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：在过去的几十年中，无限维全纯函数因其与参数微分方程及计算不确定性量化密切相关而得到了深入研究。由于构建物理过程复杂数学模型的代理模型具有重要的实际意义，从有限个样本中逼近此类函数尤其受到关注。在先前的工作[5]中，我们研究了从 $m$ 个（可能为自适应的）样本中逼近无限维超立方体 $[-1,1]^{\mathbb{N}}$ 上所谓的 Banach 值 $(\boldsymbol{b},\varepsilon)$-全纯函数的问题。特别地，我们推导了此类函数类的自适应 $m$ 宽度下界，结果表明形式为 $m^{1/2-1/p}$ 的特定代数收敛率是可能达到的最佳速率，且与采样-恢复对的选择无关。在本工作中，我们继续这一研究，重点关注采样点是从某个概率测度中独立同分布抽取的点态求值的实际情形。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的张量积 Jacobi（超球）测度，可以达到相同的收敛率（至多相差一个小的多对数或代数因子）。我们的重构映射基于使用相应正交归一化 Jacobi 多项式的最小二乘与压缩感知过程。通过这一工作，我们强化并推广了过去仅针对均匀测度与 Chebyshev 测度（及相应多项式）推导出较弱非均匀保证的研究。我们还将多种最佳 $s$ 项多项式逼近误差界推广至任意的 Jacobi 多项式展开。总体而言，我们证明了独立同分布的点采样对于无限维全纯函数的恢复是接近最优的。