基于一步张量学习的多视图子空间聚类

赵晓佳; 徐婷婷; 陈勇勇; 徐勇

doi:10.16383/j.aas.c220138

基于一步张量学习的多视图子空间聚类

doi: 10.16383/j.aas.c220138

赵晓佳^1,,
徐婷婷^1,,
陈勇勇^1,,
徐勇^{1, 2,}

1.
哈尔滨工业大学(深圳)深圳市视觉目标检测与判识重点实验室深圳 518055
2.
鹏城实验室深圳 518055

基金项目: 广东省自然科学基金 (2022A1515010819), 国家自然科学基金 (62106063), 深圳市科技创新委员会 (GJHZ20210705141812038) 资助

详细信息

作者简介:
赵晓佳：哈尔滨工业大学(深圳)计算机科学与技术学院硕士研究生. 主要研究方向为多视图聚类. E-mail: 21S151152@stu.hit.edu.cn

徐婷婷：哈尔滨工业大学(深圳)计算机科学与技术学院硕士研究生. 主要研究方向为低秩张量近似. E-mail: 21S151168@stu.hit.edu.cn

陈勇勇：哈尔滨工业大学(深圳)计算机科学与技术学院助理教授. 主要研究方向为机器学习和模式识别. 本文通信作者. E-mail: YongyongChen.cn@gmail.com

徐勇：哈尔滨工业大学(深圳)计算机科学与技术学院教授. 主要研究方向为机器学习, 模式学习, 生物信息学和视频分析. E-mail: yongxu@ymail.com

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出版历程
- 收稿日期: 2022-03-04
- 录用日期: 2022-08-22
- 网络出版日期: 2022-10-20
- 刊出日期: 2023-01-07

One-step Tensor Learning for Multi-view Subspace Clustering

1.
Shenzhen Key Laboratory of Visual Object Detection and Recognition, Harbin Institute of Technology, Shenzhen 518055
2.
Peng Cheng Laboratory, Shenzhen 518055

Funds: Supported by Guangdong Natural Science Foundation (2022A1515010819), National Natural Science Foundation of China (62106063), Shenzhen Science and Technology Innovation Committee (GJHZ20210705141812038)

More Information

Author Bio:
ZHAO Xiao-Jia　Master student at School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen. Her main research interest is multi-view clustering

XU Ting-Ting　Master student at School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen. Her main research interest is low-rank tensor approximation

CHEN Yong-Yong　Assistant professor at School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen. His research interest covers machine learning and pattern recognition. Corresponding author of this paper

XU Yong　Professor at School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen. His research interest covers machine learning, pattern recognition, biometrics, and video analysis

摘要

摘要: 现有多视图子空间聚类算法通常先进行张量表示学习, 进而将学习到的表示张量融合为统一的亲和度矩阵. 然而, 因其独立地学习表示张量和亲和度矩阵, 忽略了两者之间的高度相关性. 为了解决此问题, 提出一种基于一步张量学习的多视图子空间聚类方法, 联合学习表示张量和亲和度矩阵. 具体地, 该方法对表示张量施加低秩张量约束, 以挖掘视图的高阶相关性. 利用自适应最近邻法对亲和度矩阵进行灵活重建. 使用交替方向乘子法对模型进行优化求解, 通过对真实多视图数据的实验表明, 较于最新的多视图聚类方法, 提出的算法具有更好的聚类准确性.
- 多视图子空间聚类 /
- 张量奇异值分解 /
- 一步化学习 /
- 图学习
Abstract: A surge of the existing multi-view subspace clustering algorithms generally learn the third-order tensor representation first and then fuse the learned representation tensor into a unified affinity matrix. However, since they learn the representation tensor and the affinity matrix independently, they cannot seamlessly capture their high-order correlation. To address this challenge, we propose a novel multi-view subspace clustering method based on one-step tensor learning (OTSC) to jointly learn the representation tensor and affinity matrix. Specifically, we impose the low-rank tensor constraint on the representation tensor to explore the correlation of high-order cross-views dexterously, utilize the adaptive nearest neighbor strategy to reconstruct a flexible affinity matrix, and adopt the alternating direction method of multipliers (ADMM) to optimize our model. Extensive experiments on real multi-view data demonstrated the superiority of OTSC compared to the state-of-the-art methods.
- Multi-view subspace clustering /
- tensor singular value decomposition /
- one-step learning /
- graph learning

