Higher order error estimates for regularization of inverse problems under non-additive noise
Diana-Elena Mirciu, Elena Resmerita
TL;DR
This work develops higher-order error estimates for inverse problems under non-additive noise within a variational regularization framework using a Kullback-Leibler data fidelity. It introduces a novel equality-type source condition in the Fenchel dual, and a variational dual source condition, to tightly couple dual and primal regularity and obtain higher-order rates in the Bregman distance $D_R$. The results cover general convex fidelities, with KL-specific rates for exact data, and robust rates under Poisson-type noise, including joint KL regularization. The framework clarifies the interplay between data fidelity, regularization, and source conditions, and extends to broad classes of convex data fidelities, offering guidance for parameter choice in imaging tasks such as PET where non-additive noise is prevalent.
Abstract
In this work we derive higher order error estimates for inverse problems distorted by non-additive noise, in terms of Bregman distances. The results are obtained by means of a novel source condition, inspired by the dual problem. Specifically, we focus on variational regularization having the Kullback-Leibler divergence as data-fidelity, and a convex penalty term. In this framework, we provide an interpretation of the new source condition, and present error estimates also when a variational formulation of the source condition is employed. We show that this approach can be extended to variational regularization that incorporates more general convex data fidelities.