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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.03461 |
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| _version_ | 1866929487842639872 |
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| author | Everink, Jasper Marijn Dong, Yiqiu Andersen, Martin Skovgaard |
| author_facet | Everink, Jasper Marijn Dong, Yiqiu Andersen, Martin Skovgaard |
| contents | In this work, we study the well-posedness of certain sparse regularized linear regression problems, i.e., the existence, uniqueness and continuity of the solution map with respect to the data. We focus on regularization functions that are convex piecewise linear, i.e., whose epigraph is polyhedral. This includes total variation on graphs and polyhedral constraints. We provide a geometric framework for these functions based on their connection to polyhedral sets and apply this to the study of the well-posedness of the corresponding sparse regularized linear regression problem. Particularly, we provide geometric conditions for well-posedness of the regression problem, compare these conditions to those for smooth regularization, and show the computational difficulty of verifying these conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03461 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Geometry and Well-Posedness of Sparse Regularized Linear Regression Everink, Jasper Marijn Dong, Yiqiu Andersen, Martin Skovgaard Statistics Theory Optimization and Control 52B99, 62J05, 90C31 In this work, we study the well-posedness of certain sparse regularized linear regression problems, i.e., the existence, uniqueness and continuity of the solution map with respect to the data. We focus on regularization functions that are convex piecewise linear, i.e., whose epigraph is polyhedral. This includes total variation on graphs and polyhedral constraints. We provide a geometric framework for these functions based on their connection to polyhedral sets and apply this to the study of the well-posedness of the corresponding sparse regularized linear regression problem. Particularly, we provide geometric conditions for well-posedness of the regression problem, compare these conditions to those for smooth regularization, and show the computational difficulty of verifying these conditions. |
| title | The Geometry and Well-Posedness of Sparse Regularized Linear Regression |
| topic | Statistics Theory Optimization and Control 52B99, 62J05, 90C31 |
| url | https://arxiv.org/abs/2409.03461 |