Variational Tensor-Product Splines
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911280188620800 |
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| author | Guillemet, Vincent Unser, Michael |
| author_facet | Guillemet, Vincent Unser, Michael |
| contents | Multidimensional continuous-domain inverse problems are often solved by the minimization of a loss functional, formed as the sum of a data fidelity and a regularization. In this work, we present a new construction where the regularization is itself built as the sum of two terms: i) the M norm of the regularizing operator L1 b L2, with L1 and L2 being two one-dimensional differential operators; ii) a bounded-variation norm that regularizes on the infinite-dimensional nullspace of L1 b L2. In this construction, we show that the extreme points of the solution set are the tensor product of one-dimensional splines, with a number of atoms upper-bounded in term of the number of data points. Further, when the data of the inverse problem is localized, we reveal that the term ii) must take the form of a sum of bounded-variation norms, precomposed with partial derivative of different orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17791 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variational Tensor-Product Splines Guillemet, Vincent Unser, Michael Numerical Analysis Optimization and Control Multidimensional continuous-domain inverse problems are often solved by the minimization of a loss functional, formed as the sum of a data fidelity and a regularization. In this work, we present a new construction where the regularization is itself built as the sum of two terms: i) the M norm of the regularizing operator L1 b L2, with L1 and L2 being two one-dimensional differential operators; ii) a bounded-variation norm that regularizes on the infinite-dimensional nullspace of L1 b L2. In this construction, we show that the extreme points of the solution set are the tensor product of one-dimensional splines, with a number of atoms upper-bounded in term of the number of data points. Further, when the data of the inverse problem is localized, we reveal that the term ii) must take the form of a sum of bounded-variation norms, precomposed with partial derivative of different orders. |
| title | Variational Tensor-Product Splines |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2511.17791 |