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广义余弦二维主成分分析

王肖锋 陆程昊 郦金祥 刘军

王肖锋, 陆程昊, 郦金祥, 刘军. 广义余弦二维主成分分析. 自动化学报, 2022, 48(11): 2836−2851 doi: 10.16383/j.aas.c190392
引用本文: 王肖锋, 陆程昊, 郦金祥, 刘军. 广义余弦二维主成分分析. 自动化学报, 2022, 48(11): 2836−2851 doi: 10.16383/j.aas.c190392
Wang Xiao-Feng, Lu Cheng-Hao, Li Jin-Xiang, Liu Jun. Generalized cosine two-dimensional principal component analysis. Acta Automatica Sinica, 2022, 48(11): 2836−2851 doi: 10.16383/j.aas.c190392
Citation: Wang Xiao-Feng, Lu Cheng-Hao, Li Jin-Xiang, Liu Jun. Generalized cosine two-dimensional principal component analysis. Acta Automatica Sinica, 2022, 48(11): 2836−2851 doi: 10.16383/j.aas.c190392

广义余弦二维主成分分析

doi: 10.16383/j.aas.c190392
基金项目: 国家重点研发计划(2018AAA0103004), 天津市科技计划重大专项(20YFZCGX00550)资助
详细信息
    作者简介:

    王肖锋:博士, 天津理工大学机械工程学院副教授. 2018年获得河北工业大学工学博士学位. 主要研究方向为发育机器人, 模式识别与机器学习. 本文通信作者. E-mail: wangxiaofeng@tjut.edu.cn

    陆程昊:天津大学机械工程学院硕士研究生. 2019年获得天津理工大学机械电子工程学士学位. 主要研究方向为数据降维, 机器学习与机器人学. E-mail: chenghaolu_bit@163.com

    郦金祥:天津理工大学机械工程学院硕士研究生. 2018年获得天津理工大学机械工程学士学位. 主要研究方向为数据降维, 机器学习与机器人学. E-mail: lijinxiang_go@163.com

    刘军:博士, 天津理工大学机械工程学院教授. 2002获得日本名古屋大学工学博士学位. 主要研究方向为转子故障信号的特征提取与分类识别. E-mail: liujunjp@tjut.edu.cn

Generalized Cosine Two-dimensional Principal Component Analysis

Funds: Supported by National Key Research and Development Program of China (2018AAA0103004) and Tianjin Science and Technology Planed Key Project (20YFZCGX00550)
More Information
    Author Bio:

    WANG Xiao-Feng Ph.D., associate professor at the School of Mechanical Engineering, Tianjin University of Technology. He received his Ph.D. degree from Heibei University of Technology in 2018. His research interest covers developmental robotics, pattern recognition, and machine learning. Corresponding author of this paper

    LU Cheng-Hao Master student at the School of Mechanical Engineering, Tianjin University. He received his bachelor degree from Tianjin University of Technology in 2019. His research interest covers dimensionality reduction, machine learning, and robotics

    LI Jin-Xiang Master student at the School of Mechanical Engineering, Tianjin University of Technology. He received his bachelor degree from Tianjin University of Technology in 2018. His research interest covers dimensionality reduction, machine learning, and robotics

    LIU Jun Ph.D., professor at the School of Mechanical Engineering, Tianjin University of Technology. He received his Ph.D. degree from Nagoya University, Japan in 2002. His research interest covers feature extraction and recognition for rotor fault

  • 摘要: 主成分分析(Principal component analysis, PCA) 是一种广泛应用的特征提取与数据降维方法, 其目标函数采用L2范数距离度量方式, 对离群数据及噪声敏感. 而L1范数虽然能抑制离群数据的影响, 但其重构误差并不能得到有效控制. 针对上述问题, 综合考虑投影距离最大及重构误差较小的目标优化问题, 提出一种广义余弦模型的目标函数. 通过极大化矩阵行向量的投影距离与其可调幂的2范数之间的比值, 使得其在数据降维的同时提高了鲁棒性. 在此基础上提出广义余弦二维主成分分析(Generalized cosine two dimensional PCA, GC2DPCA), 给出了其迭代贪婪的求解算法, 并对其收敛性及正交性进行理论证明. 通过选择不同的可调幂参数, GC2DPCA可应用于广泛的含离群数据的鲁棒降维. 人工数据集及多个人脸数据集的实验结果表明, 本文算法在重构误差、相关性及分类率等性能方面均得到了提升, 具有较强的抗噪能力.
  • 图  1  广义余弦模型

