Entropy and Sampling Numbers of Classes of Ridge Functions
Entropy and Sampling Numbers of Classes of Ridge Functions
复制标题
岭函数类的熵和采样数
DOI：
10.1007/s00365-014-9267-x
复制
发表时间：
2015
影响因子：
2.7
通讯作者：
J. Vybiral
中科院分区：
文献类型：
--
作者：
S. Mayer;T. Ullrich;J. Vybiral
从逼近论的角度研究了高维脊函数的性质。所考虑的函数类由脊函数组成，使得轮廓是具有光滑性（包括无限光滑性）的单变量Lipschitz类的成员，并且脊方向具有范数。首先，我们调查熵数，以量化这些脊函数类的紧性。我们表明，他们基本上是紧凑的一类单变量Lipschitz功能。其次，我们研究抽样数，并考虑两个极端的情况。在这种情况下，欧几里德单位球上的采样脊函数遭受维数灾难。而且，它与一般的多元Lipschitz函数的采样一样困难，这与熵数的结果形成鲜明对比。当我们另外假设所有可行的配置文件有一个一阶导数一致有界远离零的原点，采样脊函数的复杂性大大降低到采样单变量Lipschitz函数的复杂性。在这两者之间，抽样问题的难度有所不同，取决于和的值。令人惊讶的是，我们几乎看到了诺瓦克和沃什尼亚科夫斯基在最近的专著中介绍的整个易处理性层次。
We study the properties of ridge functionsin high dimensionsfrom the viewpoint of approximation theory. The function classes considered consist of ridge functions such that the profileis a member of a univariate Lipschitz class with smoothness(including infinite smoothness) and the ridge directionhas-norm. First, we investigate entropy numbers in order to quantify the compactness of these ridge function classes in. We show that they are essentially as compact as the class of univariate Lipschitz functions. Second, we examine sampling numbers and consider two extreme cases. In the case, sampling ridge functions on the Euclidean unit ball suffers from the curse of dimensionality. Moreover, it is as difficult as sampling general multivariate Lipschitz functions, which is in sharp contrast to the result on entropy numbers. When we additionally assume that all feasible profiles have a first derivative uniformly bounded away from zero at the origin, the complexity of sampling ridge functions reduces drastically to the complexity of sampling univariate Lipschitz functions. In between, the sampling problem’s degree of difficulty varies, depending on the values ofand. Surprisingly, we see almost the entire hierarchy of tractability levels as introduced in the recent monographs by Novak and Woźniakowski.
登录
查看更多内容
影响因子：
3
作者：
J. Creutzig;S. Dereich;T. Müller;K. Ritter
通讯作者：
K. Ritter
影响因子：
1.7
作者：
V. Maiorov
通讯作者：
V. Maiorov
影响因子：
2.7
作者：
A. Cohen;I. Daubechies;R. DeVore;G. Kerkyacharian;D. Picard
通讯作者：
D. Picard
根据函数的投影对函数进行最优重构
影响因子：
2.5
作者：
F. B.;Logan And;A. L.;Shepp
通讯作者：
Shepp
DOI：
10.1142/9789812836021_0020
发表时间：
2008
期刊：
J. Approx. Theory
影响因子：
--
作者：
H. Flad;W. Hackbusch;B. Khoromskij;R. Schneider
通讯作者：
R. Schneider