{"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/on-the-robustness-of-the-successive","title":"On the Robustness of the Successive Projection Algorithm","arxiv_id":"2411.16195","date":"2024-11-25","proceeding":null,"authors":["Giovanni Barbarino","Nicolas Gillis"],"abstract":"The successive projection algorithm (SPA) is a workhorse algorithm to learn the $r$ vertices of the convex hull of a set of $(r-1)$-dimensional data points, a.k.a. a latent simplex, which has numerous applications in data science. In this paper, we revisit the robustness to noise of SPA and several of its variants. In particular, when $r \\geq 3$, we prove the tightness of the existing error bounds for SPA and for two more robust preconditioned variants of SPA. We also provide significantly improved error bounds for SPA, by a factor proportional to the conditioning of the $r$ vertices, in two special cases: for the first extracted vertex, and when $r \\leq 2$. We then provide further improvements for the error bounds of a translated version of SPA proposed by Arora et al. (''A practical algorithm for topic modeling with provable guarantees'', ICML, 2013) in two special cases: for the first two extracted vertices, and when $r \\leq 3$. Finally, we propose a new more robust variant of SPA that first shifts and lifts the data points in order to minimize the conditioning of the problem. We illustrate our results on synthetic data.","url_abs":"https://arxiv.org/abs/2411.16195v2","url_pdf":"https://arxiv.org/pdf/2411.16195v2.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":"on-the-robustness-of-the-successive","repo_url":"https://gitlab.com/ngillis/robustSPA","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[{"method_slug":"set","method_name":"SET"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2411.16195","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}