多线性鲁棒主成分分析

doi:10.3969/j.issn.0372-2112.2014.08.004

PDF(3765 KB)

电子学报 ›› 2014, Vol. 42 ›› Issue (8) : 1480-1486. DOI: 10.3969/j.issn.0372-2112.2014.08.004

学术论文

多线性鲁棒主成分分析

史加荣¹, 周水生², 郑秀云¹

作者信息 +

Multilinear Robust Principal Component Analysis

SHI Jia-rong¹, ZHOU Shui-sheng², ZHENG Xiu-yun¹

Author information +

文章历史 +

摘要

鲁棒主成分分析（RPCA）是恢复低秩与稀疏成分的一种非常有效的方法.本文将RPCA推广到张量情形，提出了多线性鲁棒主成分分析（MRPCA）框架.首先建立了MRPCA模型，即最小化张量核范数与l₁范数的加权组合.然后使用增广拉格朗日乘子法求解上述张量核范数优化问题.实验结果证实：对于具有多线性结构的数据，MRPCA比RPCA更加鲁棒.

Abstract

Robust principal component analysis(RPCA)is a very effective method to recover both the low-rank and sparse components.This paper extends RPCA to the case of tensor and proposes a framework of multilinear robust principal component analysis(MRPCA).First,it establishes the model of MRPCA which minimizes a weighted combination of the tensor nuclear norm and l₁ norm.Then,it employs the augmented Lagrange multipliers algorithm to solve the above nuclear norm optimization problem.Experimental results demonstrate that MRPCA is more robust than RPCA for the data with multilinear structure.

导出引用

史加荣, 周水生, 郑秀云. 多线性鲁棒主成分分析[J]. 电子学报, 2014, 42(8): 1480-1486. https://doi.org/10.3969/j.issn.0372-2112.2014.08.004

SHI Jia-rong, ZHOU Shui-sheng, ZHENG Xiu-yun. Multilinear Robust Principal Component Analysis[J]. Acta Electronica Sinica, 2014, 42(8): 1480-1486. https://doi.org/10.3969/j.issn.0372-2112.2014.08.004

中图分类号： TP391

参考文献

[1] Wright J,Ganesh A,Rao S,et al.Robust principal component analysis:exact recovery of corrupted low-rank matrices via convex optimization[A].Proc Neural Information Processing Systems[C].British Columbia,Canada,2009.2080-2088.
[2] Candès E J,Li X,Ma Y,et al.Robust principal component analysis?[J].Journal of the ACM,2011,58(3):1-37.
[3] Recht B,Fazel M,Parrilo P A.Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization[J].SIAM Review,2010,52(3):471-501.
[4] Cai J F,Candès E J,Shen Z W.A singular value thresholding algorithm for matrix completion[J].SIAM Journal on Optimization,2010,20(4):1956-1982.
[5] Toh K C,Yun S.An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems[J].Pacific Journal of Optimization,2010,6(3):615-640.
[6] Lin Z C,Chen M,Wu L,et al.The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices[R].Technical Report UILU-ENG-09-2215,UIUC,October,2009.
[7] Yuan X,Yang J.Sparse and low-rank matrix decomposition via alternating direction methods[R].Dept of Mathematics,Hong Kong Baptist University,2009.
[8] Xu H,Caramanis C,Sanghavi S.Robust PCA via outlier pursuit[J].IEEE Transactions on Information Theory,2012,58(5):3047-3064.
[9] Tang G G,Nehorai A.The stability of low-rank matrix reconstruction:a constrained singular value view[J].IEEE Transactions on Information Theory,2012,58(9):6079-6092.
[10] Liu G C,Lin Z C,Yan S C,et al.Robust recovery of subspace structures by low-rank representation[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,2013,35(1):171-184.
[11] 胡正平,李静.基于低秩子空间恢复的联合稀疏表示人脸识别算法[J].电子学报,2013,41(5):987-991. Hu Zhengping,Li Jing.Face recognition of joint sparse representation based on low-rank subspace recovery[J].Acta Electronica Sinica,2013,41(5):987-991.(in Chinese)
[12] Yan S C,Xu D,Yang Q,et al.Multilinear discriminant analysis for face recognition[J].IEEE Transactions on Image Processing,2007,16(1):212-220.
[13] Shi J R,Jiao L C,Shang F H.Metric learning for high-dimensional tensor data[J].Chinese Journal of Electronics,2011,20(3):495-498.
[14] Kolda T G,Bader B W.Tensor decompositions and applications[J].SIAM Review,2009,51(3):455-500.
[15] Lu H,Plataniotis K N,Venetsanopoulos A N.MPCA:multilinear principal component analysis of tensor objects[J].IEEE Transactions on Neural Networks,2008,19(1):18-39.
[16] 温静,李洁,高新波.基于增量张量子空间学习的自适应目标跟踪[J].电子学报,2009,37(7):1618-1623. Wen Jing,Li Jie,Gao Xinbo.Adaptive object tracking with incremental tensor subspace learning[J].Acta Electronica Sinica,2009,37(7):1618-1623.(in Chinese)
[17] 史加荣,焦李成,尚凡华.张量补全算法及其在人脸识别中的应用[J].模式识别与人工智能,2011,24(2):255-261. Shi Jiarong,Jiao Licheng,Shang Fanhua.Tensor completion algorithm and its applications in face recognition[J].Pattern Recognition and Artifical Intelligence,2011,24(2):255-261.(in Chinese)
[18] Liu J,Musialski P,Wonka P,et al.Tensor completion for estimating missing values in visual data[J].IEEE Transactions on Pattern Analysis and Machine Intelligence,2013,35(1):208-220.
[19] 史加荣,郑秀云,魏宗田,等.低秩矩阵恢复算法综述[J].计算机应用研究,2013,30(6):1601-1605. Shi Jiarong,Zheng Xiuyun,Wei Zongtian,et al.Survey on algorithms of low-rank matrix recovery[J].Application Research of Computers,2013,30(6):1601-1605.(in Chinese)