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

arXiv: https://arxiv.org/abs/2206.13629 · HTML: https://mlsanity.com/paper/2206.13629/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 19 references that are in the map.
- First posted on arXiv 27 Jun 2022.

## Builds on (most cited first, 2 of 2)

- [Sign-Perturbed Sums: A New System Identification Approach for Constructing Exact Non-Asymptotic Confidence Regions in Linear Regression Models](https://mlsanity.com/paper/1807.08216/) (2018)
- [Distribution-Free Uncertainty Quantification for Kernel Methods by Gradient Perturbations](https://mlsanity.com/paper/1812.09632/) (2018)

## Built on by (most cited first, 8 of 8)

- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Improving Kernel-Based Nonasymptotic Simultaneous Confidence Bands](https://mlsanity.com/paper/2401.15791/) (2024)
- [Optimal kernel regression bounds under energy-bounded noise](https://mlsanity.com/paper/2505.22235/) (2025)
- [Sample Complexity of the Sign-Perturbed Sums Identification Method: Scalar Case](https://mlsanity.com/paper/2401.15792/) (2024)
- [Derandomizing Simultaneous Confidence Regions for Band-Limited Functions by Improved Norm Bounds and Majority-Voting Schemes](https://mlsanity.com/paper/2506.17764/) (2025)
- [Single Image Inpainting and Super-Resolution with Simultaneous Uncertainty Guarantees by Universal Reproducing Kernels](https://mlsanity.com/paper/2506.23221/) (2025)
- [Non-asymptotic confidence regions on RKHS. The Paley-Wiener and standard Sobolev space cases](https://mlsanity.com/paper/2507.06657/) (2025)
- [MiNCE: Nonparametric, Strongly Consistent Confidence Envelopes for Band-Limited Functions and their Smoothed Spectra](https://mlsanity.com/paper/2609.09436/) (2026)

## Abstract (arXiv)

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.