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图 1 多视图数据示例

Fig. 1 Examples of multi-view data

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图 2 张量奇异值分解示例

Fig. 2 Examples of t-SVD

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图 3 基于一步张量学习的多视图子空间聚类结构图

Fig. 3 The framework of the one-step tensor learning for multi-view subspace clustering (OTSC)

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图 4 OTSC性能的消融实验

Fig. 4 Ablation experiment of OTSC performance

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图 5 根据$ACC$和$NMI$调整数据集ORL参数$\alpha$与$\gamma$

Fig. 5 Parameters tuning in terms of $ACC$ and $NMI$ on ORL

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图 6 参数$K$对数据集ORL的$ACC$影响

Fig. 6 The effect of parameter $K$ on the $ACC$ of ORL.

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图 7 收敛性曲线 ((a) BBCSport; (b) Extended YaleB)

Fig. 7 The convergence curves and ACC versus iterations on ((a) BBCSport; (b) Extended YaleB)

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表 1 符号与定义

Table 1 Notations and definitions

符号	定义
$\boldsymbol{x}, X, {\cal{X}}$	向量, 矩阵, 张量
1	单位向量
$I$	单位矩阵
${\cal{I}}$	单位张量
$n$	样本个数
$V$	视图个数
$d_v$	第$v$个视图的特征维度
$X^v\in {\bf{R}}^{d_v \times n}$	第$v$个视图的特征矩阵
${\cal{Z}}\in{\bf{R}}^{n\times n\times V}$	表示张量
$A\in {\bf{R}}^{n \times n}$	亲和度矩阵
$E^v\in {\bf{R}}^{n \times n}$	噪声矩阵
$\\|\cdot\\|_{2,1}$	$l_{2,1}$范数
$\\|\cdot\\|_{\rm{F}}$	Frobenius范数
$\\|\cdot\\|_\infty$	无穷范数
$\\|\cdot\\|_{*}$	矩阵核范数
$\\|\cdot\\|_{\circledast}$	张量核范数
${\rm{FFT}}$	快速傅里叶分解

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表 2 真实多视图数据集信息

Table 2 Summary of all real-world multi-view databases

数据集	样本数量	类别	视图	种类
Extended YaleB	640	38	3	面部图像
ORL	400	40	3	面部图像
3Sources	169	6	3	新闻故事
BBCSport	544	5	2	新闻故事
UCI-Digits	2000	10	3	手写数字
COIL_20	1440	20	3	通用对象

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表 3 参数设置

Table 3 Parameter setting

数据集	$\alpha$	$\gamma$	$K$
Extended YaleB	1	0.005	5
ORL	0.1	0.05	12
3Sources	0.1	50	8
BBCSport	0.05	5	8
UCI-Digits	0.2	2	15
COIL_20	0.05	1	5

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表 4 数据集Extended YaleB、ORL的聚类结果

Table 4 Clustering results (mean$ \pm $standard deviation) on Extended YaleB and ORL

