{"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/lattice-partition-recovery-with-dyadic-cart","title":"Lattice partition recovery with dyadic CART","arxiv_id":"2105.13504","date":"2021-05-27","proceeding":"NeurIPS 2021 12","authors":["Oscar Hernan Madrid Padilla","Yi Yu","Alessandro Rinaldo"],"abstract":"We study piece-wise constant signals corrupted by additive Gaussian noise over a $d$-dimensional lattice. Data of this form naturally arise in a host of applications, and the tasks of signal detection or testing, de-noising and estimation have been studied extensively in the statistical and signal processing literature. In this paper we consider instead the problem of partition recovery, i.e.~of estimating the partition of the lattice induced by the constancy regions of the unknown signal, using the computationally-efficient dyadic classification and regression tree (DCART) methodology proposed by \\citep{donoho1997cart}. We prove that, under appropriate regularity conditions on the shape of the partition elements, a DCART-based procedure consistently estimates the underlying partition at a rate of order $\\sigma^2 k^* \\log (N)/\\kappa^2$, where $k^*$ is the minimal number of rectangular sub-graphs obtained using recursive dyadic partitions supporting the signal partition, $\\sigma^2$ is the noise variance, $\\kappa$ is the minimal magnitude of the signal difference among contiguous elements of the partition and $N$ is the size of the lattice. Furthermore, under stronger assumptions, our method attains a sharper estimation error of order $\\sigma^2\\log(N)/\\kappa^2$, independent of $k^*$, which we show to be minimax rate optimal. Our theoretical guarantees further extend to the partition estimator based on the optimal regression tree estimator (ORT) of \\cite{chatterjee2019adaptive} and to the one obtained through an NP-hard exhaustive search method. We corroborate our theoretical findings and the effectiveness of DCART for partition recovery in simulations.","url_abs":"https://arxiv.org/abs/2105.13504v2","url_pdf":"https://arxiv.org/pdf/2105.13504v2.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":"lattice-partition-recovery-with-dyadic-cart","repo_url":"https://github.com/hernanmp/partition_recovery","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"regression-1","task_name":"regression"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}