Eignets for function approximation on manifolds
Eignets for function approximation on manifolds
复制标题
流形上函数逼近的 Eignet
DOI：
10.1016/j.acha.2009.08.006
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
H. Mhaskar
中科院分区：
文献类型：
--
作者：
H. Mhaskar
设X是紧致光滑连通的无边界黎曼流形，G：X×X→R是核.类似于径向基函数网络，特征网络是形式为∑j= 1 MajG（○，yj）的表达式，其中aj∈R，yj∈X，1 <$j <$M。本文描述了一种确定性的、通用的算法，用于构造Lp（μ;X）中的函数的特征网，该特征网对一般的测度类μ和核G进行逼近.我们的算法产生线性算子。利用中心之间的最小间隔yj作为逼近的代价，我们给出了本征值逼近度的光滑模估计，并通过一个匡威定理证明了这些估计对每个函数都是最好的。我们也给出了特征值范数项的系数ajin的估计。最后，我们证明，如果任何序列的eignets满足最佳估计的光滑函数的近似程度，测量方面的最小分离，然后衍生物的eignets也近似相应的衍生物的目标函数的最佳方式。
Let X be a compact, smooth, connected, Riemannian manifold without boundary, G:X×X→R be a kernel. Analogous to a radial basis function network, an eignet is an expression of the form ∑j=1MajG(○,yj), where aj∈R, yj∈X, 1⩽j⩽M. We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in Lp(μ;X) for a general class of measures μ and kernels G. Our algorithm yields linear operators. Using the minimal separation among the centers yjas the cost of approximation, we give modulus of smoothness estimates for the degree of approximation by our eignets, and show by means of a converse theorem that these are the best possible for every individual function. We also give estimates on the coefficients ajin terms of the norm of the eignet. Finally, we demonstrate that if any sequence of eignets satisfies the optimal estimates for the degree of approximation of a smooth function, measured in terms of the minimal separation, then the derivatives of the eignets also approximate the corresponding derivatives of the target function in an optimal manner.