Average Kernel Sizes -- Computable Sharp Accuracy Bounds for Inverse Problems

Abstract

The reconstruction of an unknown quantity from noisy measurements is a mathematical problem relevant in most applied sciences, for example, in medical imaging, radar inverse scattering, or astronomy. This underlying mathematical problem is often an ill-posed (non-linear) reconstruction problem, referred to as an ill-posed inverse problem. To tackle such problems, there exist a myriad of methods to design approximate inverse maps, ranging from optimization-based approaches, such as compressed sensing, over Bayesian approaches, to data-driven techniques such as deep learning. For all stable approximate inverse maps, there are accuracy limits that are strictly larger than zero for ill-posed inverse problems, due to the accuracy-stability tradeoff [Gottschling et al., SIAM Review, 67.1 (2025)] and [Colbrook et al., Proceedings of the National Academy of Sciences, 119.12 (2022)]. The variety of methods that aim to solve such problems begs for a unifying approach to help scientists choose the approximate inverse map that obtains this theoretical optimum. Up to now there do not exist computable accuracy bounds to this optimum that are applicable to all inverse problems. We provide computable sharp accuracy bounds to the reconstruction error of solution methods to inverse problems. The bounds are method-independent and purely depend on the dataset of signals, the forward model of the inverse problem, and the noise model. To facilitate the use in scientific applications, we provide an algorithmic framework and an accompanying software library to compute these accuracy bounds. We demonstrate the validity of the algorithms on two inverse problems from different domains: fluorescence localization microscopy and super-resolution of multi-spectral satellite data. Computing the accuracy bounds for a problem before solving it, enables a fundamental shift towards optimizing datasets and forward models.