Over the last decade, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples $m$. Our work focuses on providing theoretical approximation guarantees for the class of $(\boldsymbol{b},\varepsilon)$-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of $m$-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.

翻译：过去十年间，基于样本对无限维函数进行逼近在计算科学与工程领域（尤其是计算不确定性量化）中日益受到关注。其主要原因是参数化微分方程的解函数在化学、经济学、工程学和物理学等多个领域具有重要应用。尽管获取此类函数的精确可靠逼近本身具有难度，当前基准方法利用这些函数往往属于特定全纯函数类这一特性，在无限维空间中实现了关于（可能自适应的）样本数$m$的代数收敛速度。本文针对$(\boldsymbol{b},\varepsilon)$-全纯函数类提供了理论逼近保证，证明这类代数速度对于无限维中的Banach值函数而言是最优的。我们通过将问题简化为离散情形，结合$m$-宽度、盖尔范德宽度和科尔莫戈罗夫宽度理论建立下界。本研究探讨了两种情形：已知各向异性与未知各向异性，分别对应变量相对重要性已知和未知的情况。本文的关键结论是：在后者情形下，即使样本可自适应选取，若无变量间的固有排序，基于有限样本的逼近将无法实现。最后，我们在两种情形下均提出了逼近最优的非自适应（随机）采样与恢复策略，其收敛速度与下界速率基本一致。