基于低秩张量图学习的不完整多视角聚类

文杰; 颜珂; 张正; 徐勇

doi:10.16383/j.aas.c200519

基于低秩张量图学习的不完整多视角聚类

doi: 10.16383/j.aas.c200519

文杰^1,,
颜珂^2,,
张正^{1, 3,},
徐勇^{1, 3,}

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.
北京理工大学计算机科学与技术学院北京 100080
3.
鹏城实验室深圳 518055

基金项目: 深圳市基础研究项目(JCYJ20190806142416685), 国家自然科学基金(62006059, 62002085), 中国博士后科学基金(2020M681099), 中国博士后创新人才支持计划(BX20190100), 深圳市科技创新委员会重点实验室组建项目(ZDSYS20190902093015527)资助

详细信息

作者简介:
文杰：哈尔滨工业大学(深圳)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: jiewen_pr@126.com

颜珂：北京理工大学博士后. 主要研究方向为生物信息学和模式识别. E-mail: yanke401@163.com

张正：哈尔滨工业大学(深圳)助理教授. 主要研究方向为机器学习, 计算机视觉和多媒体分析. E-mail: darrenzz219@gmail.com

徐勇：哈尔滨工业大学(深圳)教授. 主要研究方向为机器学习, 模式识别, 生物特征和视频分析. E-mail: yongxu@ymail.com

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出版历程
- 收稿日期: 2020-07-09
- 修回日期: 2020-11-24
- 网络出版日期: 2021-03-27
- 刊出日期: 2023-07-20

Low-rank Tensor Graph Learning Based Incomplete Multi-view Clustering

WEN Jie^1
,,
YAN Ke^2
,,
ZHANG Zheng^{1, 3
,},
XU Yong^{1, 3
,}

1.
Shenzhen Key Laboratory of Visual Object Detection and Recognition, Harbin Institute of Technology (Shenzhen), Shenzhen 518055
2.
School of Computer Science and Technology, Beijing Institute of Technology, Beijing 100080
3.
Peng Cheng Laboratory, Shenzhen 518055

Funds: Supported by Shenzhen Fundamental Research Fund (JCYJ20190806142416685), National Natural Science Foundation of China (62006059, 62002085), China Postdoctoral Science Foundation (2020M681099), China Postdoctoral Program for Innovative Talents (BX20190100), and Establishment of Key Laboratory of Shenzhen Science and Technology Innovation Committee (ZDSYS20190902093015527)

More Information

Author Bio:
WEN Jie　Assistant professor at Harbin Institute of Technology (Shenzhen). His research interest covers machine learning and pattern recognition. Corresponding author of this paper

YAN Ke　Postdoctor at Beijing Institute of Technology. His research interest covers bioinformatics and pattern recognition

ZHANG Zheng　Assistant professor at Harbin Institute of Technology (Shenzhen). His research interest covers machine learning, computer vision, and multimedia analytics

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

摘要

摘要: 传统多视角聚类都基于视角完备假设, 要求所有样本的视角信息完整, 不能处理存在部分视角缺失情形下的不完整多视角聚类任务. 为解决该问题, 提出一种基于低秩张量图学习的不完整多视角聚类方法. 为了恢复相似图中缺失视角所对应的样本关联信息, 该方法将低秩张量图约束和视角内在图保持约束融入到多视角谱聚类模型. 通过在一个统一模型中同时挖掘视角间的互补信息和视角内未缺失样例的关联信息, 所提出的方法能够得到表征样例邻接关系的完整相似图和视角间一致的最优聚类指示矩阵. 与12种不完整多视角聚类方法进行实验对比, 实验结果表明所提出的方法在多种视角缺失率下的5个数据集上获得了最好的聚类性能.
- 多视角聚类 /
- 视角缺失 /
- 不完整多视角聚类 /
- 图学习
Abstract: Conventional multi-view clustering methods all assume that the give multi-view data is complete, i.e., all views are fully observed, which are not applicable to the incomplete multi-view clustering case with missing-views. To address this issue, we propose a method, called low-rank tensor graph learning (LASAR). To recover the missing connections corresponding to the missing views in the graph space, the proposed method integrates the low-rank tensor graph constraint and intra-view graph constraint into the multi-view spectral clustering framework. By exploring the inter-view information and intra-view information with respect to the observed views simultaneously, the proposed method can obtain the optimal completed graphs of all views and the optimal clustering indicator matrix shared by all views. Experimental results on five datasets with different missing-view rates show that the proposed method obtains better performance than 12 state-of-the-art incomplete multi-view clustering methods.
- Multi-view clustering /
- view missing /
- incomplete multi-view clustering /
- graph learning
注释:

1) 1¹ 三阶以上的张量通常称之为高阶张量, 在高阶张量空间挖掘的信息称之为高阶信息^{[3, 22]}.

