基于截断核范数的低秩矩阵恢复算法研究
Low-Rank Matrix Recovery Algorithm Based on Truncated Nuclear Norm
DOI: 10.12677/pm.2026.166159, PDF,   
作者: 张志红*：太原师范学院数学与统计学院，山西 晋中；付亚茹#：太原师范学院数学与统计学院，山西 晋中；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，山西 晋中
关键词: 低秩矩阵恢复；截断核范数；奇异值分解；ALM；Low-Rank Matrix Recovery； Truncated Nuclear Norm； Singular Value Decomposition； ALM
摘要: 本文研究基于截断核范数(Truncated Nuclear Norm, TNN)的低秩矩阵恢复模型及其求解算法。传统核范数对所有奇异值施加均匀的软阈值惩罚，易造成大奇异值的过度收缩，影响图像主体结构的还原精度；截断核范数仅惩罚较小的奇异值，保留前r个主导奇异值不受惩罚，能更精准地刻画矩阵的低秩结构。本文针对稀疏随机噪声去除任务，构建了基于截断核范数的低秩矩阵恢复模型，并采用增广拉格朗日乘子法(ALM)设计了相应的求解算法Alg.1.在不同尺寸灰度图像上的实验结果表明，与基于标准核范数的ALM方法和加速近端梯度(APG)方法相比，本文所提算法在PSNR、SSIM与相对误差等指标上整体最优，视觉恢复效果更清晰。
Abstract: This paper investigates a low-rank matrix recovery model based on the Truncated Nuclear Norm (TNN) and its solution algorithm. The conventional nuclear norm imposes uniform soft-threshold shrinkage on all singular values, which easily causes excessive shrinkage of the dominant singular values and degrades the recovery accuracy of the principal image structure. In contrast, the truncated nuclear norm only penalizes the smaller singular values while keeping the first r leading singular values unaffected, thereby characterizing the low-rank structure more precisely. For the sparse random noise removal task, we construct a TNN-based low-rank recovery model and design a solution algorithm Alg.1 based on the Augmented Lagrange Multiplier (ALM) method. Numerical experiments on gray images of various sizes demonstrate that the proposed algorithm overall outperforms the standard nuclear-norm-based ALM and the Accelerated Proximal Gradient (APG) method in terms of PSNR, SSIM and relative error, producing clearer visual restoration.
文章引用：张志红, 付亚茹. 基于截断核范数的低秩矩阵恢复算法研究[J]. 理论数学, 2026, 16(6): 82-90. https://doi.org/10.12677/pm.2026.166159

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