{"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/tensor-balancing-on-statistical-manifold","title":"Tensor Balancing on Statistical Manifold","arxiv_id":"1702.08142","date":"2017-02-27","proceeding":"ICML 2017 8","authors":["Mahito Sugiyama","Hiroyuki Nakahara","Koji Tsuda"],"abstract":"We solve tensor balancing, rescaling an Nth order nonnegative tensor by\nmultiplying N tensors of order N - 1 so that every fiber sums to one. This\ngeneralizes a fundamental process of matrix balancing used to compare matrices\nin a wide range of applications from biology to economics. We present an\nefficient balancing algorithm with quadratic convergence using Newton's method\nand show in numerical experiments that the proposed algorithm is several orders\nof magnitude faster than existing ones. To theoretically prove the correctness\nof the algorithm, we model tensors as probability distributions in a\nstatistical manifold and realize tensor balancing as projection onto a\nsubmanifold. The key to our algorithm is that the gradient of the manifold,\nused as a Jacobian matrix in Newton's method, can be analytically obtained\nusing the Moebius inversion formula, the essential of combinatorial\nmathematics. Our model is not limited to tensor balancing, but has a wide\napplicability as it includes various statistical and machine learning models\nsuch as weighted DAGs and Boltzmann machines.","url_abs":"http://arxiv.org/abs/1702.08142v3","url_pdf":"http://arxiv.org/pdf/1702.08142v3.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":"tensor-balancing-on-statistical-manifold","repo_url":"https://github.com/mahito-sugiyama/newton-balancing","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1702.08142","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}