Nonparametric, Nonasymptotic Confidence Bands with Paley-Wiener Kernels for Band-Limited Functions

TL;DR

This paper presents a nonparametric, distribution-free method for constructing finite-sample confidence bands for band-limited functions using Paley-Wiener kernels, applicable in both noiseless and noisy observation scenarios.

Contribution

It introduces a novel approach leveraging Paley-Wiener RKHS to create non-asymptotic confidence bands for band-limited functions without assuming specific noise distributions or parametric models.

Findings

Method provides finite-sample guarantees.

Effective in noiseless and noisy settings.

Numerical experiments validate the approach.

Abstract

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.