Orthogonal multi-view tensor-based learning for clustering
Introduction
Spectral clustering is one of the most classic clustering algorithms [1], [2]. The performance of spectral clustering has been widely demonstrated in many artificial intelligence fields such as pedestrian re-identification [3], [4] and image segmentation [5], [6]. Despite the promising performance achieved by spectral clustering, the target data are usually limited to one single view or model in these classic clustering tasks. In practice, real-world data are often acquired in different views [7]. For example, in the mission of the web news category, the data can be composed of context texts, news pictures, and video clips at the same time. In the mission of pedestrian re-identification, the same object can be photographed at different angles and under different lighting conditions. To make full use of multi-view data, the research on clustering has been extended to study how to use multi-view data to improve the performance of clustering.
The comprehensive studies on multi-view learning have shown that multi-view data conclude two kinds of information, that is, consensus information and complementary information. The consensus information is the maximized agreement from distinct multiple views, while the complementary information is based on the assumption that each view of data can provide unique information for learning. In other words, multi-view data can provide more accurate and comprehensive information compared to single-view data. For multi-view clustering, the main idea is to boost the clustering performance by combining the multi-view information. The previous studies [8], [9], [10] have proven that simply combing the multi-view data cannot boost the clustering performance, but even reduce the clustering performance. Therefore, the challenge of multi-view clustering lies in the way of fusing multi-view clustering information.
Many multi-view spectral clustering methods have been proposed recently and achieved promising performance [11], [12], [13], [14], [15], [16], [8], [17]. Most of them are designed to seek numerical consistency among the constructed multi-view similarity matrices [16], [17], [11], [15]. For example, Robust Multi-view Spectral Clustering [16] (RMSC) pursues a shared low-rank matrix from multi-view similarity matrices, which reflects the underlying consensus clustering information among multiple views. Some other methods average or sum the multi-view similarity matrices. For example, Essential Tensor Learning for Multi-view Spectral Clustering [17] (ETLMSC) explores the principal components of multi-view representations by t-SVD based tensor nuclear norm, then sums the optimized multi-view similarity matrices. However, the similarity matrices from multiple views may vary a lot in values. In fact, they tend to be similar in clustering structure rather than be numerically uniform. Thus, pursuing numerical uniform in multi-view similarity matrices or directly averaging/summing the multi-view similarity matrices is not proper. Besides, only the partial information of similarity matrices with consistent values is considered in most existing methods, while the complementary information of each individual view and the high-order correlation among multiple views is always ignored. In this paper, we focus on multi-view spectral clustering methods, and we hope to learn a similarity matrix for multi-view spectral clustering, which can intrinsically reflect the underlying consistent clustering structure of multi-view data. Furthermore, the complementary information and high-order correlation among multiple views can also be well explored.
In order to achieve the above goals, we propose a novel Orthogonal Multi-view Tensor-based Learning (OMTL) for clustering in this paper. The whole framework of OMTL is shown in Fig. 1. In OMTL, an orthogonal matrix factorization is first introduced and conducted on the recovered low-rank matrices to eliminate the view-specific statistic distribution. It should be noted that the decomposed matrix on the right shares the same clustering structure with the original similarity matrix under the effect of orthogonal matrix factorization. To further explore the high-order correlations and pursue consistency of clustering structure among multiple views, we introduce the tensor-Singular Value Decomposition (t-SVD) based low-rank tensor constraint [18] in OMTL. The desired similarity matrix can be learned by conducting tensor rotation and the t-SVD based nuclear norm minimization on the 3-order tensor formed by the decomposed matrices. To solve the objective function of OMTL, an effective altering direction minimization algorithm based on Augmented Lagrangian Multiplier [19] (ALM) is designed in this paper as well. Comprehensive experiments on seven benchmark datasets show the superior performance of the proposed OMTL.
The main contributions of this paper are summarized as follows:
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We propose a novel multi-view spectral clustering method named Orthogonal Multi-view Tensor-based Learning (OMTL) for clustering. An orthogonal matrix factorization is introduced in our proposed method to eliminate the diversity of statistic distribution in multiple views and preserve the intrinsic clustering structure in each individual view.
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We further employ the t-SVD based low-rank tensor constraint to explore the high-order correlation and pursue multi-view low-rank constraints among multiple views.
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A corresponding optimization schedule is proposed in this paper to find the optimal solution of our proposed objective function, which is based on Altering Direction Method for Multipliers (ADMM). Extensive experiments on seven benchmark datasets demonstrate the superior performance of our proposed method compared to nine state-of-the-art multi-view clustering methods and two baselines.
