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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.Abstract: Principal component analysis (PCA) is an important method for processing high-dimensional data. In recent years, PCA models based on various norms have been extensively studied to improve the robustness. However, on the one hand, these algorithms do not consider the relationship between reconstruction error and covariance; on the other hand, they lack the uncertainty of considering the principal component to the data description. Aiming at these problems, this paper proposes a new robust PCA algorithm. Firstly, the $L_{2, p}$-norm is used to measure the reconstruction error and the description variance of the projection data. Based on the reconstruction error and the description variance, the adaptive probability error minimization model is established to calculate the uncertainty of the principal component's description of the data. Based on the uncertainty, the adaptive probability weighting PCA is established. The corresponding optimization method is designed. The experimental results of artificial data sets, UCI data sets and face databases show that RPCA-PW is superior than other PCA algorithms.
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Key words:
- Principle component analysis (PCA) /
- weighted principal component analysis (WPCA) /
- dimensionality reduction /
- robustness
1) 本文责任编委 白翔 -
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 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
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 
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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
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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
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