{"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/efficient-certifiably-optimal-clustering-with","title":"Efficient, Certifiably Optimal Clustering with Applications to Latent Variable Graphical Models","arxiv_id":"1806.00530","date":"2018-06-01","proceeding":null,"authors":["Carson Eisenach","Han Liu"],"abstract":"Motivated by the task of clustering either $d$ variables or $d$ points into\n$K$ groups, we investigate efficient algorithms to solve the Peng-Wei (P-W)\n$K$-means semi-definite programming (SDP) relaxation. The P-W SDP has been\nshown in the literature to have good statistical properties in a variety of\nsettings, but remains intractable to solve in practice. To this end we propose\nFORCE, a new algorithm to solve this SDP relaxation. Compared to the naive\ninterior point method, our method reduces the computational complexity of\nsolving the SDP from $\\tilde{O}(d^7\\log\\epsilon^{-1})$ to\n$\\tilde{O}(d^{6}K^{-2}\\epsilon^{-1})$ arithmetic operations for an\n$\\epsilon$-optimal solution. Our method combines a primal first-order method\nwith a dual optimality certificate search, which when successful, allows for\nearly termination of the primal method. We show for certain variable clustering\nproblems that, with high probability, FORCE is guaranteed to find the optimal\nsolution to the SDP relaxation and provide a certificate of exact optimality.\nAs verified by our numerical experiments, this allows FORCE to solve the P-W\nSDP with dimensions in the hundreds in only tens of seconds. For a variation of\nthe P-W SDP where $K$ is not known a priori a slight modification of FORCE\nreduces the computational complexity of solving this problem as well: from\n$\\tilde{O}(d^7\\log\\epsilon^{-1})$ using a standard SDP solver to\n$\\tilde{O}(d^{4}\\epsilon^{-1})$.","url_abs":"http://arxiv.org/abs/1806.00530v3","url_pdf":"http://arxiv.org/pdf/1806.00530v3.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":"efficient-certifiably-optimal-clustering-with","repo_url":"https://github.com/ceisenach/R_GFORCE","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"efficient-certifiably-optimal-clustering-with","repo_url":"https://github.com/cran/GFORCE","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}