Learning sparsity-promoting regularizers for linear inverse problems
This addresses the challenge of sparse regularization in infinite dimensions for researchers in inverse problems, offering a data-driven extension to Tikhonov regularization.
The paper tackles the problem of learning sparsity-promoting regularizers for linear inverse problems by developing a bilevel optimization framework to select an optimal synthesis operator, resulting in theoretical guarantees and sample complexity bounds.
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 examples, including compact perturbations of a known operator and the problem of learning the mother wavelet, showcasing its flexibility in incorporating prior knowledge into the regularization framework. 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.