Harmonic Analysis of Neural Networks

It is known that superpositions of ridge functions (single hidden-layer feedforward neural networks) may give good approximations to certain kinds of multivariate functions. It remains unclear, however, how to effectively obtain such approximations. In this paper, we use ideas from harmonic analysis to attack this question. We introduce a special admissibility condition for neural activation functions. The new condition is not satisfied by the sigmoid activation in current use by the neural networks community; instead, our condition requires that the neural activation function be oscillatory. Using an admissible neuron we construct linear transforms which represent quite general functionsfas a superposition of ridge functions. We develop

• • • a continuous transform which satisfies a Parseval-like relation;
• • • a discrete transform which satisfies frame bounds.

Both transforms representfin a stable and effective way. The discrete transform is more challenging to construct and involves an interesting new discretization of time–frequency–direction space in order to obtain frame bounds for functions inL²(A) whereAis a compact set of Rⁿ. Ideas underlying these representations are related to Littlewood–Paley theory, wavelet analysis, and group representation theory.     

时滞细胞神经网络的稳定性分析

研究具有时滞的细胞神经网络的稳定性问题,通过构造合适的Lyapunov函数及不等式分析技巧,给出了时滞细胞神经网络全局稳定的新的充分判据,这些结论推广了已知文献中的结果。     

非对称细胞神经网络的稳定性分析

This paper describes the problem of stability for one-dimensional Cellular Neural Networks(CNNs). A sufficient condition is presented to ensure complete stability for a class of special CNN's with nonsymmetric templates, where the parameter in the output function is greater than or equal to zero. The main method is analysising the property of the equilibrium point of the CNNs system.     

Cohen-Grossberg神经网络的全局指数稳定性分析

本文研究了CohenGrossberg神经网络模型的指数稳定性.为避免构造Lyapunov函数的困难,我们采用广义相对Dalquist数方法来分析神经网络的稳定性.借助这一方法,我们不但得到了CohenGrossberg神经网络模型平衡解的存在性、唯一性和全局指数稳定性的新的充分条件,而且给出了神经网络的指数衰减估计.所获结论改进了已有文献的相关结果.     

一类具有时滞的Cohen-Grossberg神经网络的指数稳定性

考虑一类具有时滞的Cohen-Grossberg神经网络,利用Lyapunov方法和微分不等式理论,得到了其全局指数稳定性的判别准则.该准则引入了更多的参数,更便于系统的设计与分析.     

一类一维混沌映射的拓扑条件

20世纪中期以来,人们在物理、天文、气象等领域中发现了大量的混沌现象.这些新发现引发了近几十年来对混沌现象的研究.由于它的困难程度和在解决实际问题中的巨大价值,对混沌现象的研究成为动力系统乃至数学中的一个长期的前沿和热点研究方向.混沌现象最本质的特征是初值敏感性,保证有初值敏感性的一个充分条件是系统具有正Lyapunov指数.因此研究系统是否具有正Lyapunov指数成为研究系统是否出现混沌的重要方法.从拓扑角度给出了一类一维映射出现混沌现象的充分条件.从拓扑的角度来研究,将加深对此类映射出现混沌的机理的认识.研究此类映射,最重要的是研究临界点、临界点轨道及它们的相互关系.我们采用临界点的逆像建立拓扑工具,使用这一拓扑工具分析临界点轨道与临界点的复杂关系,研究临界点逆轨道的运动形态、相应开集的拓扑特征,进而导出系统出现混沌的拓扑特征及它与Lyapunov指数之间的关系.     

Topological Design of Survivable Mesh-Based Transport Networks

The advent of Sonet and DWDM mesh-restorable networks which contain explicit reservations of spare capacity for restoration presents a new problem in topological network design. On the one hand, the routing of working flows wants a sparse tree-like graph for minimization of the classic fixed charge plus routing (FCR) costs. On the other hand, restorability requires a closed (bi-connected) and preferably high-degree topology for efficient sharing of spare capacity allocations (SCA) for restoration over non-simultaneous failure scenarios. These diametrically opposed considerations underlie the determination of an optimum physical facilities graph for a broadband network provider. Standalone instances of each constituent problem are NP-hard. The full problem of simultaneously optimizing mesh-restorable topology, routing, and sparing is therefore very difficult computationally. Following a comprehensive survey of prior work on topological design problems, we provide a {1–0} MIP formulation for the complete mesh-restorable design problem and also propose a novel three-stage heuristic. The heuristic is based on the hypothesis that the union set of edges obtained from separate FCR and SCA sub-problems constitutes an effective topology space within which to solve a restricted instance of the full problem. Where fully optimal reference solutions are obtainable the heuristic shows less than 8% gaps but runs in minutes as opposed to days. In other test cases the reference problem cannot be solved to optimality and we can only report that heuristic results obtained in minutes are not improved upon by CPLEX running the full problem for 6 to 18 hours. The computational behavior we observe gives insight for further work based on an appreciation of the problem as embodying unexpectedly difficult feasibility apects, as well as optimality aspects.     

