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鲁棒自适应概率加权主成分分析

高云龙 罗斯哲 潘金艳 陈柏华 张逸松

高云龙, 罗斯哲, 潘金艳, 陈柏华, 张逸松. 鲁棒自适应概率加权主成分分析.自动化学报, 2021, 47(4): 825-838 doi: 10.16383/j.aas.c180743
引用本文: 高云龙, 罗斯哲, 潘金艳, 陈柏华, 张逸松. 鲁棒自适应概率加权主成分分析.自动化学报, 2021, 47(4): 825-838 doi: 10.16383/j.aas.c180743
Gao Yun-Long, Luo Si-Zhe, Pan Jin-Yan, Chen Bai-Hua, Zhang Yi-Song. Robust PCA using adaptive probability weighting. Acta Automatica Sinica, 2021, 47(4): 825-838 doi: 10.16383/j.aas.c180743
Citation: Gao Yun-Long, Luo Si-Zhe, Pan Jin-Yan, Chen Bai-Hua, Zhang Yi-Song. Robust PCA using adaptive probability weighting. Acta Automatica Sinica, 2021, 47(4): 825-838 doi: 10.16383/j.aas.c180743

鲁棒自适应概率加权主成分分析

doi: 10.16383/j.aas.c180743
基金项目: 

国家自然科学基金 61203176

福建省自然科学基金 2013J05098

福建省自然科学基金 2016J01756

详细信息
    作者简介:

    罗斯哲  厦门大学硕士研究生. 主要研究方向为机器学习与维数约简.E-mail: sizheluo@foxmail.com

    潘金艳  集美大学副教授. 主要研究方向为最优化方法和数据挖掘.E-mail: jypan@jmu.edu.cn

    陈柏华  厦门大学博士研究生. 主要研究方向为机器学习和数据降维.E-mail: chenbaihua001@163.com

    张逸松  厦门大学硕士研究生. 主要研究方向为机器学习和数据聚类.E-mail: yisongzhang@foxmail.com

    通讯作者:

    高云龙  厦门大学副教授. 2005年获得兰州大学计算机科学专业硕士学位. 2011年获得西安交通大学控制科学与工程专业博士学位. 主要研究方向为模式识别, 时间序列分析. 本文通信作者.E-mail: gaoyl@xmu.edu.cn

Robust PCA Using Adaptive Probability Weighting

Funds: 

National Natural Science Foundation of China 61203176

Fujian Provincial Natural Science Foundation 2013J05098

Fujian Provincial Natural Science Foundation 2016J01756

More Information
    Author Bio:

    LUO Si-Zhe  Master student at Xiamen University. His research interest covers machine learning and dimensionality reduction

    PAN Jin-Yan  Associate professor at Jimei University. Her research interest covers optimization method and data mining

    CHEN Bai-Hua  Ph. D. candidate at Xiamen University. His research interest covers machine learning and data dimensionality reduction

    ZHANG Yi-Song  Master student at Xiamen University. His research interest covers machine learning and data cluster analysis

    Corresponding author: GAO Yun-Long  Associate professor at Xiamen University. He received the master degree in computer science from Lanzhou University in 2005, and the Ph. D. degree in control science and engineering from Xian Jiaotong University in 2011. His research interest covers pattern recognition and time series analysis. Corresponding author of this paper
  • 摘要: 主成分分析(Principal component analysis, PCA) 是处理高维数据的重要方法. 近年来, 基于各种范数的PCA模型得到广泛研究, 用以提高PCA对噪声的鲁棒性. 但是这些算法一方面没有考虑重建误差和投影数据描述方差之间的关系; 另一方面也缺少确定样本点可靠性(不确定性)的度量机制. 针对这些问题, 本文提出一种新的鲁棒PCA模型. 首先采用$L_{2, p}$模来度量重建误差和投影数据的描述方差. 基于重建误差和描述方差之间的关系建立自适应概率误差极小化模型, 据此计算主成分对于数据描述的不确定性, 进而提出了鲁棒自适应概率加权PCA模型(RPCA-PW). 此外, 本文还设计了对应的求解优化方案. 对人工数据集、UCI数据集和人脸数据库的实验结果表明, RPCA-PW在整体上优于其他PCA算法.
    Recommended by Associate Editor BAI Xiang
    1)  本文责任编委 白翔
  • 图  1  人工数据集上的鲁棒性实验

    Fig.  1  Robustness experiment on artificial data set

    图  2  $p$在不同取值下的目标函数取值变化曲线

    Fig.  2  Objective function values under different $p$

    图  3  三种算法在双高斯人工数据集上的投影结果

    Fig.  3  Projection result of three algorithms on two-Gaussian artificial dataset

    图  4  Extended Yale B和AR人脸数据库原始图像与加入三种不同噪声后对应的图像(从上至下分别为黑白噪声块、高斯噪声和椒盐噪声

    Fig.  4  Original images in Extended Yale B and AR face database and the corresponding images with three different noise types (top to bottom are noise block, Gaussian noise and salt-and-pepper noise respectively)

    图  5  不同维度下的各算法平均识别准确率

    Fig.  5  Recognition accuracy with different reduced dimensions

    图  6  人脸重构效果图, 每一列从左到右依次是原图、PCA、PCA-L21、RPCA-OM、Angle PCA、HQ-PCA、$L{_{2, p}}$-PCA $p=0.5$、$L_{2, p}$-PCA $p=1$、RPCA-PW $p=0.5$、RPCA-PW $p=1$

