Adaptive Gaussian inverse regression with partially unknown operator

TL;DR

This paper develops a data-driven orthogonal series estimator for inverse problems involving unknown operators, achieving minimax optimality under Gaussian noise in both the function and operator observations.

Contribution

It introduces a fully data-driven method combining model selection and Lepski's technique to adaptively estimate functions in ill-posed inverse problems with unknown operators.

Findings

Estimator attains minimax rates in various noise regimes.

Method adapts to unknown operator singular values and function smoothness.

Illustrations include Sobolev spaces and different degrees of ill-posedness.

Abstract

This work deals with the ill-posed inverse problem of reconstructing a function $f$ given implicitly as the solution of $g = Af$, where $A$ is a compact linear operator with unknown singular values and known eigenfunctions. We observe the function $g$ and the singular values of the operator subject to Gaussian white noise with respective noise levels $\varepsilon$ and $\sigma$. We develop a minimax theory in terms of both noise levels and propose an orthogonal series estimator attaining the minimax rates. This estimator requires the optimal choice of a dimension parameter depending on certain characteristics of $f$ and $A$. This work addresses the fully data-driven choice of the dimension parameter combining model selection with Lepski's method. We show that the fully data-driven estimator preserves minimax optimality over a wide range of classes for $f$ and $A$ and noise levels $\varepsilon$ and $\sigma$. The results are illustrated considering Sobolev spaces and mildly and severely ill-posed inverse problems.