{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/adaptive-piecewise-polynomial-estimation-via","title":"Adaptive piecewise polynomial estimation via trend filtering","arxiv_id":"1304.2986","date":"2013-04-10","proceeding":null,"authors":["Ryan J. Tibshirani"],"abstract":"We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev.\n51 (2009) 339-360] for nonparametric regression. The trend filtering estimate\nis defined as the minimizer of a penalized least squares criterion, in which\nthe penalty term sums the absolute $k$th order discrete derivatives over the\ninput points. Perhaps not surprisingly, trend filtering estimates appear to\nhave the structure of $k$th degree spline functions, with adaptively chosen\nknot points (we say ``appear'' here as trend filtering estimates are not really\nfunctions over continuous domains, and are only defined over the discrete set\nof inputs). This brings to mind comparisons to other nonparametric regression\ntools that also produce adaptive splines; in particular, we compare trend\nfiltering to smoothing splines, which penalize the sum of squared derivatives\nacross input points, and to locally adaptive regression splines [Ann. Statist.\n25 (1997) 387-413], which penalize the total variation of the $k$th derivative.\nEmpirically, we discover that trend filtering estimates adapt to the local\nlevel of smoothness much better than smoothing splines, and further, they\nexhibit a remarkable similarity to locally adaptive regression splines. We also\nprovide theoretical support for these empirical findings; most notably, we\nprove that (with the right choice of tuning parameter) the trend filtering\nestimate converges to the true underlying function at the minimax rate for\nfunctions whose $k$th derivative is of bounded variation. This is done via an\nasymptotic pairing of trend filtering and locally adaptive regression splines,\nwhich have already been shown to converge at the minimax rate [Ann. Statist. 25\n(1997) 387-413]. At the core of this argument is a new result tying together\nthe fitted values of two lasso problems that share the same outcome vector, but\nhave different predictor matrices.","url_abs":"http://arxiv.org/abs/1304.2986v2","url_pdf":"http://arxiv.org/pdf/1304.2986v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"adaptive-piecewise-polynomial-estimation-via","repo_url":"https://github.com/capolitsch/trendfiltering","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"regression-1","task_name":"regression"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1304.2986","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}