On Learning the Optimal Regularization Parameter in Inverse Problems
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917029606326272 |
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| author | Rodriguez, Jonathan Chirinos De Vito, Ernesto Molinari, Cesare Rosasco, Lorenzo Villa, Silvia |
| author_facet | Rodriguez, Jonathan Chirinos De Vito, Ernesto Molinari, Cesare Rosasco, Lorenzo Villa, Silvia |
| contents | Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require less a priori knowledge, but their theoretical analysis is limited. In this paper, we propose and study a statistical machine learning approach, based on empirical risk minimization. Our main contribution is a theoretical analysis, showing that, provided with enough data, this approach can reach sharp rates while being essentially adaptive to the noise and smoothness of the problem. Numerical simulations corroborate and illustrate the theoretical findings. Our results are a step towards grounding theoretically data-driven approaches to inverse problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_15845 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Learning the Optimal Regularization Parameter in Inverse Problems Rodriguez, Jonathan Chirinos De Vito, Ernesto Molinari, Cesare Rosasco, Lorenzo Villa, Silvia Statistics Theory Optimization and Control 65J20, 47N10, 65K10, 62G05 Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require less a priori knowledge, but their theoretical analysis is limited. In this paper, we propose and study a statistical machine learning approach, based on empirical risk minimization. Our main contribution is a theoretical analysis, showing that, provided with enough data, this approach can reach sharp rates while being essentially adaptive to the noise and smoothness of the problem. Numerical simulations corroborate and illustrate the theoretical findings. Our results are a step towards grounding theoretically data-driven approaches to inverse problems. |
| title | On Learning the Optimal Regularization Parameter in Inverse Problems |
| topic | Statistics Theory Optimization and Control 65J20, 47N10, 65K10, 62G05 |
| url | https://arxiv.org/abs/2311.15845 |