数据	类型	方法	$ACC$	$NMI$	$AR$	$F$-$score$	$Precision$	$Recall$
Extended YaleB	单视图方法	SSC_best	0.587±0.003	0.534±0.003	0.430±0.005	0.487±0.004	0.451±0.002	0.509±0.007
		LRR_best	0.615±0.013	0.627±0.040	0.451±0.002	0.508±0.004	0.481±0.002	0.539±0.001
		RSS_best	0.742±0.001	0.787±0.000	0.685±0.001	0.717±0.001	0.704±0.001	0.730±0.000
	多视图方法	RMSC	0.210±0.013	0.157±0.019	0.060±0.014	0.155±0.012	0.151±0.012	0.159±0.013
		DiMSC	0.615±0.003	0.636±0.002	0.453±0.005	0.504±0.006	0.481±0.004	0.534±0.004
		LT-MSC	0.626±0.010	0.637±0.003	0.459±0.030	0.521±0.006	0.485±0.001	0.539±0.002
		MLAN	0.346±0.011	0.352±0.015	0.093±0.009	0.213±0.023	0.159±0.018	0.321±0.013
		t-SVD	0.652±0.000	0.667±0.004	0.500±0.003	0.550±0.002	0.514±0.004	0.590±0.004
		GMC	0.434±0.000	0.449±0.000	0.157±0.000	0.265±0.000	0.204±0.000	0.378±0.000
		LMSC	0.598±0.005	0.568±0.004	0.354±0.007	0.423±0.006	0.390±0.006	0.463±0.005
		SCMV-3DT	0.410±0.001	0.413±0.002	0.185±0.002	0.276±0.001	0.244±0.002	0.318±0.001
		LRTG	0.954±0.000	0.905±0.000	0.899±0.000	0.909±0.000	0.908±0.000	0.911±0.000
		WTNNM	0.648±0.005	0.661±0.002	0.501±0.000	0.552±0.000	0.533±0.000	0.573±0.000
		GLTA	0.571±0.002	0.630±0.005	0.510±0.005	0.560±0.004	0.544±0.004	0.576±0.006
	本方法	OTSC	0.969±0.001	0.934±0.001	0.931±0.002	0.937±0.002	0.935±0.002	0.939±0.002
	本方法	WOTSC	0.972±0.000	0.943±0.000	0.938±0.000	0.944±0.000	0.942±0.000	0.946±0.000
ORL	单视图方法	SSC_best	0.765±0.008	0.893±0.007	0.694±0.013	0.682±0.012	0.673±0.007	0.764±0.005
		LRR_best	0.773±0.003	0.895±0.006	0.724±0.020	0.731±0.004	0.701±0.001	0.754±0.002
		RSS_best	0.846±0.024	0.938±0.007	0.798±0.023	0.803±0.023	0.759±0.030	0.852±0.017
	多视图方法	RMSC	0.723±0.007	0.872±0.012	0.645±0.003	0.654±0.007	0.607±0.009	0.709±0.004
		DiMSC	0.838±0.001	0.940±0.003	0.802±0.000	0.807±0.003	0.764±0.012	0.856±0.004
		LT-MSC	0.795±0.007	0.930±0.003	0.750±0.003	0.768±0.004	0.766±0.009	0.837±0.005
		MLAN	0.705±0.02	0.854±0.018	0.384±0.010	0.376±0.015	0.254±0.021	0.721±0.020
		t-SVD	0.970±0.003	0.993±0.002	0.967±0.002	0.968±0.003	0.946±0.004	0.991±0.003
		GMC	0.633±0.000	0.857±0.000	0.337±0.000	0.360±0.000	0.232±0.000	0.801±0.000
		LMSC	0.877±0.024	0.949±0.006	0.839±0.022	0.843±0.021	0.806±0.027	0.884±0.017
		SCMV-3DT	0.839±0.012	0.908±0.007	0.763±0.018	0.769±0.017	0.747±0.020	0.792±0.016
		LRTG	0.933±0.003	0.970±0.002	0.905±0.005	0.908±0.005	0.888±0.004	0.928±0.007
		WTNNM	0.967±0.000	0.992±0.000	0.960±0.000	0.952±0.000	0.946±0.000	0.968±0.000
		GLTA	0.976±0.002	0.994±0.006	0.958±0.024	0.963±0.019	0.952±0.035	0.989±0.012
	本方法	OTSC	0.983±0.002	0.988±0.001	0.964±0.003	0.965±0.003	0.958±0.004	0.972±0.001
	本方法	WOTSC	0.938±0.000	0.972±0.000	0.907±0.000	0.909±0.000	0.885±0.000	0.936±0.000

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表 5 数据集3Sources、UCI-Digits的聚类结果

Table 5 Clustering results (mean $ \pm $ standard deviation) on 3Sources and UCI-Digits