A Comparison of Discriminant Analysis versus Artificial Neural Networks

Artificial Neural Network (ANN) techniques have recently been applied to many different fields and have demonstrated their capabilities in solving complex problems. In a business environment, the techniques have been applied to predict bond ratings and stock price performance. In these applications, ANN techniques outperformed widely-used multivariate statistical techniques. The purpose of this paper is to compare the ANN method with the Discriminant Analysis (DA) method in order to understand the merits of ANN that are responsible for the higher level of performance. The paper provides an overview of the basic concepts of ANN techniques in order to enhance the understanding of this emerging technique. The similarities and differences between ANN and DA techniques in representing their models are described. This study also proposes a method to overcome the limitations of the ANN approach, Finally, a case study using a data set in a business environment demonstrates the superiority of ANN over DA as a method of classification of observations.     

Stability Analysis of Gradient-Based Neural Networks for Optimization Problems

The paper introduces a new approach to analyze the stability of neural network models without using any Lyapunov function. With the new approach, we investigate the stability properties of the general gradient-based neural network model for optimization problems. Our discussion includes both isolated equilibrium points and connected equilibrium sets which could be unbounded. For a general optimization problem, if the objective function is bounded below and its gradient is Lipschitz continuous, we prove that (a) any trajectory of the gradient-based neural network converges to an equilibrium point, and (b) the Lyapunov stability is equivalent to the asymptotical stability in the gradient-based neural networks. For a convex optimization problem, under the same assumptions, we show that any trajectory of gradient-based neural networks will converge to an asymptotically stable equilibrium point of the neural networks. For a general nonlinear objective function, we propose a refined gradient-based neural network, whose trajectory with any arbitrary initial point will converge to an equilibrium point, which satisfies the second order necessary optimality conditions for optimization problems. Promising simulation results of a refined gradient-based neural network on some problems are also reported.     

不分明化拓扑群的一致结构

本文引入了不分明化拓扑群的左、右、双一致结构，讨论了此类一致结构在一致连续下的一些性质，给出了此类结构在其子群上的相对结构和在其乘积上的来积结构。     

数学规划的求解与广义Hopfield神经网络

神经网络技术最为成功的应用领域之一是用于求解优化问题,本文就近年来的求解优化问题的神经网络方法进行了综述     

时变时滞BAM神经网络系统的指数稳定性分析

利用Lyapunov稳定性理论和线性矩阵不等式技术,得到了保证时变时滞BAM神经网络系统指数稳定性的时滞依赖稳定性准则.所给的准则可用Matlab中的LMI控制工具箱进行验证.仿真实例进一步说明了结果的有效性.     

局部域反馈神经网络的全局收敛性

本文针对具有-般投影映射的连续型局部域反馈神经网络,给出了在一可验证条件下的临界全局收敛性分析.所获得的结果极大地改进了现有典型的收敛性结论.数值试验进-步证实了所得结果的正确性和可行性.     

神经网络在非线性规划中的应用

本文对蔡少棠[1 ] ,[2 ] 等提出的非线性规划神经网络作了改进 ,使得该神经网络适合于通用VLSI技术实现 .     

I-fuzzy拓扑空间中的pre-重域结构

首先在I-fuzzy拓扑空间框架下引入了I-fuzzy pre-开集,I-fuzzy pre-重域,及I-fuzzy pre-闭包等概念,进而分别研究了它们的性质,最后在I-fuzzy拓扑空间中讨论了I-fuzzy pre-网收敛.     

The Topological Structure of Subseries Convergence

Let E be an abstract set, G be an Abelian topological group, FEG and be afamily of subsets of F. Let the sequence {xj}E be subseries aF convergent. In this paper,we present a sufficient and necessary condition for {xj} to be also subseries convergent in thetopology of uniform convergence on the sets in ,     

解线性不等式的神经网络

本文提出两个解线性不等式的Hopfiedl-Tank型的神经网络。第一个网络模拟同时松弛投影方法，第二个网络是二次规划方法。当线性不等式的解集非空时，这两个方法都给出该线性不等式的解。同时我们还给出了这两个网络的数值模拟。     

On learning of Sigmoid Neural Networks

The Probably Approximately Correct (PAC) learning theory creates a framework to assess the learning properties of static models for which the data are assumed to be independently and identically distributed (i.i.d.). One important family of dynamic models to which the conventional PAC learning can not be applied is nonlinear Finite Impulse Response (FIR) models. The present article, using an extension of PAC learning that covers learning with m-dependent data, the learning properties of FIR modeling with sigmoid neural networks are evaluated. These results include upper bounds on the size of the data set required to train FIR sigmoid neural networks, provided that the input data are uniformly distributed. © 2001 John Wiley & Sons, Inc.     

区间联想神经网络的稳定性

本文用矩阵测度研究了一类区间联想神经络的稳定性,给出了其稳定的判别准则,推广和改进了文[1-2]的工作。