N-Widths and ε-Dimensions for High-Dimensional Approximations
N-Widths and ε-Dimensions for High-Dimensional Approximations
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DOI：
10.1007/s10208-013-9149-9
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发表时间：
2013-12-01
影响因子：
3
通讯作者：
Ullrich, Tino
中科院分区：
文献类型：
--
作者：
Dinh Dung;Ullrich, Tino
本文研究了具有各向异性光滑性的周期d-变量函数类W的线性三角双曲交叉逼近、Kolmogorov n-宽度d（n）（W，H（gamma））和ε-维数n（n）（W，H（gamma）），其中d可以很大.我们感兴趣的是找到d（n）（W，H（gamma））和n（n）（W，H（gamma））分别作为两个变量n，d和n，d的函数的精确依赖性。回想一下，n，近似子空间的维数，是研究n向无穷大收敛速度的主要参数。然而，当d较大时，参数d可能严重影响该速率。我们构造线性近似的函数W的三角多项式的频率从双曲交叉和证明上界测量的误差在各向同性Sobolev空间H（伽玛）。此外，为了证明所提出的近似的最优性，我们证明了相应的n-宽度d（n）（W，H（gamma））和ε-维数n（n）（W，H（gamma））的上界和下界.一些收到的结果意味着维数灾难可以在一些相关的情况下被打破。
In this paper, we study linear trigonometric hyperbolic cross approximations, Kolmogorov n-widths d (n) (W,H (gamma) ), and epsilon-dimensions n (epsilon) (W,H (gamma) ) of periodic d-variate function classes W with anisotropic smoothness, where d may be large. We are interested in finding the accurate dependence of d (n) (W,H (gamma) ) and n (epsilon) (W,H (gamma) ) as a function of two variables n, d and epsilon, d, respectively. Recall that n, the dimension of the approximating subspace, is the main parameter in the study of convergence rates with respect to n going to infinity. However, the parameter d may seriously affect this rate when d is large. We construct linear approximations of functions from W by trigonometric polynomials with frequencies from hyperbolic crosses and prove upper bounds for the error measured in isotropic Sobolev spaces H (gamma) . Furthermore, in order to show the optimality of the proposed approximation, we prove upper and lower bounds of the corresponding n-widths d (n) (W,H (gamma) ) and epsilon-dimensions n (epsilon) (W,H (gamma) ). Some of the received results imply that the curse of dimensionality can be broken in some relevant situations.