{"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/an-alternative-to-em-for-gaussian-mixture","title":"An Alternative to EM for Gaussian Mixture Models: Batch and Stochastic Riemannian Optimization","arxiv_id":"1706.03267","date":"2017-06-10","proceeding":null,"authors":["Reshad Hosseini","Suvrit Sra"],"abstract":"We consider maximum likelihood estimation for Gaussian Mixture Models (Gmms).\nThis task is almost invariably solved (in theory and practice) via the\nExpectation Maximization (EM) algorithm. EM owes its success to various\nfactors, of which is its ability to fulfill positive definiteness constraints\nin closed form is of key importance. We propose an alternative to EM by\nappealing to the rich Riemannian geometry of positive definite matrices, using\nwhich we cast Gmm parameter estimation as a Riemannian optimization problem.\nSurprisingly, such an out-of-the-box Riemannian formulation completely fails\nand proves much inferior to EM. This motivates us to take a closer look at the\nproblem geometry, and derive a better formulation that is much more amenable to\nRiemannian optimization. We then develop (Riemannian) batch and stochastic\ngradient algorithms that outperform EM, often substantially. We provide a\nnon-asymptotic convergence analysis for our stochastic method, which is also\nthe first (to our knowledge) such global analysis for Riemannian stochastic\ngradient. Numerous empirical results are included to demonstrate the\neffectiveness of our methods.","url_abs":"http://arxiv.org/abs/1706.03267v1","url_pdf":"http://arxiv.org/pdf/1706.03267v1.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":"an-alternative-to-em-for-gaussian-mixture","repo_url":"https://github.com/utvisionlab/mixest","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"density-estimation","task_name":"Density Estimation"},{"task_slug":"riemannian-optimization","task_name":"Riemannian optimization"},{"task_slug":"parameter-estimation","task_name":"parameter estimation"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1706.03267","atlas_url":"https://app.syntology.ai/?focus=1706.03267","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}