数据	类型	方法	$ACC$	$NMI$	$AR$	$F$-$score$	$Precision$	$Recall$
3Sources	单视图方法	SSC_best	0.762±0.003	0.694±0.003	0.658±0.004	0.743±0.003	0.769±0.001	0.719±0.005
		LRR_best	0.647±0.033	0.542±0.018	0.486±0.028	0.608±0.033	0.594±0.031	0.636±0.096
		RSS_best	0.722±0.000	0.601±0.000	0.533±0.000	0.634±0.000	0.679±0.000	0.595±0.000
	多视图方法	RMSC	0.583±0.022	0.630±0.011	0.455±0.031	0.557±0.025	0.635±0.029	0.497±0.028
		DiMSC	0.795±0.004	0.727±0.010	0.661±0.005	0.748±0.004	0.711±0.005	0.788±0.003
		LT-MSC	0.781±0.000	0.698±0.003	0.651±0.003	0.734±0.002	0.716±0.008	0.754±0.005
		MLAN	0.775±0.015	0.676±0.005	0.580±0.008	0.666±0.007	0.756±0.003	0.594±0.009
		t-SVD	0.781±0.000	0.678±0.000	0.658±0.000	0.745±0.000	0.683±0.000	0.818±0.000
		GMC	0.693±0.000	0.622±0.000	0.443±0.000	0.605±0.000	0.484±0.000	0.804±0.000
		LMSC	0.912±0.006	0.826±0.007	0.842±0.011	0.887±0.008	0.873±0.007	0.877±0.012
		SCMV-3DT	0.440±0.020	0.386±0.009	0.226±0.012	0.411±0.009	0.399±0.012	0.425±0.016
		LRTG	0.947±0.000	0.865±0.000	0.881±0.000	0.909±0.000	0.911±0.000	0.906±0.000
		WTNNM	0.793±0.000	0.692±0.000	0.679±0.000	0.761±0.010	0.693±0.000	0.845±0.000
		GLTA	0.859±0.008	0.753±0.015	0.713±0.014	0.775±0.011	0.827±0.009	0.730±0.013
	本方法	OTSC	0.953±0.000	0.880±0.000	0.893±0.000	0.918±0.000	0.914±0.000	0.922±0.000
	本方法	WOTSC	0.947±0.000	0.867±0.000	0.888±0.000	0.914±0.000	0.909±0.000	0.920±0.000
UCI-Digits	单视图方法	SSC_best	0.815±0.011	0.840±0.001	0.770±0.005	0.794±0.004	0.747±0.010	0.848±0.004
		LRR_best	0.871±0.001	0.768±0.002	0.736±0.002	0.763±0.002	0.759±0.002	0.767±0.002
		RSS_best	0.819±0.000	0.863±0.000	0.787±0.000	0.810±0.000	0.756±0.000	0.872±0.000
	多视图方法	RMSC	0.915±0.024	0.822±0.008	0.789±0.014	0.811±0.012	0.797±0.017	0.826±0.006
		DiMSC	0.703±0.010	0.772±0.006	0.652±0.006	0.695±0.006	0.673±0.005	0.718±0.007
		LT-MSC	0.803±0.001	0.775±0.001	0.725±0.001	0.753±0.001	0.739±0.001	0.767±0.001
		MLAN	0.874±0.000	0.910±0.000	0.847±0.000	0.864±0.000	0.797±0.000	0.943±0.000
		t-SVD	0.955±0.000	0.932±0.000	0.924±0.000	0.932±0.000	0.930±0.000	0.934±0.000
		GMC	0.736±0.000	0.815±0.000	0.678±0.000	0.713±0.000	0.644±0.000	0.799±0.000
		LMSC	0.893±0.000	0.815±0.000	0.783±0.000	0.805±0.000	0.798±0.000	0.812±0.000
		SCMV-3DT	0.930±0.001	0.861±0.001	0.846±0.001	0.861±0.001	0.859±0.001	0.864±0.001
		LRTG	0.981±0.000	0.953±0.000	0.957±0.000	0.961±0.000	0.961±0.000	0.962±0.000
		WTNNM	0.998±0.000	0.993±0.000	0.994±0.000	0.995±0.010	0.998±0.000	0.995±0.000
		GLTA	0.997±0.000	0.992±0.000	0.993±0.000	0.994±0.000	0.994±0.000	0.994±0.000
	本方法	OTSC	0.983±0.001	0.958±0.001	0.962±0.001	0.966±0.001	0.965±0.000	0.966±0.002
	本方法	WOTSC	0.983±0.000	0.958±0.000	0.962±0.000	0.966±0.000	0.965±0.000	0.966±0.000

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表 6 数据集BBCSport、COIL-20的聚类结果

Table 6 Clustering results (mean $ \pm $ standard deviation) on BBCSport and COIL-20