    Fig.  1  Generalized cosine model

    图  2  人工数据集的平均重构误差

    Fig.  2  ARCE on the artificial datasets

    图  3  人脸数据集示例

    Fig.  3  Examples of face datasets

    图  4  40%遮盖噪声示例

    Fig.  4  Examples with 40% occluded noise

    图  5  60%遮盖噪声示例

    Fig.  5  Examples with 60% occluded noise

    图  6  哑噪声示例

    Fig.  6  Examples with dummy noise

    图  7  不同遮盖噪声下的平均重构误差

    Fig.  7  ARCE with different occluded noises

    图  8  不同哑噪声下的平均重构误差

    Fig.  8  ARCE with different dummy noises

    图  9  主成分相关性

    Fig.  9  Correlations between principal components

    表  1  Yale数据集下平均分类率 (40%遮盖噪声) (%)

    Table  1  Average classification rate under Yale dataset (40% occluded noise) (%)

    NPCPCAPCA-$L_1$2DPCA2DPCA-$L_1$2DPCA-
    $L_1$(non-greedy)
    Angle-
    2DPCA
    GC2DPCA
    S = 0.5
    GC2DPCA
    S = 1
    GC2DPCA
    S = 1.5
    GC2DPCA
    S = 2
    1078.977.884.484.472.973.384.284.784.784.9
    2075.376.482.283.174.270.983.383.383.383.3
    3072.073.179.380.074.273.180.780.280.280.2
    4071.671.676.277.173.874.077.878.077.878.0
    5072.071.874.475.174.275.875.375.876.276.0
    下载: 导出CSV

    表  2  ORL数据集下平均分类率 (40%遮盖噪声) (%)

    Table  2  Average classification rate under ORL dataset (40% occluded noise) (%)

    NPCPCAPCA-$L_1$2DPCA2DPCA-$L_1$2DPCA-
    $L_1$(non-greedy)
    Angle-
    2DPCA
    GC2DPCA
    S = 0.5
    GC2DPCA
    S = 1
    GC2DPCA
    S = 1.5
    GC2DPCA
    S = 2
    1079.979.388.188.177.280.388.388.588.688.5
    2081.280.485.985.98280.686.386.486.486.4
    3081.682.385.185.584.883.385.585.685.785.7
    4081.982.885.485.585.785.485.585.585.585.5
    5081.783.285.685.585.685.585.485.685.685.6
    下载: 导出CSV

    表  3  FERET数据集下平均分类率 (40%遮盖噪声) (%)

    Table  3  Average classification rate under FERET dataset (40% occluded noise) (%)

    NPCPCAPCA-$L_1$2DPCA2DPCA-$L_1$2DPCA-
    $L_1$(non-greedy)
    Angle-
    2DPCA
    GC2DPCA
    S = 0.5
    GC2DPCA
    S = 1
    GC2DPCA
    S = 1.5
    GC2DPCA
    S = 2
    1050.552.558.058.547.851.059.059.860.060.0
    2050.051.052.553.550.851.354.054.354.554.3
    3051.351.351.550.851.852.850.851.851.851.8
    4048.849.851.350.850.550.550.850.850.850.8
    5048.049.551.350.850.850.850.050.050.050.0
    下载: 导出CSV

    表  4  Yale数据集下平均分类率 (60%遮盖噪声) (%)

    Table  4  Average classification rate under Yale dataset (60% occluded noise) (%)

