{"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-ring-decomposition","title":"Tensor Ring Decomposition","arxiv_id":"1606.05535","date":"2016-06-17","proceeding":null,"authors":["Qibin Zhao","Guoxu Zhou","Shengli Xie","Liqing Zhang","Andrzej Cichocki"],"abstract":"Tensor networks have in recent years emerged as the powerful tools for\nsolving the large-scale optimization problems. One of the most popular tensor\nnetwork is tensor train (TT) decomposition that acts as the building blocks for\nthe complicated tensor networks. However, the TT decomposition highly depends\non permutations of tensor dimensions, due to its strictly sequential\nmultilinear products over latent cores, which leads to difficulties in finding\nthe optimal TT representation. In this paper, we introduce a fundamental tensor\ndecomposition model to represent a large dimensional tensor by a circular\nmultilinear products over a sequence of low dimensional cores, which can be\ngraphically interpreted as a cyclic interconnection of 3rd-order tensors, and\nthus termed as tensor ring (TR) decomposition. The key advantage of TR model is\nthe circular dimensional permutation invariance which is gained by employing\nthe trace operation and treating the latent cores equivalently. TR model can be\nviewed as a linear combination of TT decompositions, thus obtaining the\npowerful and generalized representation abilities. For optimization of latent\ncores, we present four different algorithms based on the sequential SVDs, ALS\nscheme, and block-wise ALS techniques. Furthermore, the mathematical properties\nof TR model are investigated, which shows that the basic multilinear algebra\ncan be performed efficiently by using TR representaions and the classical\ntensor decompositions can be conveniently transformed into the TR\nrepresentation. Finally, the experiments on both synthetic signals and\nreal-world datasets were conducted to evaluate the performance of different\nalgorithms.","url_abs":"http://arxiv.org/abs/1606.05535v1","url_pdf":"http://arxiv.org/pdf/1606.05535v1.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-ring-decomposition","repo_url":"https://github.com/zhaoxile/reproducible-tensor-completion-state-of-the-art","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"tensor-decomposition","task_name":"Tensor Decomposition"},{"task_slug":"tensor-networks","task_name":"Tensor Networks"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1606.05535","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}