{"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-vertex-enumeration-for-convex-hulls-in","title":"Robust Vertex Enumeration for Convex Hulls in High Dimensions","arxiv_id":"1802.01515","date":"2018-02-05","proceeding":null,"authors":["Pranjal Awasthi","Bahman Kalantari","Yikai Zhang"],"abstract":"Computation of the vertices of the convex hull of a set $S$ of $n$ points in $\\mathbb{R} ^m$ is a fundamental problem in computational geometry, optimization, machine learning and more. We present \"All Vertex Triangle Algorithm\" (AVTA), a robust and efficient algorithm for computing the subset $\\overline S$ of all $K$ vertices of $conv(S)$, the convex hull of $S$. If $\\Gamma_*$ is the minimum of the distances from each vertex to the convex hull of the remaining vertices, given any $\\gamma \\leq \\gamma_* = \\Gamma_*/R$, $R$ the diameter of $S$, $AVTA$ computes $\\overline S$ in $O(nK(m+ \\gamma^{-2}))$ operations. If $\\gamma_*$ is unknown but $K$ is known, AVTA computes $\\overline S$ in $O(nK(m+ \\gamma_*^{-2})) \\log(\\gamma_*^{-1})$ operations. More generally, given $t \\in (0,1)$, AVTA computes a subset $\\overline S^t$ of $\\overline S$ in $O(n |\\overline S^t|(m+ t^{-2}))$ operations, where the distance between any $p \\in conv(S)$ to $conv(\\overline S^t)$ is at most $t R$. Next we consider AVTA where input is $S_\\varepsilon$, an $\\varepsilon$ perturbation of $S$. Assuming a bound on $\\varepsilon$ in terms of the minimum of the distances of vertices of $conv(S)$ to the convex hull of the remaining point of $S$, we derive analogous complexity bounds for computing $\\overline S_\\varepsilon$. We also analyze AVTA under random projections of $S$ or $S_\\varepsilon$. Finally, via AVTA we design new practical algorithms for two popular machine learning problems: topic modeling and non-negative matrix factorization. For topic models AVTA leads to significantly better reconstruction of the topic-word matrix than state of the art approaches~\\cite{arora2013practical, bansal2014provable}. For non-negative matrix AVTA is competitive with existing methods~\\cite{arora2012computing}. Empirically AVTA is robust and can handle larger amounts of noise than existing methods.","url_abs":"http://arxiv.org/abs/1802.01515v2","url_pdf":"http://arxiv.org/pdf/1802.01515v2.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-vertex-enumeration-for-convex-hulls-in","repo_url":"https://github.com/yikaizhang/AVTA","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}