论文标题
在繁殖内核希尔伯特空间中线性功能的实验设计
Experimental Design for Linear Functionals in Reproducing Kernel Hilbert Spaces
论文作者
论文摘要
最佳实验设计旨在确定实验最有用的分配，以推断未知的统计数量。在这项工作中，我们研究了{\ em估计在复制核Hilbert空间（RKHSS）}中线性函数的最佳设计。在估计性条件下，已在线性回归设置中进行了广泛的研究，该问题允许估计没有偏差的参数。我们将此框架推广到RKHSS，并允许线性函数仅被近似推断，即具有固定偏置。这种情况捕获了许多重要的现代应用，例如对微分方程的梯度图，积分和解决方案的估计。我们提供用于构建线性函数偏置感知设计的算法。我们在高斯噪声下为固定和自适应设计提供了非反应置信度集，从而使我们能够以有界误差的估计，并具有很高的概率。
Optimal experimental design seeks to determine the most informative allocation of experiments to infer an unknown statistical quantity. In this work, we investigate the optimal design of experiments for {\em estimation of linear functionals in reproducing kernel Hilbert spaces (RKHSs)}. This problem has been extensively studied in the linear regression setting under an estimability condition, which allows estimating parameters without bias. We generalize this framework to RKHSs, and allow for the linear functional to be only approximately inferred, i.e., with a fixed bias. This scenario captures many important modern applications, such as estimation of gradient maps, integrals, and solutions to differential equations. We provide algorithms for constructing bias-aware designs for linear functionals. We derive non-asymptotic confidence sets for fixed and adaptive designs under sub-Gaussian noise, enabling us to certify estimation with bounded error with high probability.