Section snippets
Related work
Spectral clustering [2] is one of the most popular and classic clustering algorithms. For spectral clustering, the first foremost step is to construct the similarity matrix from original data. The standard spectral clustering method is designed to partition the Laplacian graph (which is learned from the similarity matrix) into several sub-graphs, where each sub-graph represents a cluster. Many multi-view clustering methods are derived from spectral clustering and are different in the way of
Notations and preliminaries
Before presenting our proposed method, we need to briefly introduce the t-SVD based tensor nuclear norm. To help the readers understand the t-SVD based tensor nuclear norm, some related definitions and frequently used notions are introduced in this section.
For convenience, we summarize the frequently used notations in Table 1. In this paper, the 3-order tensor, that is, is mainly considered. For the tensor , the vector along its i-th mode is defined as mode-i fiber. Accordingly,
The proposed method
In this section, we present our tensor based multi-view spectral clustering model: Orthogonal Multi-view Tensor-based Learning (OMTL) for clustering. As our proposed method is a spectral clustering based method, we first deal with the multi-view data by computing multi-view similarity matrices.
Given multi-view dataset , where N is the number of the instances and M denotes the number of the views, the multi-view similarity matrices are separately
Experiments
In this section, we design a series of experiments on several benchmark datasets to demonstrate the superiority of OMTL. We first evaluate the clustering performance of OMTL with several state-of-the-art multi-view clustering methods and baselines by running them on adopted benchmark datasets. Next, we discuss the parameter sensitivity and convergence properties of OMTL via specialized experiments.
Conclusion
In this paper, we propose a novel orthogonal multi-view tensor-based learning method for spectral clustering, which is abbreviated as OMTL. In OMTL, the robust principle component analysis is utilized to recover a low-rank similarity matrix from each view, which aims to learn the underlying true clustering information in the associated view. In order to eliminate the view-specific statistic distribution of multi-view similarity matrices, we introduce an orthogonal matrix factorization. As a
CRediT authorship contribution statement
Shuangxun Ma: Conceptualization, Methodology, Investigation, Writing – original draft. Yuehu Liu: Supervision, Resources, Writing – review & editing. Guangcan Liu: Conceptualization, Writing – review & editing. Qinghai Zheng: Methodology, Writing – review & editing. Chi Zhang: Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work is supported by the National Key R&D Program of China under Grant No. 2018AAA0102504.
Shuangxun Ma received his B.E. degree from Northwestern Polytechnical University, China and the M. Eng. degree from Lanzhou University, China. He is currently a doctoral candidate of Xi’an Jiaotong University. His research interests include machine learning and computer vision.
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Cited by (0)
Shuangxun Ma received his B.E. degree from Northwestern Polytechnical University, China and the M. Eng. degree from Lanzhou University, China. He is currently a doctoral candidate of Xi’an Jiaotong University. His research interests include machine learning and computer vision.
Yuehu Liu received B.E. and M.E. degrees from Xi’an Jiaotong University, Xi’an, China, in 1984 and 1989, respectively, and a Ph.D. degree in Electrical Engineering from Keio University, Tokyo, Japan, in 2000. He is currently a Professor at Xi’an Jiaotong University. His research interests are focused on computer vision, computer graphics, and simulation testing for autonomous vehicles. He is a member of the IEEE and the IEICE.
Guangcan Liu received the bachelor’s degree in mathematics, and the PhD degree in computer science and engineering from Shanghai Jiao Tong University, China, in 2004 and 2010, respectively. He was a post-doctoral researcher with the National University of Singapore, Singapore, from 2011 to 2012, the University of Illinois at Urbana-Champaign, Champaign, IL, USA, from 2012 to 2013, Cornell University, Ithaca, NY, USA, from 2013 to 2014, and Rutgers University, Piscataway, NJ, USA, in 2014. Since 2014, he has been a professor with the School of Information and Control, Nanjing University of Information Science and Technology, Nanjing, China. His research interests are pattern recognition and signal processing. He obtained the National Excellent Youth Fund in 2016 and was designated as the global Highly Cited Researchers in 2017.
Qinghai Zheng received his B.S. degree and M.S. degree from Xi’an Jiaotong University, China, in 2015 and 2018. He is currently a doctoral candidate of Xi’an Jiaotong University. His research interests include machine learning, data mining and multiview learning.
Chi Zhang received B.E. and Ph.D. degrees from Xi’an Jiaotong University, Xi’an, China, in 2011 and 2021, respectively. He is currently an Assistant Professor at the College of Artificial Intelligence, Xi’an Jiaotong University. His research interests include computer vision, machine learning and intelligence testing for vision systems.