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图 1 基于低秩张量图学习的不完整多视角聚类框图

Fig. 1 The flow chart of the low-rank tensor graph learning based incomplete multi-view clustering

下载: 全尺寸图片幻灯片

图 2 多视角样本分布典型范例, 其中${\boldsymbol{S}}_{a,b}^{\left( v \right)}$表示a、b两样本的第$v$个视角特征的相似度

Fig. 2 Example of the distribution of multi-view samples, where ${\boldsymbol{S}}_{a,b}^{\left( v \right)}$ denotes the similarity degree of samples $a$ and $b$ in the $v$-th view

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图 3 各方法在不同视角缺失率下的Handwritten、不同视角配对率下的Animal和不同视角配对率下的Reuters数据集上的聚类Purity (%)

Fig. 3 The clustering Purities (%) of different methods on Handwritten dataset with different missing-view rates, Animal dataset with different paired-view rates, and Reuters dataset with different paired-view rates

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图 4 在视角缺失率为30%的Handwritten和Caltech7数据集上, 所提出LASAR方法的聚类NMI (%) 与超参数${\lambda _1}$和${\lambda _2}$的关系

Fig. 4 Clustering NMI (%) of the proposed LASAR v.s. hyper-parameters ${\lambda _1}$ and ${\lambda _2}$ on Handwritten and Caltech7 datasets with a missing-view rate of 30%

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图 5 本文模型和其两种退化模型在视角缺失率或配对率均为30%的4个数据集上的聚类NMI和聚类ACC

Fig. 5 Clustering NMIs and ACCs of the proposed method and two degraded models on the four datasets with a missing-view rate or paired-view rate of 30%

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图 6 在视角缺失率为30%的Handwritten和Caltech7数据集上, LASAR的目标函数值和聚类NMI (%)与迭代步数之间的关系

Fig. 6 Objective function value and clustering NMI (%) of LASAR v.s. the iterations on Handwritten and Caltech7 datasets with a missing-view rate of 30%

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图 7 本文方法在视角缺失率为70%的Handwritten数据集上得到的第1个视角和第5个视角的相似图

Fig. 7 Two similarity graphs, corresponding to the 1st view and the 5th view, obtained by the proposed method on Handwritten dataset with a missing-view rate of 70%

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表 1 各方法在不同视角缺失率下的Caltech7和BBCSport数据集上的实验结果

Table 1 Experimental results of different methods on Caltech7 and BBCSport datasets with different missing-view rates

数据集	方法	NMI (%)			ACC (%)			Purity (%)
数据集	方法	0.1	0.3	0.5	0.1	0.3	0.5	0.1	0.3	0.5
Caltech7	MultiNMF+均值	30.16	28.52	28.88	46.39	40.61	37.92	79.80	78.43	78.03
	MultiNMF+KNN	36.38	34.27	34.19	50.71	43.98	42.66	80.68	80.88	80.17
	CCo_MVSC+零值	35.53	34.18	31.52	38.47	38.16	37.35	78.33	77.89	76.21
	CCo_MVSC+KNN	37.38	37.18	37.33	39.06	39.62	37.34	80.25	80.36	81.45
	MIC	33.71	27.35	20.44	44.07	38.01	35.80	78.12	73.31	68.26
	OMVC	28.13	25.32	18.76	40.88	36.82	33.28	79.21	77.73	74.05
	DAIMC	44.61	38.45	36.28	48.29	47.46	44.89	83.32	76.83	75.50
	OPIMC	42.98	41.54	35.98	49.24	48.34	44.12	84.89	83.70	80.64
	UEAF	39.44	31.07	24.02	50.82	42.71	36.32	81.49	78.26	76.29
	IMKKM-IK-MKC	24.09	23.45	22.91	36.54	34.87	36.05	72.98	73.82	72.52
	IMVSC_AGL	44.05	42.61	37.35	54.72	54.61	51.78	84.27	83.98	82.31
	GM-PMVC	60.26	52.35	47.95	62.65	54.95	47.65	91.26	84.49	83.93
	LASAR	66.08	62.80	52.76	70.29	69.02	66.46	92.76	90.27	84.84
BBCSport	MultiNMF+均值	23.48	18.25	14.79	48.58	42.75	40.34	47.17	44.66	43.10
	MultiNMF+KNN	33.41	29.17	32.66	56.03	54.14	54.66	58.62	55.51	58.27
	CCo_MVSC+零值	62.79	57.69	39.55	72.76	70.06	61.38	81.49	78.62	66.72
	CCo_MVSC+KNN	59.24	56.25	38.97	69.02	69.83	60.06	80.92	78.56	68.62
	MIC	29.90	25.84	24.01	51.21	46.21	46.03	55.00	51.72	52.41
	OMVC	30.64	41.57	40.63	53.33	51.38	48.79	56.49	59.20	57.47
	DAIMC	56.62	50.17	37.89	68.62	63.45	56.89	76.90	71.72	61.03
	OPIMC	35.66	31.56	21.75	54.14	52.93	45.69	58.28	56.72	50.86
	UEAF	70.71	68.25	55.13	78.22	77.24	69.31	87.41	87.07	77.07
	IMKKM-IK-MKC	72.91	64.42	53.52	77.55	75.66	67.07	88.76	84.03	77.00
	IMVSC_AGL	70.46	66.11	54.57	76.41	74.48	69.31	87.41	85.00	78.10
	GM-PMVC	70.34	66.26	48.93	74.66	73.48	60.07	85.17	84.48	72.68
	LASAR	77.09	70.30	57.04	85.00	77.41	73.28	90.00	88.62	78.79

