Minimax Goodness-of-Fit Testing in Ill-Posed Inverse Problems with Partially Unknown Operators

TL;DR

This paper investigates the problem of goodness-of-fit testing in ill-posed inverse problems where the operator is only partially known, deriving minimax rates and bounds in a Gaussian sequence model.

Contribution

It introduces a framework for minimax goodness-of-fit testing with partially unknown operators in ill-posed inverse problems, providing non-asymptotic bounds and detailed examples.

Findings

Derived lower and upper bounds for the minimax separation radius.

Established minimax rates for different types of ill-posed problems.

Analyzed examples with ordinary-smooth and super-smooth sequences.

Abstract

We consider a Gaussian sequence model that contains ill-posed inverse problems as special cases. We assume that the associated operator is partially unknown in the sense that its singular functions are known and the corresponding singular values are unknown but observed with Gaussian noise. For the considered model, we study the minimax goodness-of-fit testing problem. Working with certain ellipsoids in the space of squared-summable sequences of real numbers, with a ball of positive radius removed, we obtain lower and upper bounds for the minimax separation radius in the non-asymptotic framework, i.e., for fixed values of the involved noise levels. Examples of mildly and severely ill-posed inverse problems with ellipsoids of ordinary-smooth and super-smooth sequences are examined in detail and minimax rates of goodness-of-fit testing are obtained for illustrative purposes.