标题： 从高度噪声数据中学习函数的混合最小二乘法 显示英文标题

标题： Hybrid least squares for learning functions from highly noisy data

摘要： 鉴于对条件期望高效估计的需求，我们考虑了一个数据严重污染的最小二乘函数逼近问题。 在小噪声情况下表现强大的现有方法在存在大噪声时则次优。 我们提出了一种混合方法，将Christoffel采样与某些类型的最优实验设计相结合，以解决此问题。 我们证明了所提出的算法在样本点生成和噪声平滑方面具有适当的最优性，与现有方法相比，提高了计算效率和样本复杂度。 我们还将该算法扩展到具有类似理论保证的凸约束情形。 当目标函数定义为随机场的期望时，我们将方法扩展为利用自适应随机子空间，并建立了自适应过程的逼近能力结果。 我们的理论发现得到了在合成数据以及计算金融中更具挑战性的随机模拟问题上的数值研究的支持。 显示英文摘要

摘要： Motivated by the need for efficient estimation of conditional expectations, we consider a least-squares function approximation problem with heavily polluted data. Existing methods that are powerful in the small noise regime are suboptimal when large noise is present. We propose a hybrid approach that combines Christoffel sampling with certain types of optimal experimental design to address this issue. We show that the proposed algorithm enjoys appropriate optimality properties for both sample point generation and noise mollification, leading to improved computational efficiency and sample complexity compared to existing methods. We also extend the algorithm to convex-constrained settings with similar theoretical guarantees. When the target function is defined as the expectation of a random field, we extend our approach to leverage adaptive random subspaces and establish results on the approximation capacity of the adaptive procedure. Our theoretical findings are supported by numerical studies on both synthetic data and on a more challenging stochastic simulation problem in computational finance.