On accuracy and existence of approximate decoders for ill-posed inverse problems

Abstract

Based on work by Cohen, Damen and Devore \and Bourrier et. al., we propose a framework that highlights the importance of knowing the measurement model $F$ and model class $\mathcal{M}_1$, for solving ill-posed (non-)linear inverse problems. Previous work has assumed that the measurement model is injective on the model class $\mathcal{M}_1$ and we obviate the need for this assumption. We establish fundamental upper and lower bounds on the reconstruction accuracy of an inverse problem in terms of the kernel size. The key definition introduced in this work, the kernel size of an inverse problem, only requires the measurement model $F$ and model class $\mathcal{M}_1$ to be computed. Thus, it is applicable in deep learning (DL) based settings where $\mathcal{M}_1$ can be an arbitrary data set.