{"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/a-fast-numerical-method-for-max-convolution","title":"A fast numerical method for max-convolution and the application to efficient max-product inference in Bayesian networks","arxiv_id":"1501.02627","date":"2015-01-12","proceeding":null,"authors":["Oliver Serang"],"abstract":"Observations depending on sums of random variables are common throughout many\nfields; however, no efficient solution is currently known for performing\nmax-product inference on these sums of general discrete distributions\n(max-product inference can be used to obtain maximum a posteriori estimates).\nThe limiting step to max-product inference is the max-convolution problem\n(sometimes presented in log-transformed form and denoted as \"infimal\nconvolution\", \"min-convolution\", or \"convolution on the tropical semiring\"),\nfor which no O(k log(k)) method is currently known. Here I present a O(k\nlog(k)) numerical method for estimating the max-convolution of two nonnegative\nvectors (e.g., two probability mass functions), where k is the length of the\nlarger vector. This numerical max-convolution method is then demonstrated by\nperforming fast max-product inference on a convolution tree, a data structure\nfor performing fast inference given information on the sum of n discrete random\nvariables in O(n k log(n k) log(n) ) steps (where each random variable has an\narbitrary prior distribution on k contiguous possible states). The numerical\nmax-convolution method can be applied to specialized classes of hidden Markov\nmodels to reduce the runtime of computing the Viterbi path from n k^2 to n k\nlog(k), and has potential application to the all-pairs shortest paths problem.","url_abs":"http://arxiv.org/abs/1501.02627v2","url_pdf":"http://arxiv.org/pdf/1501.02627v2.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":"a-fast-numerical-method-for-max-convolution","repo_url":"https://bitbucket.org/orserang/fast-numerical-max-convolution","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[{"method_slug":"convolution","method_name":"Convolution"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}