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表 2 各方法在不同视角缺失率下的Handwritten及不同视角配对率下的Animal和Reuters数据集上的实验结果

Table 2 Experimental results of different methods on Handwritten dataset with different missing-view rates, Animal and Reuters datasets with different paired-view rates

数据集	方法	NMI (%)				ACC (%)
数据集	方法	0.1	0.3	0.5	0.7	0.1	0.3	0.5	0.7
Handwritten	MultiNMF+均值	72.05	60.11	41.99	20.88	82.35	71.74	52.03	31.85
	MultiNMF+KNN	74.23	74.77	73.81	67.93	83.64	83.89	83.53	78.33
	CCo_MVSC+零值	70.90	68.23	62.86	53.14	74.61	73.17	70.15	64.62
	CCo_MVSC+KNN	73.03	73.77	73.06	67.66	76.76	76.79	76.20	75.52
	MIC	70.84	65.39	52.95	34.71	77.59	73.29	61.27	41.34
	OMVC	56.72	44.99	35.16	25.83	65.04	55.00	36.40	29.80
	DAIMC	79.78	76.65	68.77	47.10	88.86	86.73	81.92	60.44
	OPIMC	77.26	73.74	66.57	51.86	80.20	76.45	69.50	56.66
	UEAF	77.74	69.37	55.09	50.56	85.80	76.11	65.39	61.11
	IMKKM-IK-MKC	69.43	65.42	59.04	47.36	71.78	69.07	66.08	55.55
	IMVSC_AGL	93.68	90.55	86.39	75.44	97.15	95.50	93.19	84.08
	GM-PMVC	99.84	99.64	99.12	97.14	99.94	99.87	99.66	98.87
	LASAR	99.92	99.81	99.32	99.13	99.97	99.93	99.73	99.68
Animal	MultiNMF+均值	48.25	53.51	56.30	61.48	42.04	46.33	48.62	52.92
	MultiNMF+KNN	53.68	55.58	58.85	60.68	45.15	47.89	51.51	53.16
	CCo_MVSC+零值	52.19	55.31	58.31	61.71	48.12	52.03	54.72	56.73
	CCo_MVSC+KNN	49.13	57.96	61.37	63.86	47.71	53.58	56.79	58.41
	MIC	48.96	52.79	55.69	59.30	39.44	43.38	45.88	49.15
	OMVC	44.66	50.77	53.11	55.38	30.66	42.51	43.98	46.39
	DAIMC	49.33	55.03	59.36	62.76	42.76	50.18	53.87	56.42
	OPIMC	44.29	52.34	58.51	62.04	37.86	46.33	53.14	53.88
	UEAF	54.88	62.10	64.27	68.62	46.76	52.45	58.15	62.71
	IMKKM-IK-MKC	51.47	57.21	62.14	66.50	46.05	53.61	58.48	61.85
	IMVSC_AGL	56.38	59.65	63.26	66.71	51.61	54.94	57.97	60.73
	GM-PMVC	55.57	57.62	61.25	64.55	47.84	50.89	53.99	57.81
	LASAR	65.79	69.61	90.26	92.97	58.05	65.67	86.28	88.05
Reuters	MultiNMF+均值	19.58	18.82	18.56	19.51	37.68	38.31	37.27	37.58
	MultiNMF+KNN	19.91	23.17	22.50	22.21	37.95	38.08	37.47	37.35
	CCo_MVSC+零值	17.04	18.66	18.10	20.11	34.66	37.39	37.46	40.01
	CCo_MVSC+KNN	17.64	19.33	19.67	20.95	35.42	37.79	39.41	41.06
	MIC	18.72	21.30	22.78	24.34	38.47	40.85	41.37	43.05
	OMVC	18.05	19.77	21.37	22.97	36.78	37.44	39.45	41.09
	DAIMC	26.53	29.98	30.27	31.55	43.32	46.67	47.78	48.89
	OPIMC	19.02	22.47	23.82	25.39	40.58	42.19	43.87	44.43
	UEAF	24.77	26.85	27.14	28.36	41.45	44.95	46.87	47.92
	IMKKM-IK-MKC	24.30	26.75	26.88	28.67	41.13	45.71	46.40	47.29
	IMVSC_AGL	25.75	28.41	28.39	29.79	42.38	46.29	47.82	48.91
	GM-PMVC	26.22	29.35	30.71	30.89	43.15	46.88	48.19	50.44
	LASAR	28.34	32.01	31.27	32.77	45.38	48.14	50.80	52.77

