{"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/riemannian-stochastic-variance-reduced","title":"Riemannian stochastic variance reduced gradient on Grassmann manifold","arxiv_id":"1605.07367","date":"2016-05-24","proceeding":null,"authors":["Hiroyuki Kasai","Hiroyuki Sato","Bamdev Mishra"],"abstract":"Stochastic variance reduction algorithms have recently become popular for\nminimizing the average of a large, but finite, number of loss functions. In\nthis paper, we propose a novel Riemannian extension of the Euclidean stochastic\nvariance reduced gradient algorithm (R-SVRG) to a compact manifold search\nspace. To this end, we show the developments on the Grassmann manifold. The key\nchallenges of averaging, addition, and subtraction of multiple gradients are\naddressed with notions like logarithm mapping and parallel translation of\nvectors on the Grassmann manifold. We present a global convergence analysis of\nthe proposed algorithm with decay step-sizes and a local convergence rate\nanalysis under fixed step-size with some natural assumptions. The proposed\nalgorithm is applied on a number of problems on the Grassmann manifold like\nprincipal components analysis, low-rank matrix completion, and the Karcher mean\ncomputation. In all these cases, the proposed algorithm outperforms the\nstandard Riemannian stochastic gradient descent algorithm.","url_abs":"http://arxiv.org/abs/1605.07367v3","url_pdf":"http://arxiv.org/pdf/1605.07367v3.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":"riemannian-stochastic-variance-reduced","repo_url":"https://github.com/hiroyuki-kasai/RSOpt","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"low-rank-matrix-completion","task_name":"Low-Rank Matrix Completion"},{"task_slug":"matrix-completion","task_name":"Matrix Completion"},{"task_slug":"riemannian-optimization","task_name":"Riemannian optimization"},{"task_slug":"translation","task_name":"Translation"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1605.07367","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}