{"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/robust-topological-inference-distance-to-a","title":"Robust Topological Inference: Distance To a Measure and Kernel Distance","arxiv_id":"1412.7197","date":"2014-12-22","proceeding":null,"authors":["Frédéric Chazal","Brittany T. Fasy","Fabrizio Lecci","Bertrand Michel","Alessandro Rinaldo","Larry Wasserman"],"abstract":"Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the distance function (the distance of any point x to S). Given a sample from P we can infer the persistent homology using an empirical version of the distance function. However, the empirical distance function is highly non-robust to noise and outliers. Even one outlier is deadly. The distance-to-a-measure (DTM), introduced by Chazal et al. (2011), and the kernel distance, introduced by Phillips et al. (2014), are smooth functions that provide useful topological information but are robust to noise and outliers. Chazal et al. (2014) derived concentration bounds for DTM. Building on these results, we derive limiting distributions and confidence sets, and we propose a method for choosing tuning parameters.","url_abs":"https://arxiv.org/abs/1412.7197v1","url_pdf":"https://arxiv.org/pdf/1412.7197v1.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"robust-topological-inference-distance-to-a","repo_url":"https://github.com/MathieuCarriere/sklearn-tda","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"robust-topological-inference-distance-to-a","repo_url":"https://github.com/MathieuCarriere/sklearn_tda","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1412.7197","atlas_url":"https://app.syntology.ai/?focus=1412.7197","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}