数据	类型	方法	$ACC$	$NMI$	$AR$	$F$-$score$	$Precision$	$Recall$
BBCSport	单视图方法	SSC_best	0.627±0.003	0.534±0.008	0.364±0.007	0.565±0.005	0.427±0.004	0.834±0.004
		LRR_best	0.836±0.001	0.698±0.002	0.705±0.001	0.776±0.001	0.768±0.001	0.784±0.001
		RSS_best	0.878±0.000	0.714±0.000	0.717±0.000	0.784±0.000	0.787±0.000	0.782±0.000
	多视图方法	RMSC	0.826±0.001	0.666±0.001	0.637±0.001	0.719±0.001	0.766±0.001	0.677±0.001
		DiMSC	0.922±0.000	0.785±0.000	0.813±0.000	0.858±0.000	0.846±0.000	0.872±0.000
		LT-MSC	0.460±0.046	0.222±0.028	0.167±0.043	0.428±0.014	0.328±0.028	0.629±0.053
		MLAN	0.721±0.000	0.779±0.000	0.591±0.000	0.714±0.000	0.567±0.000	0.962±0.000
		t-SVD	0.879±0.000	0.765±0.000	0.784±0.000	0.834±0.000	0.863±0.000	0.807±0.000
		GMC	0.807±0.000	0.760±0.000	0.722±0.000	0.794±0.000	0.727±0.000	0.875±0.000
		LMSC	0.847±0.003	0.739±0.001	0.749±0.001	0.810±0.001	0.799±0.001	0.822±0.001
		SCMV-3DT	0.980±0.000	0.929±0.000	0.935±0.000	0.950±0.000	0.959±0.000	0.942±0.000
		LRTG	0.943±0.005	0.869±0.009	0.840±0.012	0.879±0.000	0.866±0.006	0.892±0.014
		WTNNM	0.963±0.000	0.900±0.000	0.908±0.000	0.930±0.000	0.950±0.000	0.911±0.000
		GLTA	1.000±0.000	1.000±0.000	1.000±0.000	1.000±0.000	1.000±0.000	1.000±0.000
	本方法	OTSC	0.970±0.000	0.914±0.000	0.911±0.000	0.933±0.000	0.928±0.000	0.937±0.000
	本方法	WOTSC	0.985±0.000	0.950±0.000	0.957±0.000	0.967±0.000	0.963±0.000	0.971±0.000
COIL-20	单视图方法	SSC_best	0.803±0.022	0.935±0.009	0.798±0.022	0.809±0.013	0.734±0.027	0.804±0.028
		LRR_best	0.761±0.003	0.829±0.006	0.720±0.020	0.734±0.006	0.717±0.003	0.751±0.002
		RSS_best	0.837±0.012	0.930±0.006	0.789±0.005	0.800±0.005	0.717±0.012	0.897±0.017
	多视图方法	RMSC	0.685±0.045	0.800±0.017	0.637±0.044	0.656±0.042	0.620±0.057	0.698±0.026
		DiMSC	0.778±0.022	0.846±0.002	0.732±0.005	0.745±0.005	0.739±0.007	0.751±0.003
		LT-MSC	0.804±0.011	0.860±0.002	0.748±0.004	0.760±0.007	0.741±0.009	0.776±0.006
		MLAN	0.862±0.011	0.961±0.004	0.835±0.006	0.844±0.013	0.758±0.008	0.953±0.007
		t-SVD	0.830±0.000	0.884±0.005	0.786±0.003	0.800±0.004	0.785±0.007	0.808±0.001
		GMC	0.791±0.001	0.941±0.000	0.782±0.000	0.794±0.000	0.694±0.000	0.929±0.000
		LMSC	0.806±0.013	0.862±0.007	0.765±0.014	0.776±0.013	0.770±0.013	0.783±0.013
		SCMV-3DT	0.701±0.028	0.810±0.009	0.635±0.003	0.654±0.029	0.614±0.039	0.702±0.018
		LRTG	0.927±0.000	0.976±0.000	0.928±0.000	0.932±0.000	0.905±0.000	0.961±0.000
		WTNNM	0.902±0.000	0.945±0.000	0.893±0.000	0.898±0.010	0.897±0.000	0.900±0.000
		GLTA	0.903±0.006	0.946±0.001	0.891±0.007	0.897±0.006	0.893±0.013	0.900±0.001
	本方法	OTSC	0.936±0.004	0.983±0.004	0.938±0.006	0.941±0.006	0.906±0.007	0.979±0.006
	本方法	WOTSC	0.960±0.026	0.976±0.004	0.934±0.025	0.938±0.024	0.918±0.042	0.959±0.004

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