Spectral Function Space Learning and Numerical Linear Algebra Networks for Solving 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_ | 1866911996169945088 |
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| author | Aspri, Andrea Frischauf, Leon Scherzer, Otmar |
| author_facet | Aspri, Andrea Frischauf, Leon Scherzer, Otmar |
| contents | We consider solving a probably ill-conditioned linear operator equation, where the operator is not modeled by physical laws but is specified via training pairs (consisting of images and data) of the input-output relation of the operator. We derive a stable method for computing the operator, which consists of first a Gram-Schmidt orthonormalization of images and a principal component analysis of the data. This two-step algorithm provides a spectral decomposition of the linear operator. Moreover, we show that both Gram-Schmidt and principal component analysis can be written as a deep neural network, which relates this procedure to de-and encoder networks. Therefore, we call the two-step algorithm a linear algebra network. Finally, we provide numerical simulations showing the strategy is feasible for reconstructing spectral functions and for solving operator equations without explicitly exploiting the physical model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10690 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Function Space Learning and Numerical Linear Algebra Networks for Solving Linear Inverse Problems Aspri, Andrea Frischauf, Leon Scherzer, Otmar Numerical Analysis Functional Analysis We consider solving a probably ill-conditioned linear operator equation, where the operator is not modeled by physical laws but is specified via training pairs (consisting of images and data) of the input-output relation of the operator. We derive a stable method for computing the operator, which consists of first a Gram-Schmidt orthonormalization of images and a principal component analysis of the data. This two-step algorithm provides a spectral decomposition of the linear operator. Moreover, we show that both Gram-Schmidt and principal component analysis can be written as a deep neural network, which relates this procedure to de-and encoder networks. Therefore, we call the two-step algorithm a linear algebra network. Finally, we provide numerical simulations showing the strategy is feasible for reconstructing spectral functions and for solving operator equations without explicitly exploiting the physical model. |
| title | Spectral Function Space Learning and Numerical Linear Algebra Networks for Solving Linear Inverse Problems |
| topic | Numerical Analysis Functional Analysis |
| url | https://arxiv.org/abs/2408.10690 |