{"title":"四元张量低阶近似值","authors":"Alaeddine Zahir, Ahmed Ratnani, Khalide Jbilou","doi":"arxiv-2409.10724","DOIUrl":null,"url":null,"abstract":"In this paper, we propose a new approaches for low rank approximation of\nquaternion tensors\n\\cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method\nuses quasi-norms to approximate the tensor by a low-rank tensor using the\nQT-product \\cite{miao2023quaternion}, which generalizes the known L-product to\nN-mode quaternions. The second method involves Non-Convex norms to approximate\nthe Tucker and TT-rank for the completion problem. We demonstrate that the\nproposed methods can effectively approximate the tensor compared to the\nconvexifying of the rank, such as the nuclear norm. We provide theoretical\nresults and numerical experiments to show the efficiency of the proposed\nmethods in the Inpainting and Denoising applications.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"29 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Quaternion tensor low rank approximation\",\"authors\":\"Alaeddine Zahir, Ahmed Ratnani, Khalide Jbilou\",\"doi\":\"arxiv-2409.10724\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we propose a new approaches for low rank approximation of\\nquaternion tensors\\n\\\\cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method\\nuses quasi-norms to approximate the tensor by a low-rank tensor using the\\nQT-product \\\\cite{miao2023quaternion}, which generalizes the known L-product to\\nN-mode quaternions. The second method involves Non-Convex norms to approximate\\nthe Tucker and TT-rank for the completion problem. We demonstrate that the\\nproposed methods can effectively approximate the tensor compared to the\\nconvexifying of the rank, such as the nuclear norm. We provide theoretical\\nresults and numerical experiments to show the efficiency of the proposed\\nmethods in the Inpainting and Denoising applications.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"29 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.10724\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10724","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
In this paper, we propose a new approaches for low rank approximation of
quaternion tensors
\cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method
uses quasi-norms to approximate the tensor by a low-rank tensor using the
QT-product \cite{miao2023quaternion}, which generalizes the known L-product to
N-mode quaternions. The second method involves Non-Convex norms to approximate
the Tucker and TT-rank for the completion problem. We demonstrate that the
proposed methods can effectively approximate the tensor compared to the
convexifying of the rank, such as the nuclear norm. We provide theoretical
results and numerical experiments to show the efficiency of the proposed
methods in the Inpainting and Denoising applications.