Experimental Design for Linear Functionals in Reproducing Kernel Hilbert Spaces
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.
Code (0)
등록된 구현이 없습니다.
Tasks
Experimental DesignMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Learning with Invariance via Linear Functionals on Reproducing Kernel Hilbert Space
Incorporating invariance information is important for many learning problems. To exploit invariances, most existing methods resort to approximations that either lead to expensive optimization problems such as semi-defini…
White Functionals for Anomaly Detection in Dynamical Systems
We propose new methodologies to detect anomalies in discrete-time processes taking values in a set. The method is based on the inference of functionals whose evaluations on successive states visited by the process have l…
Anomaly DetectionApproximation of RKHS Functionals by Neural Networks
Motivated by the abundance of functional data such as time series and images, there has been a growing interest in integrating such data into neural networks and learning maps from function spaces to R (i.e., functionals…
regressionTime SeriesOn the stability test for reproducing kernel Hilbert spaces
Reproducing kernel Hilbert spaces (RKHSs) are special Hilbert spaces where all the evaluation functionals are linear and bounded. They are in one-to-one correspondence with positive definite maps called kernels. Stable R…
Featured Reproducing Kernel Banach Spaces for Learning and Neural Networks
Reproducing kernel Hilbert spaces provide a foundational framework for kernel-based learning, where regularization and interpolation problems admit finite-dimensional solutions through classical representer theorems. Man…