{"ID":2854277,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.16148","arxiv_id":"2510.16148","title":"Fitting an Escalier to a Curve","abstract":"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.","short_abstract":"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, l...","url_abs":"https://arxiv.org/abs/2510.16148","url_pdf":"https://arxiv.org/pdf/2510.16148v1","authors":"[\"Sebastien Bossu\",\"Andrew Papanicolaou\",\"Nour El Hatto\"]","published":"2025-10-17T18:33:07Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.FA\"]","methods":"[]","has_code":false}