Nonparametric, Nonasymptotic Confidence Bands with Paley-Wiener Kernels for Band-Limited Functions
This addresses the need for reliable uncertainty quantification in regression for band-limited functions, applicable in signal processing or similar domains, but appears incremental as it builds on existing kernel and perturbation methods.
The paper tackles the problem of constructing confidence bands for bounded, band-limited functions from finite samples, providing distribution-free and nonparametric methods with non-asymptotic guarantees, and demonstrates the approach through numerical experiments.
The paper introduces a method to construct confidence bands for bounded, band-limited functions based on a finite sample of input-output pairs. The approach is distribution-free w.r.t. the observation noises and only the knowledge of the input distribution is assumed. It is nonparametric, that is, it does not require a parametric model of the regression function and the regions have non-asymptotic guarantees. The algorithm is based on the theory of Paley-Wiener reproducing kernel Hilbert spaces. The paper first studies the fully observable variant, when there are no noises on the observations and only the inputs are random; then it generalizes the ideas to the noisy case using gradient-perturbation methods. Finally, numerical experiments demonstrating both cases are presented.