在线咨询
中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2025, Vol. 42 ›› Issue (2): 329-342.doi: 10.3969/j.issn.1005-3085.2025.02.008

• • 上一篇    下一篇

带有光滑$\epsilon$-迹函数的双层张量补全模型

南佳琨1,2,  王川龙1,2   

  1. 1. 太原师范学院数学与统计学院，晋中  030619
    2. 太原师范学院 山西省智能优化计算与区块链技术重点实验室，晋中 030619
  • 收稿日期:2023-10-13 接受日期:2024-06-10 出版日期:2025-04-15 发布日期:2025-06-15
  • 通讯作者: 王川龙 E-mail: clwang1964@163.com
  • 基金资助:
    国家自然科学基金(12371381)；太原师范学院研究生教育创新项目(SYYJSYC-2316).

Tensor Completion via Bi-level Optimization Models with the Smooth $\epsilon$-trace Function

NAN Jiakun1,2,  WANG Chuanlong1,2   

  1. 1. School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong 030619
    2. Shanxi Key Laboratory for Intelligent Optimization Computing and Block-chain Technology, Taiyuan Normal University, Jinzhong 030619
  • Received:2023-10-13 Accepted:2024-06-10 Online:2025-04-15 Published:2025-06-15
  • Contact: C. Wang. E-mail address: clwang1964@163.com
  • Supported by:
    The National Natural Science Foundation of China (12371381); the Graduate Education Innovation Project of Taiyuan Normal University (SYYJSYC-2316).

摘要：

提出了一种新的张量补全模型，即用光滑$\epsilon$-迹函数来代替核范数的双层优化模型。在新模型中，不需要在每次迭代时对张量的所有模式矩阵进行奇异值分解，最多需要两次奇异值分解，有效减少了张量全部模展开带来的较大计算量，提高了计算效率。最后，通过随机张量补全和图像修复的数值实验结果表明，与传统的核范数模型相比，所提出的极小极小和极小极大组合模型双层优化具有较少的CPU时间和较好的精度。

关键词: 张量补全, 双层优化, $\epsilon$-迹函数, 极小极小, 极小极大

Abstract:

In this paper, the novel optimization model for tensor completion by considering the bi-level optimization model with the smooth $\epsilon$-trace functions instead of nuclear norm is proposed. In the new model, it is not necessary to perform singular value decomposition for all modes of the tensor in each iteration, but only two singular value decomposition is needed at most, which effectively reduces the huge computation amount brought by all modes expansion of the tensor and greatly improves the computation efficiency. Finally, the experimental results of randomly generated tensor completion problem and color image inpainting problem show that the proposed bi-level (minimin and minimax combination) optimization models usually has less than CPU time and better precision than the traditional nuclear norm based model.

Key words: tensor completion, bi-level optimization, $\epsilon$-trace function, minimin, minimax

中图分类号: