This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON CYBERNETICS 1 Multiview Clustering Based on Non-Negative Matrix Factorization and Pairwise Measurements Xiumei Wang , Tianzhen Zhang, and Xinbo Gao , Senior Member, IEEE Abstract—As we all know, multiview clustering has become a hot topic in machine learning and pattern recognition. Non- negative matrix factorization (NMF) has been one popular tool in multiview clustering due to its competitiveness and interpretation. However, the existing multiview clustering methods based on NMF only consider the similarity of intra-view, while neglecting the similarity of inter-view. In this paper, we propose a novel mul- tiview clustering algorithm, named multiview clustering based on NMF and pairwise measurements, which incorporates pair- wise co-regularization and manifold regularization with NMF. In the proposed algorithm, we consider the similarity of the inter-view via pairwise co-regularization to obtain the more compact representation of multiview data space. We can also obtain the part-based representation by NMF and preserve the locally geometrical structure of the data space by utilizing the manifold regularization. Furthermore, we give the theoretical proof that the objective function of the proposed algorithm is convergent for multiview clustering. Experimental results show that the proposed algorithm outperforms the state-of-the-arts for multiview clustering. Index Terms—Manifold regularization, multiview clustering, non-negative matrix factorization (NMF), pairwise co-regularization. I. I NTRODUCTION C LUSTERING is a popular and crucial technique in machine learning and pattern recognition for its ability to capture the structure of the data space. Nowadays, with the rapid development of information technology, many real world datasets consist of multiple features or views, and different views data generally contain complementary and interaction information. Consequently, how to integrate various features or views becomes a more and more significant problem in clustering area [1]–[7]. Recently, several multiview clustering Manuscript received March 22, 2018; accepted May 15, 2018. This work was supported in part by the National Natural Science Foundation of China under Grant 61472304, Grant 61432014, and Grant 61772402, in part by the National Key Research and Development Program of China under Grant 2016QY01W0200, in part by the Key Industrial Innovation Chain in Industrial Domain under Grant 2016KTZDGY04-02, and in part by the National High-Level Talents Special Support Program of China under Grant CS31117200001. This paper was recommended by Associate Editor S. Cruces. (Corresponding author: Xinbo Gao.) The authors are with the Video and Image Processing System Lab, School of Electronic Engineering, Xidian University, Xi’an 710071, China (e-mail: wangxm@xidian.edu.cn; zhangtianzhenztz@163.com; xbgao@mail.xidian.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2018.2842052 algorithms appear in the literatures. Existing multiview clus- tering methods can be roughly divided into two categories: 1) traditional machine learning-based methods and 2) deep learning-based methods. In recent years, due to the potential of deep neural networks, deep learning-based approaches are widely used in many fields, such as pattern recognition and computer vision. Several clustering methods are based on deep learning, for instance, multiview clustering based on depth matrix decomposition [8] and multiview clustering based on auto-encoder [9], [10]. Traditional machine learning-based algorithms can be grouped into three main categories according to the essential technique: 1) spectral graph-based methods [11]–[17]; 2) K-means-based methods [18]–[20]; and 3) non-negative matrix factorization (NMF)-based methods [21]–[25]. Multiview learning methods based on spectral clustering (MVSC) extend the conventional spectral clustering from indi- vidual view to multiview data. One challenge of MVSC is the demand of constructing similarity graph or similarity matrix for each view, which is high computation cost, especially for high dimensional data. Therefore, this kind of methods would not be suitable for multiview clustering applications especially when the number of views is very large. Multiview learning methods based on K-means clustering (MVKM) are another representative algorithms. They deal with each view through standard K-means clustering algo- rithm. However, MVKM algorithms are generally conducted in the feature space, i.e., Euclidean space. Then they fail to dis- cover the geometrical structure and discriminative information of the multiview data space. Multiview clustering methods based on NMF utilize centroid-based co-regularization and NMF to find a consen- sus representation of multiview data. Ding et al. [25] showed that the minimization objective function of the spectral clus- tering algorithm can be equivalently conducted via the NMF. That is, spectral clustering is equivalent to NMF. Furthermore, they verified that NMF is equivalent to kernel K-means clus- tering. Therefore, NMF-based clustering, spectral clustering, and K-means clustering are different representation forms for the same problem with little different constraint. As a consequence, NMF-based multiview clustering methods have become a widely used clustering algorithm due to its simplicity and interpretability. In recent years, matrix factorization has received more atten- tion for data representation, and NMF is the most popular method among the matrix factorization approaches [26]–[35]. In NMF, each data point can obtain an efficient and low 2168-2267 c  2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.