Learning sparsity-promoting regularizers for linear inverse problems

TL;DR

This paper proposes a bilevel optimization framework to learn sparsity-promoting regularizers for linear inverse problems, combining theoretical guarantees with practical examples and simulations.

Contribution

It introduces a novel data-driven method for learning regularizers in infinite-dimensional inverse problems, extending prior Tikhonov regularization approaches.

Findings

The method guarantees well-posedness and provides sample complexity bounds.

The approach is validated through theoretical examples and numerical simulations.

It effectively learns regularizers that promote sparsity in solutions.

Abstract

This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as $B$, which regularizes the inverse problem while promoting sparsity in the solution. The method leverages statistical properties of the underlying data and incorporates prior knowledge through the choice of $B$. We establish the well-posedness of the optimization problem, provide theoretical guarantees for the learning process, and present sample complexity bounds. The approach is demonstrated through theoretical infinite-dimensional examples, including compact perturbations of a known operator and the problem of learning the mother wavelet, and through extensive numerical simulations. This work extends previous efforts in Tikhonov regularization by addressing non-differentiable norms and proposing a data-driven approach for sparse regularization in infinite dimensions.