Learning sparsity-promoting regularizers for linear inverse problems
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910037181464576 |
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| author | Alberti, Giovanni S. De Vito, Ernesto Helin, Tapio Lassas, Matti Ratti, Luca Santacesaria, Matteo |
| author_facet | Alberti, Giovanni S. De Vito, Ernesto Helin, Tapio Lassas, Matti Ratti, Luca Santacesaria, Matteo |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16031 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Learning sparsity-promoting regularizers for linear inverse problems Alberti, Giovanni S. De Vito, Ernesto Helin, Tapio Lassas, Matti Ratti, Luca Santacesaria, Matteo Machine Learning Statistics Theory 65J22, 68T05 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. |
| title | Learning sparsity-promoting regularizers for linear inverse problems |
| topic | Machine Learning Statistics Theory 65J22, 68T05 |
| url | https://arxiv.org/abs/2412.16031 |