We study the approximation of a square-integrable function from a finite number of evaluations on a random set of nodes according to a well-chosen distribution. This is particularly relevant when the function is assumed to belong to a reproducing kernel Hilbert space (RKHS). This work proposes to combine several natural finite-dimensional approximations based two possible probability distributions of nodes. These distributions are related to determinantal point processes, and use the kernel of the RKHS to favor RKHS-adapted regularity in the random design. While previous work on determinantal sampling relied on the RKHS norm, we prove mean-square guarantees in $L^2$ norm. We show that determinantal point processes and mixtures thereof can yield fast convergence rates. Our results also shed light on how the rate changes as more smoothness is assumed, a phenomenon known as superconvergence. Besides, determinantal sampling generalizes i.i.d. sampling from the Christoffel function which is standard in the literature. More importantly, determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling.

翻译：我们研究根据精心选择的分布，在随机节点集上通过有限次函数评估来近似平方可积函数的问题。当假设函数属于再生核希尔伯特空间（RKHS）时，这一问题尤为关键。本工作提出结合基于节点两种可能概率分布的多个自然有限维近似方法。这些分布与行列式点过程相关，并利用RKHS的核在随机设计中引入适应RKHS的正则性。先前关于行列式点过程采样的研究依赖于RKHS范数，而我们则证明了$L^2$范数下的均方保证。研究表明，行列式点过程及其混合模型能够实现快速收敛速率。我们的结果还揭示了当假设更多光滑性时速率的变化规律——这一现象被称为超收敛性。此外，行列式点过程采样推广了文献中标准使用的克里斯托费尔函数独立同分布采样。更重要的是，与独立同分布采样相比，行列式点过程采样能以更少的函数评估次数保证所谓的实例最优性性质。