A short tour of operator learning theory: Convergence rates, statistical limits, and open questions
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866911475511066624 |
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| author | Brugiapaglia, Simone Franco, Nicola Rares Nelsen, Nicholas H. |
| author_facet | Brugiapaglia, Simone Franco, Nicola Rares Nelsen, Nicholas H. |
| contents | This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory. First, it reviews error bounds for empirical risk minimization with a focus on holomorphic operators and neural network approximations. Next, it illustrates fundamental performance limits in terms of sample size by adopting a minimax perspective and considering various notions of regularity beyond holomorphy. The paper ends with a discussion on the interplay between these two perspectives and related open questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00819 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A short tour of operator learning theory: Convergence rates, statistical limits, and open questions Brugiapaglia, Simone Franco, Nicola Rares Nelsen, Nicholas H. Numerical Analysis Machine Learning Statistics Theory 68T07 (Primary), 65J15, 62C20 (Secondary) This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory. First, it reviews error bounds for empirical risk minimization with a focus on holomorphic operators and neural network approximations. Next, it illustrates fundamental performance limits in terms of sample size by adopting a minimax perspective and considering various notions of regularity beyond holomorphy. The paper ends with a discussion on the interplay between these two perspectives and related open questions. |
| title | A short tour of operator learning theory: Convergence rates, statistical limits, and open questions |
| topic | Numerical Analysis Machine Learning Statistics Theory 68T07 (Primary), 65J15, 62C20 (Secondary) |
| url | https://arxiv.org/abs/2603.00819 |