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：近十年来，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。这主要源于参数化微分方程解的泛函在化学、经济学、工程学和物理学等多个领域的重要性。尽管获取此类函数的精确可靠逼近本质上是困难的，但当前基准方法利用这些函数常属于特定全纯函数类这一特性，获得了关于（可能自适应的）样本数量$m$的代数收敛速度。本研究致力于为$(\boldsymbol{b},\varepsilon)$-全纯函数类提供理论逼近保证，证明这些代数收敛速度是巴拿赫值函数在无限维框架下的最优可能结果。我们通过将问题简化为离散形式，结合$m$-宽度、盖尔范德宽度与柯尔莫哥洛夫宽度理论建立下界。我们研究了两种情形：已知各向异性和未知各向异性，分别对应于变量相对重要性的已知与未知情况。本文的一个关键结论是：在后一种情形中，即使采用自适应采样，若无变量的内在排序，基于有限样本的逼近是无法实现的。最后，针对这两种情形，我们提出了近乎最优的非自适应（随机）采样与重构策略，其收敛速度与下界接近一致。