熵与ridge函数类的采样数
Entropy and sampling numbers of classes of ridge functions
AI总结：
研究高维ridge函数的近似理论性质，分析其熵数和采样数，发现当p=2时采样复杂度与多变量Lipschitz函数相当，但当ridge方向导数有下界时复杂度降低至单变量情况。
摘要：
我们从逼近理论的角度研究高维d中ridge函数f(x)=g(a·x)的性质。考虑的函数类由ridge函数组成，其中profileg属于单变量Lipschitz类（光滑度α>0，包括无限光滑），ridge方向a的p-范数‖a‖_p≤1。首先，我们研究熵数以量化这些ridge函数类在L∞中的紧致性。我们证明它们的紧致性与单变量Lipschitz函数相当。其次，我们考察采样数并面临两种极端情况。在p=2时，采样ridge函数在欧几里得单位球上面临维度灾难，因此与采样一般多变量Lipschitz函数一样困难，这与熵数的结果形成鲜明对比。当进一步假设所有可行profile在原点处的一阶导数有统一下界时，采样ridge函数的复杂度大幅降低至单变量Lipschitz函数的复杂度。中间情况下，采样问题的难度取决于α和p的值。令人惊讶的是，我们观察到几乎整个由Novak和Woźniakowski最近专著中引入的可处理性层次结构。
英文摘要：
We study properties of ridge functions $f(x)=g(a\cdot x)$ in high dimensions $d$ from the viewpoint of approximation theory. The considered function classes consist of ridge functions such that the profile $g$ is a member of a univariate Lipschitz class with smoothness $α> 0$ (including infinite smoothness), and the ridge direction $a$ has $p$-norm $\|a\|_p \leq 1$. First, we investigate entropy numbers in order to quantify the compactness of these ridge function classes in $L_{\infty}$. We show that they are essentially as compact as the class of univariate Lipschitz functions. Second, we examine sampling numbers and face two extreme cases. In case $p=2$, sampling ridge functions on the Euclidean unit ball faces the curse of dimensionality. It is thus as difficult as sampling general multivariate Lipschitz functions, a result 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 in the origin, then 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 of $α$ and $p$. Surprisingly, we see almost the entire hierarchy of tractability levels as introduced in the recent monographs by Novak and Woźniakowski.
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