Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such functions from finitely many samples is of particular interest, due to the practical importance of constructing surrogate models to complex mathematical models of physical processes. In a previous work, [5] we studied the approximation of so-called Banach-valued, $(\boldsymbol{b},\varepsilon)$-holomorphic functions on the infinite-dimensional hypercube $[-1,1]^{\mathbb{N}}$ from $m$ (potentially adaptive) samples. In particular, we derived lower bounds for the adaptive $m$-widths for classes of such functions, which showed that certain algebraic rates of the form $m^{1/2-1/p}$ are the best possible regardless of the sampling-recovery pair. In this work, we continue this investigation by focusing on the practical case where the samples are pointwise evaluations drawn identically and independently from a probability measure. Specifically, for Hilbert-valued $(\boldsymbol{b},\varepsilon)$-holomorphic functions, we show that the same rates can be achieved (up to a small polylogarithmic or algebraic factor) for essentially arbitrary tensor-product Jacobi (ultraspherical) measures. Our reconstruction maps are based on least squares and compressed sensing procedures using the corresponding orthonormal Jacobi polynomials. In doing so, we strengthen and generalize past work that has derived weaker nonuniform guarantees for the uniform and Chebyshev measures (and corresponding polynomials) only. We also extend various best $s$-term polynomial approximation error bounds to arbitrary Jacobi polynomial expansions. Overall, we demonstrate that i.i.d.\ pointwise samples are near-optimal for the recovery of infinite-dimensional, holomorphic functions.

翻译：无限维全纯函数因在参数化微分方程和计算不确定性量化中的相关性，在过去几十年间得到了详细研究。由于构建物理过程复杂数学模型的替代模型具有实际重要性，从有限样本中逼近此类函数尤为引人关注。在先前工作[5]中，我们研究了从$m$个（可自适应）样本逼近无限维超立方体$[-1,1]^{\mathbb{N}}$上的所谓Banach值$(\boldsymbol{b},\varepsilon)$-全纯函数。特别地，我们推导了此类函数类的自适应$m$-宽度下界，表明无论采用何种采样-恢复对，形如$m^{1/2-1/p}$的特定代数速率均为理论最优。本文延续这一研究，重点关注实际场景：样本为从概率测度独立同分布抽取的点值评估。具体而言，对于Hilbert值$(\boldsymbol{b},\varepsilon)$-全纯函数，我们证明对于几乎任意张量积Jacobi（超球面）测度，相同速率（至多相差一个小的多对数或代数因子）均可实现。我们的重建映射基于最小二乘和压缩感知方法，并利用相应的正交Jacobi多项式。通过这一研究，我们强化并推广了先前仅针对均匀测度和Chebyshev测度（及其对应多项式）推导出的较弱非均匀保证。此外，我们将各类最佳$s$项多项式逼近误差界扩展至任意Jacobi多项式展开。总体而言，我们证明独立同分布点采样对于无限维全纯函数的恢复是近乎最优的。