下载: 导出CSV

表 3 各方法在5个数据集上的运行时间(s), 其中视角缺失率或配对率为30%

Table 3 Running time (s) of different methods on the above five datasets with a missing-view rate or paired-view rate of 30%

方法	Caltech7	BBCSport	Handwritten	Animal	Reuters	计算复杂度
MultiNMF+均值	60.38	28.62	35.54	13489.18	27385.81	${\rm{O} } \left( {\tau T\sum\nolimits_{v = 1}^l { {m_v}cn} } \right)$^[10]
MultiNMF+KNN	59.05	25.46	40.69	13499.88	27408.45	${\rm{O} } \left( {\tau T\sum\nolimits_{v = 1}^l { {m_v}cn} } \right)$^[10]
CCo_MVSC+零值	3.01	0.35	5.18	174.16	489.83	${\rm{O} } \left( {\tau \left( {l{n^3} + {n^3} } \right)} \right)$
CCo_MVSC+KNN	4.36	0.46	7.69	243.77	699.74	${\rm{O}} \left( {\tau \left( {l{n^3} + {n^3}} \right)} \right)$
MIC	225.31	3.01	149.87	3693.92	22499.46	${\rm{O}} \left( {\tau T\sum\nolimits_{v = 1}^l {{m_v}cn} } \right)$^[10]
OMVC	152.54	3.17	27.77	3409.08	18154.71	${\rm{O}} \left( {\tau T\sum\nolimits_{v = 1}^l {{m_v}cn} } \right)$^[10]
DAIMC	62.74	81.18	26.72	2191.43	96628.66	${\rm{O}} \left( {\tau \left( {Tnc{m_{\max }} + lm_{\max }^3} \right)} \right)$^[17]
OPIMC	0.16	0.05	0.09	3.31	1.05	${\rm{O}} \left( {\tau lcn{m_{\max }}} \right)$^[18]
UEAF	4.64	2.36	7.05	1258.93	18599.64	${\rm{O}} \left( {\tau \left( {2{n^3} + \sum\nolimits_{v = 1}^l {{m_v}{c^2}} } \right)} \right)$^[19]
IMKKM-IK-MKC	23.68	0.22	41.22	975.38	1762.95	${\rm{O}} \left( {\tau \left( {{n^3} + l{n^3} + {l^3}} \right)} \right)$^[21]
IMVSC_AGL	173.58	2.57	81.13	5265.99	28672.45	${\rm{O}} \left( {\tau \left( {l{n^3} + {n^3} + \sum\nolimits_{v = 1}^l {n_v^3} } \right)} \right)$^[20]
GM-PMVC	607.40	14.63	602.81	50725.43	1.78×10⁵	${\rm{O}} \left( {\tau l\left( {{n^3} + kdn + {k^3} + {k^2}d} \right)} \right)$^[22]
LASAR	73.87	1.09	115.60	2352.88	5677.46	${\rm{O}} \left( {\tau \left( {l{n^2}\log \left( n \right){\rm{ + }}{l^2}{n^2} + c{n^2}} \right)} \right)$
注: 本表未考虑K-means聚类算法的计算复杂度. 表中, T表示子循环迭代步数, m_max表示所有视角的最大特征维度, n_v表示第$v$个视角的未缺失样例数, d表示多视角数据特征的平均维度, k表示矩阵分解模型中的隐层表征维度.

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