具有随机隐单元的渐增前馈神经网络的$L^2(R^d)$逼近能力

This paper studies approximation capability to L2(Rd) functions of incremental constructive feedforward neural networks(FNN) with random hidden units.Two kinds of therelayered feedforward neural networks are considered:radial basis function(RBF) neural networks and translation and dilation invariant(TDI) neural networks.In comparison with conventional methods that existence approach is mainly used in approximation theories for neural networks,we follow a constructive approach to prove that one may simply randomly choose parameters of hidden units and then adjust the weights between the hidden units and the output unit to make the neural network approximate any function in L2(Rd) to any accuracy.Our result shows given any non-zero activation function g :R+→R and g(x Rd) ∈ L2(Rd) for RBF hidden units,or any non-zero activation function g(x) ∈ L2(Rd) for TDI hidden units,the incremental network function fn with randomly generated hidden units converges to any target function in L2(Rd) with probability one as the number of hidden units n→∞,if one only properly adjusts the weights between the hidden units and output unit.     

单隐层神经网络与最佳多项式逼近

研究单隐层神经网络逼近问题．以最佳多项式逼近为度量,用构造性方法估计单隐层神经网络逼近连续函数的速度．所获结果表明:对定义在紧集上的任何连续函数,均可以构造一个单隐层神经网络逼近该函数,并且其逼近速度不超过该函数的最佳多项式逼近的二倍．     

Integral operators with operator-valued kernels

Under fairly mild measurability and integrability conditions on operator-valued kernels, boundedness results for integral operators on Bochner spaces L_p(X) are given. In particular, these results are applied to convolutions operators.     

Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures

The density of polynomials is straightforward to prove in Sobolev spaces W^k,p((a,b)), but there exist only partial results in weighted Sobolev spaces; here we improve some of these theorems. The situation is more complicated in infinite intervals, even for weighted L^p spaces; besides, in the present paper we have proved some other results for weighted Sobolev spaces in infinite intervals.     

关于随机函数的Weierstrass逼近定理

定义了随机系数多项式的概念之后,将函数的Weierstrass逼近定理推广到了随机函数上.     

Cardaliguet-Eurrard型神经网络算子逼近

研究多维Cardaliguet-Eurrard型神经网络算子的逼近问题.分别给出该神经网络算子逼近连续函数与可导函数的速度估计,建立了Jackson型不等式.     

Fluctuations in the homogenization of semilinear equations with random potentials

We study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by homogenized solution, the Green’s function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential. These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations.     

Global behavior of homogeneous random neural systems

We define a simple form of homogeneous neural network model whose characteristics are expressed in terms of probabilistic assumptions. The networks considered operate in an asynchronous manner and receive the influence of the environment in the form of external stimulations. The operation of the network is described by means of a Markovian process whose steady-statesolution yields several global measures of the network's activity. Three different types of external stimulations are investigated, which represent possible input mechanisms. The analytical results obtained concern the macroscopic viewpoint and provide a quick insight into the structure of the network's behavior.     

Limitations of the approximation capabilities of neural networks with one hidden layer

Lets1 be an integer andW be the class of all functions having integrable partial derivatives on [0, 1] ^s . We are interested in the minimum number of neurons in a neural network with a single hidden layer required in order to provide a mean approximation order of a preassigned>0 to each function inW. We prove that this number cannot be if a spline-like localization is required. This cannot be improved even if one allows different neurons to evaluate different activation functions, even depending upon the target function. Nevertheless, for any>0, a network with neurons can be constructed to provide this order of approximation, with localization. Analogous results are also valid for otherL ^p norms.The research of this author was supported by NSF Grant # DMS 92-0698.The research of this author was supported, in part, by AFOSR Grant #F49620-93-1-0150 and by NSF Grant #DMS 9404513.     

On the Generalization Problem

The generalization problem considered in this paper assumes that a limited amount of input and output data from a system is available, and that from this information an estimate of the output produced by another input is required. The ideas arose in the study of neural networks, but apply equally to any approximation approach. The main result is that the type of neural network to be used for generalization should be determined by the prior knowledge about the nature of the output from the system. Without such information, either of two networks matching the training data is equally likely to be the better at estimating the output generated by the same system at a new input. Therefore, the search for an optimum generalization network for use on all problems is inappropriate.For both (0, 1) and accurate real outputs, it is shown that simple approximations exist that fit the data, so these will be equally likely to generalize better than more sophisticated networks, unless prior knowledge is available that excludes them. For noisy real outputs, it is shown that the standard least squares approach forces the neural network to approximate an incorrect process; an alternative approach is outlined, which again is much easier to learn and use.     

Approximation by radial bases and neural networks

In this paper, we study approximation by radial basis functions including Gaussian, multiquadric, and thin plate spline functions, and derive order of approximation under certain conditions. Moreover, neural networks are also constructed by wavelet recovery formula and wavelet frames.     

球面神经网络的构造与逼近

研究球面神经网络的构造与逼近问题.利用球面广义的de la Vallee Poussin平均、球面求积公式及改进的单变量Cardaliaguet-Euvrard神经网络算子,构造具logistic激活函数的单隐层前向网络,并给出了Jackson型误差估计.