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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.16031 |
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Table of Contents:
- 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.