Low Density Separation Density Sensitive Distance-based Spectral Clustering Algorithm
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摘要: 本文提出一种基于低密度分割密度敏感距离的谱聚类算法, 该算法首先使用低密度分割密度敏感距离计算相似度矩阵, 该距离测度通过指数函数和伸缩因子实现放大不同流形体数据间的距离和缩短同一流形体数据间距离的目的, 从而有效反映数据分布的全局一致性和局部一致性特征.另外, 算法通过增加相对密度敏感项来考虑数据的局部分布特征, 从而有效避免孤立噪声和"桥"噪声的影响.文中最后给出了基于SC (Scattering criteria)指标的k近邻图k值选取办法和基于谱熵贡献率的特征向量选取方法.实验部分, 讨论了参数选择对算法性能的影响并给出取值建议, 通过与其他流行谱聚类算法聚类结果的对比分析, 表明本文提出的基于低密度分割密度敏感距离的谱聚类算法聚类性能明显优于其他算法.Abstract: This paper proposes a low density separation density sensitive distance-based spectral clustering algorithm.First, the algorithm applies the low-density separation density sensitive distance to calculate the similarity matrix. By the exponential function and flexibility factor, we can achieve increasing the distance between difierent manifold data and decreasing the distance between the same manifold data, which can efiectively reflect the global consistency and local consistency of data distribution. In addition, by adding relative density sensitive term to take into account the local distribution characteristics of the data, isolated noise and "bridge" noise are efiectively avoided. Finally, we provide the method of selecting k-value of k nearest neighbor graph based on SC (Scattering criteria) index and the method of extracting eigenvector based on spectral entropy contribution rate. In the experimental part, the efiect of parameter selection on the performance of the proposed technique is discussed and some suggestions about the determination of the parameters are given. Compared with the state-of-the-art spectral clustering algorithms, the analysis results demonstrate that the proposed low density separation density sensitive distance-based spectral clustering algorithm performs well on artiflcial and UCI benchmark datasets
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Key words:
- Spectral clustering /
- low density separation /
- Euclidean distance /
- density sensitive /
- robustness
1) 本文责任编委 曾志刚 -
图 4 Cirlcetwopoints (Eyes)数据集相似度矩阵
Fig. 4 Cirlcetwopoints (Eyes) dataset similarity matrix
图 7 Spiral数据集5种谱聚类算法的聚类结果图
Fig. 7 Clustering results of five different spectral clustering algorithms on Spiral dataset
图 8 Spiral数据集前两个特征向量分布情况
Fig. 8 The first two eigenvectors distribution on Spiral dataset
图 9 Three-circles数据集5种谱聚类算法的聚类结果图
Fig. 9 Clustering results of five different spectral clustering algorithms on Three-circles dataset
图 10 Three-circles数据集前三个特征向量分布情况
Fig. 10 The first three eigenvectors distribution on Three-circles dataset
图 11 Ring数据集5种谱聚类算法的聚类结果图
Fig. 11 Clustering results of five different spectral clustering algorithms on Ring dataset
图 12 Ring数据集前两个特征向量分布情况
Fig. 12 The first two eigenvectors distribution on Ring dataset
图 13 Eyes数据集5种谱聚类算法的聚类结果图
Fig. 13 Clustering results of five different spectral clustering algorithms on Eyes dataset
图 14 Eyes数据集前三个特征向量分布情况
Fig. 14 The first three eigenvectors distribution on Eyes dataset
图 15 不同ρ值下8个人工数据集关于NMI的误差棒图
Fig. 15 NMI performance metrics on 8 synthetic datasets with different ρ values
图 16 不同ρ值下8个人工数据集关于RI的误差棒图
Fig. 16 RI performance metrics on 8 synthetic datasets with different ρ values
图 17 Square数据集的SC性能指标随k值变化图
Fig. 17 SC performance metric obtained by the proposed approach on Square dataset with different k values
图 18 Square数据集的NMI + RI性能指标随k值变化图
