一种张量补全的惯性Bregman邻近凸差算法

doi:10.15960/j.cnki.issn.1007-6093.2026.03.008

运筹学学报（中英文） ›› 2026, Vol. 30 ›› Issue (3): 133-152.doi: 10.15960/j.cnki.issn.1007-6093.2026.03.008

所属专题：运筹与管理前沿：模型、算法及其应用专辑

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一种张量补全的惯性Bregman邻近凸差算法

胡象鑫¹, 闫喜红^1†, 兰泽¹, 穆晓芳², 王川龙¹

1. 太原师范学院智能优化计算与区块链技术山西省重点实验室, 山西晋中 030619;
2. 山西能源学院计算机与信息工程系, 山西晋中 030619

收稿日期:2026-06-17 发布日期:2026-09-12
通讯作者: 闫喜红 E-mail:yanxihong@tynu.edu.cn
基金资助:
国家自然科学基金(No.12371381)

Inertial Bregman proximal DC algorithm for tensor completion

HU Xiangxin¹, YAN Xihong^1†, LAN Ze¹, MU Xiaofang², WANG Chuanlong¹

1. Key Laboratory of Intelligent Optimization Computing and Blockchain Technology, Taiyuan Normal University, Jinzhong 030619, Shanxi, China;
2. Department of Computer and Information Engineering, Shanxi Energy Institute, Jinzhong 030619, Shanxi, China

Received:2026-06-17 Published:2026-09-12

摘要/Abstract

摘要： 针对高维数据在采集、传输和存储过程中易出现条目缺失的问题,本文研究基于Tucker秩的张量补全问题。为减弱核范数松弛对较大奇异值的过度惩罚,并提高凸差模型的求解效率,本文在邻近凸差算法框架下提出一种惯性Bregman邻近凸差算法。该算法引入惯性外推策略,并以Bregman距离诱导的邻近项替代传统Frobenius范数邻近项,从而更灵活地刻画张量补全问题的结构信息。理论方面,在适当的水平集紧性假设和Kurdyka-Łojasiewicz性质下,证明了算法生成序列的收敛性,并说明其极限点为相应优化模型的稳定点。脑MRI图像恢复实验表明,与邻近凸差算法相比,所提算法具有更高的恢复精度;与高精度低秩张量补全算法相比,所提算法在计算时间上具有优势。

关键词: 张量补全, Tucker秩, 凸差模型, Bregman距离, 邻近梯度算法

Abstract: This paper studies the tensor completion problem based on Tucker rank for high-dimensional data whose entries are prone to missing during acquisition, transmission, and storage. To alleviate the over-penalization of large singular values caused by nuclear norm relaxation and improve the solution efficiency of the difference-of-convex model, an inertial Bregman proximal difference-of-convex algorithm is proposed within the proximal difference-of-convex algorithm framework. The proposed algorithm introduces an inertial extrapolation strategy and replaces the traditional Frobenius norm proximal term with a Bregman-distance-induced proximal term, thereby characterizing the structural information of tensor completion problems more flexibly. On the theoretical side, under a suitable compactness assumption on the level set and the Kurdyka-Łojasiewicz property, the convergence of the sequence generated by the algorithm is proved, and its limit points are shown to be stationary points of the corresponding optimization model. Experiments on brain MRI image recovery show that, compared with the proximal difference-of-convex algorithm, the proposed algorithm achieves higher recovery accuracy; compared with the high-accuracy low-rank tensor completion algorithm, the proposed algorithm has an advantage in computation time.

Key words: tensor completion, Tucker rank, DC model, Bregman distance, proximal gradient algorithm

中图分类号:

TP391.41

胡象鑫, 闫喜红, 兰泽, 穆晓芳, 王川龙. 一种张量补全的惯性Bregman邻近凸差算法[J]. 运筹学学报（中英文）, 2026, 30(3): 133-152.

HU Xiangxin, YAN Xihong, LAN Ze, MU Xiaofang, WANG Chuanlong. Inertial Bregman proximal DC algorithm for tensor completion[J]. Operations Research Transactions, 2026, 30(3): 133-152.

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一种张量补全的惯性Bregman邻近凸差算法

Inertial Bregman proximal DC algorithm for tensor completion

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