simulation results substantiate thetheoretical analysis and efficacy of ZNN on linear time-varyingmatrix equation solving.
35419934
MATLAB Simulation of Gradient-Based Neural Network for Online Matrix Inversion
This paper investigates the simulation of a gradient-based recurrent neural network for online solution of the matrix-inverse problem. Several important techniques are employed as follows to simulate such a neural system. 1) Kronecker product of matrices is introduced to transform a matrix-differential-equation (MDE) to a vector-differential-equation (VDE); i.e., finally, a standard ordinary-differential-equation (ODE) is obtained. 2) MATLAB routine "ode45" is introduced to solve the transformed initial-value ODE problem. 3) In addition to various implementation errors, different kinds of activation functions are simulated to show the characteristics of such a neural network. Simulation results substantiate the theoretical analysis and efficacy of the gradient-based neural network for online constant matrix inversion.
1213875
A simplified LVI-based primal-dual neural network for repetitive motion planning of PA10 robot manipulator starting from different initial states
This paper presents a simplified primal-dual neural network based on linear variational inequalities (LVI) for online repetitive motion planning of PA10 robot manipulator. To do this, a drift-free criterion is exploited in the form of a quadratic function. In addition, the repetitive-motion-planning scheme could incorporate the joint limits and joint velocity limits simultaneously. Such a scheme is finally reformulated as a time-varying quadratic program (QP). As a QP real-time solver, the simplified LVI-based primal-dual neural network (LVI-PDNN) is designed based on the QP-LVI conversion and Karush-Kuhn-Tucker (KKT) conditions. It has a simple piecewise-linear dynamics and could globally exponentially converge to the optimal solution of strictly-convex quadratic-programs. The simplified LVI-PDNN model is simulated based on PA10 robot arm, and simulation results show the effective remedy of the joint angle drift problem of PA10 robot.
11395736
Repetitive motion of redundant robots planned by three kinds of recurrent neural networks and illustrated with a four-link planar manipulator's straight-line example
In this paper, a dual neural network, LVI (linear variational inequalities)-based primal-dual neural network and simplified LVI-based primal-dual neural network are presented for online repetitive motion planning (RMP) of redundant robot manipulators (with a four-link planar manipulator as an example). To do this, a drift-free criterion is exploited in the form of a quadratic performance index. In addition, the repetitive-motion-planning scheme could incorporate the joint physical limits such as joint limits and joint velocity limits simultaneously. Such a scheme is finally reformulated as a quadratic program (QP). As QP real-time solvers, the aforementioned three kinds of neural networks all have piecewise-linear dynamics and could globally exponentially converge to the optimal solution of strictly-convex quadratic-programs. Furthermore, the neural-network based RMP scheme is simulated based on a four-link planar robot manipulator. Computer-simulation results substantiate the theoretical analysis and also show the effective remedy of the joint angle drift problem of robot manipulators.
15986478
MATLAB Simulink Modeling and Simulation of Zhang Neural Network for Online Time-Varying Matrix Inversion
Recently, a special kind of recurrent neural networks (RNN) with implicit dynamics has been proposed by Zhang et al for online time-varying problems solving (such as time-varying matrix inversion). Such a neural-dynamic system is elegantly designed by defining a matrix-valued error function rather than the usual scalar-valued norm-based error function. Its computational error can be made decrease to zero globally and exponentially. For the final purpose of field programmable gate array (FPGA) and application-specific integrated circuit (ASIC) realization, we investigate in this paper the MATLAB Simulink modeling and simulative verification of such a Zhang neural network (ZNN). By using click-and-drag mouse operations, it is easier to model and simulate in comparison with MATLAB coding. Both convergence and robustness properties of such a ZNN model are analyzed, which substantiate the effectiveness of Zhang neural network on inverting the time-varying matrices.
14878951
The orienteering problem in the plane revisited
We consider the <i>orienteering problem</i>: Given a set <i>P</i> of <i>n</i> points in the plane, a starting point <i>r</i> ∈ <i>P</i>, and a length constraint <i>B</i>, one needs to find a tour starting at <i>r</i> that visits as many points of <i>P</i> as possible and of length not exceeding <i>B</i>. We present a (1−ε)-approximation algorithm for this problem that runs in <i>n</i><sup><i>O</i>(1/ε)</sup> time, and visits at least (1−ε)<i>k</i><sub>opt</sub> points of <i>P</i>, where <i>k</i><sub>opt</sub> is the number of points visited by the optimal solution. This is the first polynomial time approximation scheme (PTAS) for this problem. The algorithm also works in higher dimensions.
6083596
Improved learning algorithms for mixture of experts in multiclass classification
Mixture of experts (ME) is a modular neural network architecture for supervised learning. A double-loop Expectation-Maximization (EM) algorithm has been introduced to the ME architecture for adjusting the parameters and the iteratively reweighted least squares (IRLS) algorithm is used to perform maximization in the inner loop [Jordan, M.I., Jacobs, R.A. (1994). Hierarchical mixture of experts and the EM algorithm, Neural Computation, 6(2), 181-214]. However, it is reported in literature that the IRLS algorithm is of instability and the ME architecture trained by the EM algorithm, where IRLS algorithm is used in the inner loop, often produces the poor performance in multiclass classification. In this paper, the reason of this instability is explored. We find out that due to an implicitly imposed incorrect assumption on parameter independence in multiclass classification, an incomplete Hessian matrix is used in that IRLS algorithm. Based on this finding, we apply the Newton-Raphson method to the inner loop of the EM algorithm in the case of multiclass classification, where the exact Hessian matrix is adopted. To tackle the expensive computation of the Hessian matrix and its inverse, we propose an approximation to the Newton-Raphson algorithm based on a so-called generalized Bernoulli density. The Newton-Raphson algorithm and its approximation have been applied to synthetic data, benchmark, and real-world multiclass classification tasks. For comparison, the IRLS algorithm and a quasi-Newton algorithm called BFGS have also been applied to the same tasks. Simulation results have shown that the use of the proposed learning algorithms avoids the instability problem and makes the ME architecture produce good performance in multiclass classification. In particular, our approximation algorithm leads to fast learning. In addition, the limitation of our approximation algorithm is also empirically investigated in this paper.
28337697
Trends in Neural Computation
Hyperbolic Function Networks for Pattern Classification.- Variable Selection for the Linear Support Vector Machine.- Selecting Data for Fast Support Vector Machines Training.- Universal Approach to Study Delayed Dynamical Systems.- A Hippocampus-Neocortex Model for Chaotic Association.- Latent Attractors: A General Paradigm for Context-Dependent Neural Computation.- Learning Mechanisms in Networks of Spiking Neurons.- GTSOM: Game Theoretic Self-organizing Maps.- How to Generate Different Neural Networks.- A Gradient-Based Forward Greedy Algorithm for Space Gaussian Process Regression.- An Evolved Recurrent Neural Network and Its Application.- A Min-Max Modular Network with Gaussian-Zero-Crossing Function.- Combining Competitive Learning Networks of Various Representations for Sequential Data Clustering.- Modular Neural Networks and Their Applications in Biometrics.- Performance Analysis of Dynamic Cell Structures.- Short Term Electric Load Forecasting: A Tutorial.- Performance Improvement for Formation-Keeping Control Using a Neural Network HJI Approach.- A Robust Blind Neural Equalizer Based on Higher-Order Cumulants.- The Artificial Neural Network Applied to Servo Control System.- Robot Localization Using Vision.
1757195
Combining linear discriminant functions with neural networks for supervised learning
A novel supervised learning method is proposed by combining linear discriminant functions with neural networks. The proposed method results in a tree-structured hybrid architecture. Due to constructive learning, the binary tree hierarchical architecture is automatically generated by a controlled growing process for a specific supervised learning task. Unlike the classic decision tree, the linear discriminant functions are merely employed in the intermediate level of the tree for heuristically partitioning a large and complicated task into several smaller and simpler subtasks in the proposed method. These subtasks are dealt with by component neural networks at the leaves of the tree accordingly. For constructive learning, growing and credit-assignment algorithms are developed to serve for the hybrid architecture. The proposed architecture provides an efficient way to apply existing neural networks (e.g. multi-layered perceptron) for solving a large scale problem. We have already applied the proposed method to a universal approximation problem and several benchmark classification problems in order to evaluate its performance. Simulation results have shown that the proposed method yields better results and faster training in comparison with the multilayered perceptron.
6566254
On k-Median clustering in high dimensions
We study approximation algorithms for <i>k</i>-median clustering. We obtain small coresets for <i>k</i>-median clustering in metric spaces as well as in Euclidean spaces. Specifically, in R<sup>d</sup>, those coresets are of size with only <i>polynomial</i> dependency on <i>d</i>. This leads to a (1 + ε)-approximation algorithm for <i>k</i>-median clustering in R<sup>d</sup>, with running time <i>O</i>(<i>ndk</i> +2<sup>(k/ε)<sup><i>o</i>(1)</sup></sup><i>d</i><sup>2</sup><i>n</i>σ), for any σ > 0. This is an improvement over previous results [5, 20, 21]. We also provide fast constant factor approximation algorithms for <i>k</i>-median clustering in finite metric spaces.We use those coresets to compute (1 + ε)-approximation <i>k</i>-median clustering in the streaming model of computation, using only <i>O</i>(<i>k</i><sup>2</sup><i>de</i><sup>-2</sup>log<sup>8</sup> <i>n</i>) space, where the points are taken from R<sup>d</sup>. This is the first streaming algorithm, for this problem, that has space complexity with only <i>polynomial</i> dependency on the dimension.