Fig. 18 NMI + RI performance metrics on Square dataset with different k values
图 19 k = 10时Square数据集聚类的结果图
Fig. 19 The clustering result on Square dataset when k = 10
图 21 IRIS数据集的NMI+RI性能指标随k值变化图
Fig. 21 NMI + RI performance metrics on IRIS dataset with different k values
图 23 WINE数据集的NMI+RI性能指标随k值变化图
Fig. 23 NMI + RI performance metrics on WINE dataset with different k values
图 24 5种算法的NMI性能指标鲁棒性比较结果
Fig. 24 Comparison results of NMI performance index of five algorithms robustness
图 25 5种算法的RI性能指标鲁棒性比较结果
Fig. 25 Comparison results of RI performance index of five algorithms robustness
表 1 不同谱聚类算法对8种人工合成数据集关于NMI和RI的统计结果
Table 1 Statistics of different clustering algorithms on eight synthetic datasets in terms of NMI and RI
数据集 NJW-SC DS-SC DP-SC DSGD-SC LDSDSD-SC Spiral 0.5635 ± 0 0.0290 ± 0 1 ± 0 0.0138 ± 0 1 ± 0 0.6720 ± 0 0.0390 ± 0 1 ± 0 8.0080× 10-6 ± 0 1 ± 0 Three-circles 1 ± 0 0.0456 ± 0 1 ± 0 0.0495 ± 0 1 ± 0 1 ± 0 0.0195 ± 0 1 ± 0 0.0069 ± 0 1 ± 0 Two-moons 0.9595 ± 0 0.1316 ± 0 1 ± 0 0.0334 ± 0 1 ± 0 0.9799 ± 0 0.1722 ± 0 1 ± 0 0.0013 ± 0 1 ± 0 Ring 1 ± 0 0.0277 ± 0 1 ± 0 0.0277 ± 0 1 ± 0 1 ± 0 0.0040 ± 0 1 ± 0 0.0040 ± 0 1 ± 0 Eyes 0.4998 ± 0 0.5913 ± 0 0.5117 ± 0 0.0508 ± 0 1 ± 0 0.4642 ± 0 0.6210 ± 0 0.4366 ± 0 0.0060 ± 0 1 ± 0 Three-Gausses 1 ± 0 1 ± 0 1 ± 0 0.0332 ± 0 1 ± 0 1 ± 0 1 ± 0 1 ± 0 4.4894× 10-5 ± 0 1 ± 0 Four-lines 1 ± 0 0.7677 ± 0 0.8955 ± 0 0.0293 ± 0 1 ± 0 1 ± 0 0.6729 ± 0 0.7725 ± 0 0.0005 ± 0 1 ± 0 Square 0.7978 ± 0 0.7830 ± 0 0.2335 ± 0.0017 0.2284 ± 0.1978 0.7636 ± 0 0.8348 ± 0 0.8227 ± 0 0.8817 ± 0.1113 0.2054 ± 0.1781 0.8038 ± 0 
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表 2 实验数据集描述
Table 2 Description of experimental datasets
数据集 属性 样本个数 聚类数 WINE 13 178 3 IRIS 4 150 3 BREAST CANCER 9 683 2 GLASS 10 214 6 CAR 12 84 8 VEHICLE 18 846 4 LETTER (A, B) 16 1 555 2 LETTER (C, D) 16 1 541 2
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表 3 不同谱聚类算法对8种UCI数据集关于NMI和RI的统计结果
Table 3 Statistics of different clustering algorithms oneight UCI datasets in terms of NMI and RI
数据集 NJW-SC DS-SC DP-SC DSGD-SC LDSDSD-SC WINE 0.8758 ± 0 0.8363 ± 0 0.8335 ± 0.1361 0.8399 ± 0.0640 0.8781 ± 0 (k = 20) 0.8974 ± 0 0.8515 ± 0 0.8116 ± 0.2470 0.8632 ± 0.0660 0.8991 ± 0 (k = 20) IRIS 0.6818 ± 0 0.7549 ± 0 0.7853 ± 0.0115 0.7552 ± 0.3187 0.8571 ± 0(k = 13) 0.6525 ± 0 0.7559 ± 0 0.7276 ± 0.1237 0.7251 ± 0.2611 0.8682 ± 0 (k = 13) BREAST CANCER 0.7903 ± 0 0.8143 ± 0 0.7467 ± 0.2135 0.7478 ± 0 0.7921 ± 0 (k = 20) 0.8796 ± 0 0.8909 ± 0 0.8755 ± 0.1306 0.9487 ± 0 0.8797 ± 0 (k = 20) GLASS 0.3065 ± 0 0.2692 ± 0 0.1913 ± 0.0157 0.1778 ± 0.1488 0.7411 ± 0 (k = 30) 0.1923 ± 0 0.1490 ± 0 0.1515 ± 0.0175 0.1031 ± 0.1196 0.6644 ± 0 (k = 30) CAR 0.6424 ± 0 0.7604 ± 0 0.6457 ± 0.0103 0.7126 ± 0 0.7711 ± 0 (k = 7) 0.3797 ± 0 0.5583 ± 0 0.5773 ± 0.0141 0.5571 ± 0 0.5697 ± 0 (k = 7) VEHICLE 0.1280 ± 0 0.2032 ± 0 0.1125 ± 0.0377 0.0462 ± 0.0220 0.2114 ± 0 (k = 15) 0.1006 ± 0 0.1563 ± 0 0.3759 ± 0.0435 0.0333 ± 0.0167 0.1783 ± 0 (k = 15) LETTER (A, B) 0.2593 ± 0 0.7216 ± 0 0.3103 ± 0.0016 0.7168 ± 0 0.7396 ± 0 (k = 25) 0.3367 ± 0 0.7658 ± 0 0.5667 ± 0.01005 0.5975 ± 0 0.7863 ± 0 (k = 25) LETTER (C, D) 0.0724 ± 0 0.6876 ± 0 0.5517 ± 0.1953 0.6131 ± 0 0.6535 ± 0 (k = 30) 0.0984 ± 0 0.7280 ± 0 0.6505 ± 0.0167 0.6578 ± 0 0.6864 ± 0 (k = 30)
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