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摘要: 主成分分析（Principle component analysis，PCA）是一种被广泛应用的降维方法.然而经典PCA的构造基于L2-模导致了其对离群点和噪声点敏感，同时经典PCA也不具备稀疏性的特点.针对此问题，本文提出基于Lp-模的稀疏主成分分析降维方法（LpSPCA）.LpSPCA通过极大化带有稀疏正则项的Lp-模样本方差，使得其在降维的同时保证了稀疏性和鲁棒性.LpSPCA可用简单的迭代算法求解，并且当p≥1时该算法的收敛性可在理论上保证.此外通过选择不同的p值，LpSPCA可应用于更广泛的数据类型.人工数据及人脸数据上的实验结果表明，本文所提出的LpSPCA不仅具有较好的降维效果，并且具有较强的抗噪能力.Abstract: Principle component analysis (PCA) is a widely applied dimensionality reduction method. However, the construction of classical PCA is based on L2-norm, which leads to its sensitivity to outliers and noises, as well as sparsity. To solve this problem, the paper proposes a sparse principal component analysis method based on Lp-norm for dimensionality reduction (LpSPCA). In particular, LpSPCA maximizes the Lp-norm variance with sparse regularization term, which ensures the sparseness and robustness while reducing dimensions. LpSPCA can be solved by a simple iterative algorithm, and its convergence is theoretically guaranteed when p≥1. Besides, by choosing a different p, LpSPCA can be used for more types of data sets. Experimental results on both synthetic and human face data sets demonstrate that the proposed LpSPCA not only has better dimensionality reduction ability but also has strong anti-noise property.
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Key words:
- Principal component analysis (PCA) /
- sparseness /
- robustness /
- dimensionality reduction /
- Lp-norm
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图 1 PCA和LpSPCA在人工数据集上所得到的第1个主成分及 将数据用LpSPCA、PCA、RSPCA、LpPCA投影到该主成分所张成空间上的重构误差
Fig. 1 The -rst principal components of the arti-cial data set obtained by classical PCA and LpSPCA, and reconstruction errors in the spaces spanned by the -rst principal components obtained by LpSPCA, PCA, RSPCA and LpPCA, respectively
图 2 稀疏度k对LpSPCA平均重构误差在带遮盖噪声和哑噪声的Yale 人脸数据上的影响
Fig. 2 The influence of parsity k to the ARCE of LpSPCA on the occluded or dummy Yale face database
图 3 带遮盖噪声的Yale人脸数据及LpSPCA、PCA、RSPCA和LpPCA在该数据上用前30个相应主成分的重构效果图及 将数据用各方法投影到该1 ~ 70个主成分上的重构误差
Fig. 3 Occluded Yale face database and face reconstruction pictures of the data set constructed by the first thirty principal components obtained by LpSPCA,PCA,RSPCA and LpPCA and reconstruction errors in the spaces spanned by 1 ~ 70 principal components obtained by PCA,RSPCA,LpPCA and LpSPCA,respectively
图 4 原始Yale人脸数据及LpSPCA、PCA、RSPCA和LpPCA 在带哑噪声的Yale数据上用前30个相应主成分的\\重构效果图及将数据用 各方法投影到该1 ~ 70个主成分上的重构误差
Fig. 4 Original Yale face database and face reconstruction pictures of its dummy data set constructed by the first thirty principal components obtained by LpSPCA,PCA,RSPCA and LpPCA and reconstruction errors in the spaces spanned by 1 ~ 70 principal components obtained by PCA,RSPCA,LpPCA and LpSPCA,respectively
表 1 PCA和RSPCA在人工数据集2上提取的前两个主成分
Table 1 The first two PCs extracted by PCA and RSPCA on artificial data set 2
PCA RSPCA PC1 PC2 PC1 PC2 0.0608 -0.1238 0 0.0998 -0.0527 0.0422 0 0.4954 0.0389 0.1514 0 0.4048 -0.1343 -0.1754 -0.3608 0 0.167 -0.1062 0.3679 0 0.2074 0.1841 0 0 0.2135 0.0237 0 0 -0.1254 0.1933 0.2713 0 
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表 2 LpPCA 在人工数据集2 上所提取的前两个主成分
Table 2 The first two PCs extracted by LpPCA on artificial data set 2
LpPCA (L0.5) LpPCA (L1) LpPCA (L1.5) LpPCA (L2) PC1 PC2 PC1 PC2 PC1 PC2 PC1 PC2 0.161 0.1896 -0.1924 -0.1657 0.0588 -0.1483 -0.1807 -0.1945 -0.1433 0.0855 0.1783 0.0865 -0.1833 -0.036 0.0399 0.1858 -0.1472 -0.1807 0.1511 -0.0542 -0.1771 0.0428 0.1957 -0.1054 0.0773 0.0125 -0.1056 0.131 -0.035 0.1067 0.2457 -0.0457 -0.0942 0.0566 0.1713 0.1395 -0.2173 -0.1576 -0.0184 0.0696 0.1948 -0.1646 0.05 -0.1428 -0.238 0.2375 0.1488 -0.1129 0.0831 0.1925 -0.1428 0.1499 0.2135 0.0899 0.0831 0.1925 -0.0991 0.1179 0.0085 -0.1306 -0.1254 0.1812 -0.0991 0.1179
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表 3 LpSPCA 在人工数据集2 上所提取的前两个主成分
Table 3 The first two PCs extracted by LpSPCA on artificial data set 2
LpPCA (L0.5) LpPCA (L1) LpPCA (L1.5) LpPCA (L2) PC1 PC2 PC1 PC2 PC1 PC2 PC1 PC2 0 0.2978 0 -0.4474 0 0.1285 0 0.1126 0 -0.674 0 -0.1696 0 0.4496 -0.0625 0 0 0.0282 0 -0.383 0 0.4220 0 0.4801 0.2726 0 0.2567 0 0.5954 0 0 0.4073 -0.4027 0 0.3741 0 -0.2469 0 -0.5845 0 0.3247 0 0.3693 0 0 0 -0.353 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2713 0 0 0
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