准多项式可处理性的简化准则及其应用
Simplified criterion of quasi-polynomial tractability and its applications
AI总结：
本文研究了中心随机元素序列在分离希尔伯特空间中的逼近性质，提出简化准多项式可处理性准则，并应用于平方指数核的随机元素及加权科罗波夫核。
摘要：
我们研究了中心随机元素序列$X_d$，$d\in\mathbb{N}$在分离希尔伯特空间中的逼近性质。我们关注张量积型随机元素序列，其协方差算子具有相应的张量积形式。平均情况逼近复杂度$n^{X_d}(\varepsilon)$定义为近似$X_d$所需评估任意线性泛函的最小数量，其相对$2$-平均误差不超过给定阈值$\varepsilon\in(0,1)$。$n^{X_d}(\varepsilon)$随$\varepsilon^{-1}$和$d$的增长决定序列是否可处理。本文修订了准多项式可处理性准则并提供简化版本，通过应用张量积平方指数核随机元素和扩展G. Xu(2014)关于加权科罗波夫核的结果来展示结果。
英文摘要：
We study approximation properties of sequences of centered random elements $X_d$, $d\in\mathbb{N}$, with values in separable Hilbert spaces. We focus on sequences of tensor product-type random elements, which have covariance operators of corresponding tensor product form. The average case approximation complexity $n^{X_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate $X_d$ with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. The growth of $n^{X_d}(\varepsilon)$ as a function of $\varepsilon^{-1}$ and $d$ determines whether a sequence of corresponding approximation problems for $X_d$, $d\in\mathbb{N}$, is tractable or not. Different types of tractability were studied in the paper by M. A. Lifshits, A. Papageorgiou and H. Woźniakowski (2012), where for each type the necessary and sufficient conditions were found in terms of the eigenvalues of the marginal covariance operators. We revise the criterion of quasi-polynomial tractability and provide its simplified version. We illustrate our result by applying to random elements corresponding to tensor products of squared exponential kernels. Also we extend recent result of G. Xu (2014) concerning weighted Korobov kernels.
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