Minimax signal detection in ill-posed inverse problems

TL;DR

This paper investigates the fundamental limits of detecting signals in ill-posed inverse problems across various function classes, providing both theoretical bounds and practical adaptive testing procedures.

Contribution

It introduces minimax detection rates for different ill-posed problems and constructs adaptive tests that are simple and nearly optimal.

Findings

Derived minimax detection rates for various ill-posed problems.

Constructed adaptive tests that are rate-optimal.

Analyzed error probabilities in Gaussian white noise models.

Abstract

Ill-posed inverse problems arise in various scientific fields. We consider the signal detection problem for mildly, severely and extremely ill-posed inverse problems with $l^q$-ellipsoids (bodies), $q\in(0,2]$, for Sobolev, analytic and generalized analytic classes of functions under the Gaussian white noise model. We study both rate and sharp asymptotics for the error probabilities in the minimax setup. By construction, the derived tests are, often, nonadaptive. Minimax rate-optimal adaptive tests of rather simple structure are also constructed.