    Fig.  6  Face reconstruction pictures, each column represents original image, PCA, PCA-L21, RPCA-OM, Angle PCA, HQ-PCA, $L_{2, p}$-PCA $p=0.5$, $L_{2, p}$-PCA $p=1$, RPCA-PW $p=0.5$ and RPCA-PW $p=1$ from left to right

    图  7  各算法对不同数据集的平均时间比较结果

    Fig.  7  Comparison of average iteration time for different data sets by different algorithms

    图  8  RPCA-PW收敛曲线

    Fig.  8  Convergence curves of RPCA-PW

    表  1  实验中使用的UCI数据集

    Table  1  UCI data sets used in the experiment

    数据集 维数 类别数 样本数
    Australian 14 2 690
    Cars 8 3 392
    Cleve 13 8 303
    Solar 12 6 323
    Zoo 19 7 101
    Control 60 6 600
    Crx 15 2 690
    Glass 9 6 214
    Iris 4 3 150
    Wine 13 3 178
    下载: 导出CSV

    表  2  UCI数据集上各算法的平均分类正确率(%)

    Table  2  Average classification accuracy of each algorithm on UCI data sets (%)

    数据集 Australian Cars Cleve Solar Zoo Control Crx Glass Iris Wine
    PCA 57.32 69.27 52.16 64.11 96.00 91.83 54.93 74.35 96.00 73.04
    PCA-$L$21 62.41 69.17 52.74 64.44 95.60 92.50 53.39 75.33 97.33 74.13
    RPCA-OM 63.04 69.93 52.16 65.27 95.00 92.83 56.96 74.35 96.00 74.15
    Angle PCA 60.39 69.43 52.16 65.60 94.00 84.28 59.59 73.94 95.53 74.15
    MaxEnt-PCA 60.14 69.17 52.45 64.76 95.10 92.15 59.32 74.42 95.33 73.59
    HQ-PCA 60.77 69.27 52.46 65.46 95.80 91.50 60.87 76.16 92.00 75.55
    $L{_{2, p}}$-PCA $p=0.5$ 60.68 69.60 52.84 65.42 93.40 92.10 60.29 73.56 96.47 73.59
    $p=1$ 62.39 69.58 52.16 64.51 94.90 92.50 59.43 73.73 96.67 73.59
    $p=1.5$ 62.81 69.43 52.16 65.28 95.00 92.50 58.20 75.57 96.33 73.04
    $p=2$ 62.32 69.43 52.16 64.11 96.00 91.83 54.93 75.82 96.00 73.59
    RPCA-PW $p=0.5 $ 63.77 71.21 53.76 66.85 95.00 92.50 58.84 74.35 95.33 74.15
    $p=1 $ 63.04 69.43 52.49 65.94 95.00 91.67 60.14 73.94 96.00 74.15
    $p=1.5$ 62.17 69.43 52.16 64.72 95.00 97.83 61.16 76.28 96.67 74.15
    $p=2$ 62.32 69.43 52.16 64.11 96.00 92.33 54.93 75.82 96.00 73.59
    下载: 导出CSV

    表  3  UCI数据集上各算法的重建误差

    Table  3  Reconstruction error of each algorithm on UCI data sets

    数据集 Australian Cars Cleve Solar Zoo Control Crx Glass Iris Wine
    PCA 197.53 1 979.82 5.47 2.97 1.43 115.06 6 265.42 37.91 0.67 24.49
    PCA-$L$21 197.21 1 979.16 6.90 2.90 1.56 132.84 6 545.72 38.59 1.56 34.44
    RPCA-OM 193.23 1 977.98 5.61 2.65 1.51 108.13 5 684.38 38.64 0.64 24.85
    Angle PCA 197.96 1 981.34 13.27 2.95 2.04 123.62 6 583.82 38.42 0.59 25.85
    MaxEnt-PCA 203.82 1 976.70 6.24 2.61 1.44 113.01 6 805.98 38.12 0.58 25.01
    HQ-PCA 195.79 1 929.26 5.92 2.37 1.39 131.35 6 987.68 36.38 4.31 25.85
    $L_{2, p}$-PCA $p=0.5$ 193.29 1 978.76 9.22 2.93 2.05 106.69 6 601.74 38.33 1.04 26.14
    $p=1$ 193.71 1 978.33 5.60 2.75 1.72 108.06 6 588.17 38.37 0.84 25.44
    $p=1.5 $ 193.50 1 977.93 5.50 2.69 1.44 109.69 6 807.38 37.82 0.72 25.05
    $ p=2$ 193.53 1 977.82 5.47 2.97 1.43 115.06 5 865.42 37.91 0.67 24.49
    RPCA-PW $p=0.5$ 192.93 1 976.94 5.75 2.95 1.57 106.11 6 467.83 38.43 0.52 24.12
    $p=1$ 193.27 1 979.96 6.84 2.84 1.83 110.46 6491.19 38.17 0.70 24.36
    $p=1.5$ 193.36 1 977.82 5.06 2.83 1.46 110.45 6 073.11 37.91 0.71 25.26
    $p=2$ 193.53 1 977.82 5.47 2.97 1.43 115.06 5 865.42 37.91 0.67 24.49
    下载: 导出CSV
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  • 收稿日期:  2018-11-07
  • 录用日期:  2019-04-26
  • 刊出日期:  2021-04-23

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