{"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/sets-clustering","title":"Sets Clustering","arxiv_id":"2003.04135","date":"2020-03-09","proceeding":"ICML 2020 1","authors":["Ibrahim Jubran","Murad Tukan","Alaa Maalouf","Dan Feldman"],"abstract":"The input to the \\emph{sets-$k$-means} problem is an integer $k\\geq 1$ and a set $\\mathcal{P}=\\{P_1,\\cdots,P_n\\}$ of sets in $\\mathbb{R}^d$. The goal is to compute a set $C$ of $k$ centers (points) in $\\mathbb{R}^d$ that minimizes the sum $\\sum_{P\\in \\mathcal{P}} \\min_{p\\in P, c\\in C}\\left\\| p-c \\right\\|^2$ of squared distances to these sets. An \\emph{$\\varepsilon$-core-set} for this problem is a weighted subset of $\\mathcal{P}$ that approximates this sum up to $1\\pm\\varepsilon$ factor, for \\emph{every} set $C$ of $k$ centers in $\\mathbb{R}^d$. We prove that such a core-set of $O(\\log^2{n})$ sets always exists, and can be computed in $O(n\\log{n})$ time, for every input $\\mathcal{P}$ and every fixed $d,k\\geq 1$ and $\\varepsilon \\in (0,1)$. The result easily generalized for any metric space, distances to the power of $z>0$, and M-estimators that handle outliers. Applying an inefficient but optimal algorithm on this coreset allows us to obtain the first PTAS ($1+\\varepsilon$ approximation) for the sets-$k$-means problem that takes time near linear in $n$. This is the first result even for sets-mean on the plane ($k=1$, $d=2$). Open source code and experimental results for document classification and facility locations are also provided.","url_abs":"https://arxiv.org/abs/2003.04135v1","url_pdf":"https://arxiv.org/pdf/2003.04135v1.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":"sets-clustering","repo_url":"https://github.com/muradtuk/Sets-clustering","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"},{"task_slug":"document-classification","task_name":"Document Classification"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2003.04135","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}