Optimal regularized hypothesis testing in statistical inverse problems
Remo Kretschmann, Daniel Wachsmuth, Frank Werner
TL;DR
This work develops a regularized hypothesis-testing framework for statistical inverse problems, where the quantity of interest is accessible only through an ill-posed forward operator. By allowing bias through regularized probe functions $\Phi$, the authors construct level-$α$ tests via $\Psi_{\Phi,c}$ with a data-driven critical value $c^*$, and show any probe yields valid testing, addressing issues of bias and detector power. They derive an optimal probe $\Phi^\dagger$ that minimizes a convex objective $J_{T u^\dagger}^{\mathcal{Y}}$, attaining maximal finite-sample power, and prove this regularized test dominates unregularized approaches. To handle unknown $u^\dagger$, an adaptive method based on convex optimization and a two-sample procedure yields a level-$α$ test with power tending to 1 as noise vanishes. Numerical experiments in deconvolution demonstrate substantial power gains for the optimal and adaptive tests over the unregularized test, especially in highly ill-posed or incompatible settings, highlighting the practical impact of regularized testing in inverse problems.
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 $α$ 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.