# Trust your source: quantifying source condition elements for variational regularisation methods

## TL;DR

This paper presents a practical method to compute source condition elements for variational regularisation, enabling better error estimation and convergence analysis in inverse problems, with applications in machine learning and image processing.

## Contribution

It introduces a convex minimisation-based approach to determine source condition elements, adaptable to various inverse problems and regularisation techniques.

## Key findings

- Validated on inverse problems in machine learning and image processing.
- Demonstrated ability to identify optimal sampling patterns for MRI reconstruction.
- Showed the method's flexibility in different variational regularisation contexts.

## Abstract

Source conditions are a key tool in regularisation theory that are needed to derive error estimates and convergence rates for ill-posed inverse problems. In this paper, we provide a recipe to practically compute source condition elements as the solution of convex minimisation problems that can be solved with first-order algorithms. We demonstrate the validity of our approach by testing it on two inverse problem case studies in machine learning and image processing: sparse coefficient estimation of a polynomial via LASSO regression and recovering an image from a subset of the coefficients of its discrete Fourier transform. We further demonstrate that the proposed approach can easily be modified to solve the machine learning task of identifying the optimal sampling pattern in the Fourier domain for a given image and variational regularisation method, which has applications in the context of sparsity promoting reconstruction from magnetic resonance imaging data.