On the lifting of deterministic convergence rates for inverse problems with stochastic noise

TL;DR

This paper bridges the gap between deterministic and stochastic noise models in inverse problems, extending deterministic convergence theories to stochastic settings to enable broader application of regularization methods.

Contribution

It introduces a framework to lift deterministic convergence results into stochastic inverse problems, unifying the approaches without requiring problem-specific stochastic analysis.

Findings

Established convergence rates for inverse problems with stochastic noise

Unified deterministic and stochastic noise modeling approaches

Enabled application of deterministic regularization methods to stochastic problems

Abstract

Both for the theoretical and practical treatment of Inverse Problems, the modeling of the noise is a crucial part. One either models the measurement via a deterministic worst-case error assumption or assumes a certain stochastic behavior of the noise. Although some connections between both models are known, the communities develop rather independently. In this paper we seek to bridge the gap between the deterministic and the stochastic approach and show convergence and convergence rates for Inverse Problems with stochastic noise by lifting the theory established in the deterministic setting into the stochastic one. This opens the wide field of deterministic regularization methods for stochastic problems without having to do an individual stochastic analysis for each problem.