Non-asymptotic confidence regions on RKHS. The Paley-Wiener and standard Sobolev space cases

TL;DR

This paper develops a method for constructing non-asymptotic, probabilistic confidence regions for unknown functions in RKHS, focusing on Paley-Wiener and Sobolev spaces, by estimating the RKHS norm.

Contribution

It introduces a new approach to create confidence regions in RKHS settings, specifically addressing the Paley-Wiener and Sobolev spaces, based on RKHS norm estimation.

Findings

Provides non-asymptotic confidence regions for functions in RKHS.

Reduces the problem to RKHS norm estimation.

Focuses on Paley-Wiener and Sobolev space cases.

Abstract

We consider the problem of constructing a global, probabilistic, and non-asymptotic confidence region for an unknown function observed on a random design. The unknown function is assumed to lie in a reproducing kernel Hilbert space (RKHS). We show that this construction can be reduced to accurately estimating the RKHS norm of the unknown function. Our analysis primarily focuses both on the Paley-Wiener and on the standard Sobolev space settings.