    NPCPCAPCA-$L_1$2DPCA2DPCA-$L_1$2DPCA-
    $L_1$(non-greedy)
    Angle-
    2DPCA
    GC2DPCA
    S = 0.5
    GC2DPCA
    S = 1
    GC2DPCA
    S = 1.5
    GC2DPCA
    S = 2
    1061.858.772.473.662.459.373.674.474.474.4
    2060.460.063.365.362.959.166.467.667.867.8
    3060.761.362.262.762.461.163.663.663.663.6
    4061.161.662.262.762.262.962.262.462.262.4
    5060.762.062.262.762.462.762.462.462.462.4
    下载: 导出CSV

    表  5  ORL数据集平均分类率 (60%遮盖噪声) (%)

    Table  5  Average classification rate under ORL dataset (60% occluded noise) (%)

    NPCPCAPCA-$L_1$2DPCA2DPCA-$L_1$2DPCA-
    $L_1$(non-greedy)
    Angle-
    2DPCA
    GC2DPCA
    S = 0.5
    GC2DPCA
    S = 1
    GC2DPCA
    S = 1.5
    GC2DPCA
    S = 2
    1066.065.073.674.472.167.074.674.774.674.7
    2068.167.773.474.271.370.874.174.374.374.2
    3068.769.473.774.272.472.974.474.474.374.4
    4069.070.074.474.473.575.074.574.474.374.4
    5069.270.274.174.474.274.474.374.574.474.4
    下载: 导出CSV

    表  6  FERET数据集平均分类率 (60%遮盖噪声) (%)

    Table  6  Average classification rate under FERET dataset (60% occluded noise) (%)

    NPCPCAPCA-$L_1$2DPCA2DPCA-$L_1$2DPCA-
    $L_1$(non-greedy)
    Angle-
    2DPCA
    GC2DPCA
    S = 0.5
    GC2DPCA
    S = 1
    GC2DPCA
    S = 1.5
    GC2DPCA
    S = 2
    1043.843.550.047.840.040.348.850.550.350.0
    2043.844.348.545.044.845.044.845.345.846.0
    3044.043.846.845.044.845.345.045.045.545.5
    4041.544.345.345.044.044.345.045.345.345.3
    5042.042.044.845.045.045.345.345.045.045.0
    下载: 导出CSV

    表  7  时间复杂度分析

    Table  7  Time complexity analysis

    算法 时间复杂度
    PCA ${{\rm{O}}}({Nh}^{2}{w}^{2}+{h}^{3}{w}^{2}) $
    PCA-$L_1$ ${ {\rm{O} } }({Nhwk}{t}_{{\rm{PCA}}\text{-}L_{1} })$
    2DPCA ${{\rm{O}}}({Nh}^{2}{w}+{h}^{3}) $
    2DPCA-$L_1$ ${ {\rm{O} } }({Nhwk} {t}_{{\rm{2DPCA}}\text{-}L_{1} })$
    2DPCA-$L_1$(non-greedy) ${{\rm{O}}}[({Nhw}^{2}+{wk}^{2}+{w}^{2}{k})$
    ${t}_{{\rm{2DPCA}}\text{-}L_{1}({\rm{non}}\text{-}{\rm{greedy}})}]$
    Angle-2DPCA ${ {\rm{O} } }[({Nhw}^{2}+{wk}^{2}+{w}^{2}{k}){t}_{{\rm{Angle2DPCA}}}]$
    GC2DPCA ${ {\rm{O} } }({Nhwkt}_{{\rm{GC2DPCA}}})$
    下载: 导出CSV

    表  8  各个算法所需时间对比(s)

    Table  8  Comparison of the required time by each algorithm (s)

    样本数PCAPCA-$L_1$2DPCA2DPCA-$L_1$2DPCA-$L_1$ (non-greedy)Angle-2DPCAGC2DPCA
    200.8550.5760.3102.8619.6725.5852.637
    401.6430.8820.3635.83417.5507.7535.413
    603.0721.3260.4528.61128.09616.5679.049
    804.7331.9400.50717.80035.89614.04014.883
    1006.7322.2260.62022.44988.67140.09218.517
    下载: 导出CSV
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出版历程
  • 收稿日期:  2019-05-20
  • 录用日期:  2019-10-01
  • 网络出版日期:  2022-09-16
  • 刊出日期:  2022-11-22

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