Optimal regularized hypothesis testing in statistical inverse problems

TL;DR

This paper introduces a regularized hypothesis testing framework for inverse problems, allowing biased estimators, and demonstrates its superiority over traditional methods through theoretical analysis and numerical simulations.

Contribution

It develops a novel regularized testing approach for inverse problems, providing adaptive tests with maximized power and proving their advantages over classical unregularized tests.

Findings

Regularized tests are at least as effective as unregularized tests.

The adaptive test significantly outperforms previous methods in simulations.

The approach applies under mild source-condition assumptions.

Abstract

Testing of hypotheses is a well studied topic in mathematical statistics. Recently, this issue has also been addressed in the context of Inverse Problems, where the quantity of interest is not directly accessible but only after the inversion of a (potentially) ill-posed operator. In this study, we propose a regularized approach to hypothesis testing in Inverse Problems in the sense that the underlying estimators (or test statistics) are allowed to be biased. Under mild source-condition type assumptions we derive a family of tests with prescribed level $\alpha$ and subsequently analyze how to choose the test with maximal power out of this family. As one major result we prove that regularized testing is always at least as good as (classical) unregularized testing. Furthermore, using tools from convex optimization, we provide an adaptive test by maximizing the power functional, which then outperforms previous unregularized tests in numerical simulations by several orders of magnitude.