We study the complexity of high-dimensional approximation in the $L_2$-norm when different classes of information are available; we compare the power of function evaluations with the power of arbitrary continuous linear measurements. Here, we discuss the situation when the number of linear measurements required to achieve an error $\varepsilon \in (0,1)$ in dimension $d\in\mathbb{N}$ depends only poly-logarithmically on $\varepsilon^{-1}$. This corresponds to an exponential order of convergence of the approximation error, which often happens in applications. However, it does not mean that the high-dimensional approximation problem is easy, the main difficulty usually lies within the dependence on the dimension $d$. We determine to which extent the required amount of information changes, if we allow only function evaluation instead of arbitrary linear information. It turns out that in this case we only lose very little, and we can even restrict to linear algorithms. In particular, several notions of tractability hold simultaneously for both types of available information.

翻译：我们研究了在可获得不同类型信息情况下 $L_2$ 范数中高维逼近的复杂性；比较了函数求值与任意连续线性测量的能力。这里，我们讨论当在维度 $d\in\mathbb{N}$ 中实现误差 $\varepsilon \in (0,1)$ 所需的线性测量数量仅依赖于 $\varepsilon^{-1}$ 的多对数项时的情形。这对应逼近误差的指数级收敛速度，在应用中经常出现。然而，这并不意味着高维逼近问题是容易的，主要困难通常在于对维度 $d$ 的依赖。我们确定了如果只允许函数求值而非任意线性信息时，所需信息量的变化程度。结果表明，在这种情况下我们仅损失极少，甚至可以限制在线性算法中。特别地，几种易处理性的概念对于这两种类型的信息同时成立。