Fitting an Escalier to a Curve
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917023940870144 |
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| author | Bossu, Sebastien Papanicolaou, Andrew Hatto, Nour El |
| author_facet | Bossu, Sebastien Papanicolaou, Andrew Hatto, Nour El |
| contents | We analyze the problem of fitting a fonction en escalier or multi-step function to a curve in L^2 Hilbert space. We propose a two-stage optimization approach whereby the step positions are initially fixed, corresponding to a classic linear least-squares problem with closed-form solution, and then are allowed to vary, leading to first-order conditions that can be solved recursively. We find that, subject to regularity conditions, the speed of convergence is linear as the number of steps $n$ goes to infinity, and we develop a simple algorithm to recover the global optimum fit. Our numerical results based on a sweep search implementation show promising performance in terms of speed and accuracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fitting an Escalier to a Curve Bossu, Sebastien Papanicolaou, Andrew Hatto, Nour El Optimization and Control Functional Analysis 41A99, 41-04, 49K10, 65K10 We analyze the problem of fitting a fonction en escalier or multi-step function to a curve in L^2 Hilbert space. We propose a two-stage optimization approach whereby the step positions are initially fixed, corresponding to a classic linear least-squares problem with closed-form solution, and then are allowed to vary, leading to first-order conditions that can be solved recursively. We find that, subject to regularity conditions, the speed of convergence is linear as the number of steps $n$ goes to infinity, and we develop a simple algorithm to recover the global optimum fit. Our numerical results based on a sweep search implementation show promising performance in terms of speed and accuracy. |
| title | Fitting an Escalier to a Curve |
| topic | Optimization and Control Functional Analysis 41A99, 41-04, 49K10, 65K10 |
| url | https://arxiv.org/abs/2510.16148 |