Adaptive Vector-Valued Splines for the Resolution of Inverse Problems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912468896317440 |
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| author | Guillemet, Vincent Unser, Michaël |
| author_facet | Guillemet, Vincent Unser, Michaël |
| contents | We introduce a general framework for the reconstruction of vector-valued functions from finite and possibly noisy data, acquired through a known measurement operator. The reconstruction is done by the minimization of a loss functional formed as the sum of a convex data fidelity functional and a total-variation-based regularizer involving a suitable matrix L of differential operators. Here, the total variation is a norm on the space of vector measures. These are split into two categories: inner, and outer norms. The minimization is performed over an infinite-dimensional Banach search space. When the measurement operator is weakstar-continuous over the search space, our main result is that the solution set of the loss functional is the closed convex hull of adaptive L-splines, with fewer knots than the number of measurements. We reveal the effect of the total-variation norms on the structure of the solutions and show that inner norms yield sparser solutions. We also provide an explicit description of the class of admissible measurement operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05014 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive Vector-Valued Splines for the Resolution of Inverse Problems Guillemet, Vincent Unser, Michaël Optimization and Control 41A15, 47A52, 46G10, 47N10, 93B28, 46N10 We introduce a general framework for the reconstruction of vector-valued functions from finite and possibly noisy data, acquired through a known measurement operator. The reconstruction is done by the minimization of a loss functional formed as the sum of a convex data fidelity functional and a total-variation-based regularizer involving a suitable matrix L of differential operators. Here, the total variation is a norm on the space of vector measures. These are split into two categories: inner, and outer norms. The minimization is performed over an infinite-dimensional Banach search space. When the measurement operator is weakstar-continuous over the search space, our main result is that the solution set of the loss functional is the closed convex hull of adaptive L-splines, with fewer knots than the number of measurements. We reveal the effect of the total-variation norms on the structure of the solutions and show that inner norms yield sparser solutions. We also provide an explicit description of the class of admissible measurement operators. |
| title | Adaptive Vector-Valued Splines for the Resolution of Inverse Problems |
| topic | Optimization and Control 41A15, 47A52, 46G10, 47N10, 93B28, 46N10 |
| url | https://arxiv.org/abs/2507.05014 |