Projecting onto rectangular hyperbolic paraboloids in Hilbert space
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913617229643776 |
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| author | Bauschke, Heinz H. Lal, Manish Krishan Wang, Xianfu |
| author_facet | Bauschke, Heinz H. Lal, Manish Krishan Wang, Xianfu |
| contents | In $\mathbb{R}^3$, a hyperbolic paraboloid is a classical saddle-shaped quadric surface. Recently, Elser has modeled problems arising in Deep Learning using rectangular hyperbolic paraboloids in $\mathbb{R}^n$. Motivated by his work, we provide a rigorous analysis of the associated projection. In some cases, finding this projection amounts to finding a certain root of a quintic or cubic polynomial. We also observe when the projection is not a singleton and point out connections to graphical and set convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_04878 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Projecting onto rectangular hyperbolic paraboloids in Hilbert space Bauschke, Heinz H. Lal, Manish Krishan Wang, Xianfu Optimization and Control Functional Analysis Primary 41A50, 90C26, Secondary 15A63, 46C05 In $\mathbb{R}^3$, a hyperbolic paraboloid is a classical saddle-shaped quadric surface. Recently, Elser has modeled problems arising in Deep Learning using rectangular hyperbolic paraboloids in $\mathbb{R}^n$. Motivated by his work, we provide a rigorous analysis of the associated projection. In some cases, finding this projection amounts to finding a certain root of a quintic or cubic polynomial. We also observe when the projection is not a singleton and point out connections to graphical and set convergence. |
| title | Projecting onto rectangular hyperbolic paraboloids in Hilbert space |
| topic | Optimization and Control Functional Analysis Primary 41A50, 90C26, Secondary 15A63, 46C05 |
| url | https://arxiv.org/abs/2206.04878 |