{"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/large-scale-log-determinant-computation","title":"Large-scale Log-determinant Computation through Stochastic Chebyshev Expansions","arxiv_id":"1503.06394","date":"2015-03-22","proceeding":null,"authors":["Insu Han","Dmitry Malioutov","Jinwoo Shin"],"abstract":"Logarithms of determinants of large positive definite matrices appear\nubiquitously in machine learning applications including Gaussian graphical and\nGaussian process models, partition functions of discrete graphical models,\nminimum-volume ellipsoids, metric learning and kernel learning. Log-determinant\ncomputation involves the Cholesky decomposition at the cost cubic in the number\nof variables, i.e., the matrix dimension, which makes it prohibitive for\nlarge-scale applications. We propose a linear-time randomized algorithm to\napproximate log-determinants for very large-scale positive definite and general\nnon-singular matrices using a stochastic trace approximation, called the\nHutchinson method, coupled with Chebyshev polynomial expansions that both rely\non efficient matrix-vector multiplications. We establish rigorous additive and\nmultiplicative approximation error bounds depending on the condition number of\nthe input matrix. In our experiments, the proposed algorithm can provide very\nhigh accuracy solutions at orders of magnitude faster time than the Cholesky\ndecomposition and Schur completion, and enables us to compute log-determinants\nof matrices involving tens of millions of variables.","url_abs":"http://arxiv.org/abs/1503.06394v1","url_pdf":"http://arxiv.org/pdf/1503.06394v1.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":"large-scale-log-determinant-computation","repo_url":"https://github.com/vlad17/runlmc","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}}],"tasks":[{"task_slug":"metric-learning","task_name":"Metric Learning"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1503.06394","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}