Joint Adaptive Neighbours and Metric Learning for Multi-view Subspace Clustering

Due to the existence of various views or representations in many real-world data, multi-view learning has drawn much attention recently. Multi-view spectral clustering methods based on similarity matrixes or graphs are pretty popular. Generally, these algorithms learn informative graphs by directly utilizing original data. However, in the real-world applications, original data often contain noises and outliers that lead to unreliable graphs. In addition, different views may have different contributions to data clustering. In this paper, a novelMultiview Subspace Clustering method unifying Adaptive neighbours and Metric learning (MSCAM), is proposed to address the above problems. In this method, we use the subspace representations of different views to adaptively learn a consensus similarity matrix, uncovering the subspace structure and avoiding noisy nature of original data. For all views, we also learn different Mahalanobis matrixes that parameterize the squared distances and consider the contributions of different views. Further, we constrain the graph constructed by the similarity matrix to have exact c (c is the number of clusters) connected components. An iterative algorithm is developed to solve this optimization problem. Moreover, experiments on a synthetic dataset and different real-world datasets demonstrate the effectiveness of MSCAM. Introduction In recent years, learning multi-view data has increasingly attracted research attention in many real-world applications, because data represented by different features or collected from different sources are very common. For instance, documents can have different languages; web pages can be described by different characteristics, e.g., hyperlinks and texts; images can have many descriptions with respect to different kinds of features like color or texture features. Different views or features can capture distinct perspectives of data that are complementary to each other. Thus, how to integrate these heterogeneous features and uncover the underlying structure of data is a critical problem for multi-view learning. In this paper, we focus on an unsupervised scenario, i.e., multi-view spectral clustering. In the past decades, various spectral clustering algorithms have been proposed [Shi and Malik, 2000; Ng et al., 2002; Zelnik-Manor and Perona, 2005; Von Luxburg, 2007; Nie et al., 2014; Chang et al., 2015]. These methods can achieve promising clustering performance for an individual view. However, multiple views containing different information can describe the data more accurately and improve the clustering performance. [Zhou and Burges, 2007] generalize the single-view spectral clustering normalized cut to the multi-view case. [Blaschko and Lampert, 2008] introduce the Canonical Correlation Analysis (CCA) to map multi-view data into a low-dimensional subspace. There are also some methods using co-training or coregularization strategies to integrate different information of views [Kumar and Daumé, 2011; Kumar et al., 2011]. In addition, [Cai et al., 2011] integrate heterogeneous features to learn a shared Laplacian matrix and improve model robustness with a non-negative constraint. [Wang et al., 2014] utilize the minimax optimization to obtain a universal feature embedding and a consensus clustering result. [Nie et al., 2017] simultaneously perform local structure learning and multi-view clustering in which the weight is automatically determined for each view. Recently, selfrepresentation subspace based multi-view spectral clustering methods have been developed due to the effectiveness [Cao et al., 2015; Gao et al., 2015; Yin et al., 2015; Zhang et al., 2015; Wang et al., 2017]. These methods aim to discover underlying subspaces embedded in original data for clustering accurately. Although the previousmulti-view spectral clustering methods can achieve promising performance, there still exist drawbacks. First, spectral methods need the high-quality similarity matrix. The previous methods directly learn the similarity matrix utilizing original data. However, in real-world datasets, data often contain noises and outliers, thus the similarity matrix learned from original data is unreliable. Second, different views have different contributions to data clustering. The previous methods use the Euclidean metric to learn the similarity matrix. For given data, the Euclidean distance among them is fixed, which cannot consider different contributions of views. Finally, for multi-view spectral clustering, the k-means procedure in spectral clustering requires the strict initialization, which influences the final clustering performance [Ng et al., 2002]. In this paper, we propose a novel subspace based multiview spectral clustering method, namedMulti-view Subspace Clustering unifying Adaptive neighbours and Metric learning (MSCAM) to address the aforementioned problems. In this method, we learn the subspace representations of original data for each view. By utilizing these subspace representations to adaptively learn a consensus similarity matrix, we can alleviate the influence of noises and outliers. Meanwhile, for each view, we learn the most suitable Mahalanobis matrix to parameterize the squared distance. The motivation is that due to the complexity of noises and outliers, different views have different contributions to clustering data. Thus we propose to use Mahalanobis metric to dynamically rescale data of each view. Different Mahalanobis matrixes are learned to weigh different contributions of views. Finally, we constrain the graph constructed by the similarity matrix to have exact c connected components. Here, the number of clusters is c. In this way, the learned graph can be employed to cluster directly without the k-means procedure. The main contributions of our work are as follows: • We adaptively learn a consensus similarity matrix in the subspace rather than the original space that may have noises and outliers. • Mahalanobis metric is employed to parameterize the squared distance of each view, which considers the contributions of different views to data clustering compared with the Euclidean metric. • We add a constraint on the graph constructed by the similarity matrix to replace the k-means procedure. • Extensive comparison experiments demonstrate that our MSCAM method outperforms other state-of-the-art multi-view clustering approaches. Related Work Notation Summary Lowercase letters (m,n, ...) denote scalars while bold lowercase letters (m,n, ...) denote vectors. Bold uppercase letters (M,N, ...) mean matrixes. For an arbitrary matrix M , mi means the i column of M and mij stands for the j th element in mi. M T and tr(M) denote the transpose and trace of M , respectively. ‖·‖ and ‖·‖F represent the l2 norm and Frobenius norm, respectively. For two matrixes with the same size, 〈M,N〉 represents the inner product. Moreover, 1 and I represent the vectors of all ones and identity matrix with proper sizes, respectively. Adaptive Neighbours Clustering For clustering tasks, the local correlation of original data plays an important role. Recently, many clustering methods considering the local correlation have been developed [Nie et al., 2014; Guo, 2015; Nie et al., 2016; Zhao et al., 2016]. Let X = [x1,x2, ...,xn] ∈ R d×n be the data matrix with n data points, where d is the dimension of features. The Euclidean (squared) distance is used as a measure to decide the k-nearest data of each data point. For each data point xi, all data points can be the neighbour of xi with the probability aij . Generally, a smaller distance ‖xi − xj‖ 2 2 indicates that a larger probability aij should be allocated. Therefore, the probabilities aij | n j=1 can be determined by solving the following problem min a i 1=1,0≤aij≤1 n