郝儒儒,苏志勋.张量鲁棒主成分分析的增广拉格朗日交替方向方法[J].数学研究及应用,2017,37(3):367~378
张量鲁棒主成分分析的增广拉格朗日交替方向方法
Augmented Lagrangian Alternating Direction Method for Tensor RPCA
投稿时间:2017-02-03  最后修改时间:2017-03-24
DOI:10.3770/j.issn:2095-2651.2017.03.014
中文关键词:  张量鲁棒主成分分析  交替方向法  增广拉格朗日函数  高阶奇异值分解
英文关键词:tensor RPCA  alternating direction method  augmented Lagrangian function  high-order SVD
基金项目:国家自然科学基金(Grant Nos.61572099; 61320106008; 91230103),国家科技重大专项(Grant Nos.2013ZX04005021; 2014ZX04001011).
作者单位
郝儒儒 大连理工大学数学与科学学院, 辽宁 大连 116024 
苏志勋 大连理工大学数学与科学学院, 辽宁 大连 116024 
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中文摘要:
      张量的鲁棒主成分分析是将未知的一个低秩张量与一个稀疏张量从已知的它们的和中分离出来.因为在计算机视觉与模式识别中有着广阔的应用前景,该问题在近期成为学者们的研究热点.本文提出了一种针对张量鲁棒主成分分析的新的模型,并给出交替方向极小化的求解算法,在求解过程中给出了两种秩的调整策略.针对低秩分量本文对其全部各阶展开矩阵进行低秩矩阵分解,针对稀疏分量采用软阈值收缩的策略.无论目标低秩张量为精确低秩或近似低秩,本文所提方法均可适用.本文对算法给出了一定程度上的收敛性分析,即算法迭代过程中产生的任意收敛点均满足KKT条件.如果目标低秩张量为精确低秩,当迭代终止时可对输出结果进行基于高阶奇异值分解的修正.针对人工数据和真实视频数据的数值实验表明,与同类型算法相比,本文所提方法可以得到更好的结果.
英文摘要:
      Tensor robust principal component analysis (TRPCA) problem aims to separate a low-rank tensor and a sparse tensor from their sum. This problem has recently attracted considerable research attention due to its wide range of potential applications in computer vision and pattern recognition. In this paper, we propose a new model to deal with the TRPCA problem by an alternation minimization algorithm along with two adaptive rank-adjusting strategies. For the underlying low-rank tensor, we simultaneously perform low-rank matrix factorizations to its all-mode matricizations; while for the underlying sparse tensor, a soft-threshold shrinkage scheme is applied. Our method can be used to deal with the separation between either an exact or an approximate low-rank tensor and a sparse one. We established the subsequence convergence of our algorithm in the sense that any limit point of the iterates satisfies the KKT conditions. When the iteration stops, the output will be modified by applying a high-order SVD approach to achieve an exactly low-rank final result as the accurate rank has been calculated. The numerical experiments demonstrate that our method could achieve better results than the compared methods.
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