Elsevier

Neural Networks

Volume 129, September 2020, Pages 19-30
Neural Networks

Multi-view clustering on data with partial instances and clusters

https://doi.org/10.1016/j.neunet.2020.05.021Get rights and content

Abstract

Most multi-view clustering algorithms apply to data with complete instances and clusters in the views. Recently, multi-view clustering on data with partial instances has been studied. In this paper, we study the more general version of the problem, i.e., multi-view clustering on data with partial instances and clusters in the views. We propose a non-negative matrix factorization (NMF) based algorithm. For the special case with partial instances, it introduces an instance-view-indicator matrix to indicate whether an instance exists in a view. Then, it maps the instances representing the same object to the same vector, and maps the instances representing different objects to different vectors. For the general case with partial instances and clusters, it further introduces a cluster-view-indicator matrix to indicate whether a cluster exists in a view. In each view, it also maps the instances representing the same object to the same vector, but it further makes the elements of the vector 0 if the elements correspond to missing clusters. Then it minimizes the disagreements between the approximated indicator vectors of instances representing the same object. Experimental results show that the proposed algorithm performs well on data with partial instances and clusters, and outperforms existing algorithms on data with partial instances.

Introduction

Multi-view clustering has become a hot topic and many algorithms have been proposed since the past decade. Most existing algorithms (Bickel and Scheffer, 2004, Liu et al., 2013, Luo et al., 2018, Zong, Zhang, Liu and Yu, 2018, Zong et al., 2017) are based on an ideal assumption that all of the views are complete, i.e., (1) all the instances appear in all the views and they are indexed in the same way; (2) all the clusters exist in all the views and they are formed by the same subsets of instances. However, in real-world applications, some instances or some whole clusters may be missing in some views, which results in partial instances and partial clusters. For example, both Reuters sport1 and CNN sport2 report sports news, and they can be seen as two different views. Each sport category is a cluster, and a piece of news is an instance. Under the ideal assumption, both Reuters sport and CNN sport report the same news of the same categories. However, in fact, the interests of Reuters sport and CNN sport are different. Reuters sport focuses on reporting news about golf, rugby union, tennis, cricket and formula one, while CNN sport focuses on reporting news about football, golf, tennis, sailing, motor sport, horse racing, equestrian, judo and rugby. It can be seen that only golf, tennis and rugby are interested by both of them. This observation indicates that the clusters (i.e., the sport category) are partial. In the clusters which exist in two views (e.g., news on golf), some pieces of news are reported on both websites, but the others are only reported by one of them, which shows that the instances are partial.

Recently, some researches have released the ideal assumption by allowing partial instances. PVC (Li, Jiang, & Zhou, 2014), IMVS (Yin, Wu, & Wang, 2015), DCNMF (Qian, Shen, Gu, Tang, & Ding, 2016) and IMG (Zhao, Liu, & Fu, 2016) are designed specifically for two-view data. These methods can be applied to multi-view data by conducting a series of two-view clustering, but they could not fully exploit multi-view information. MIC (Shao, He, & Yu, 2015), GPMVC (Rai, Negi, Chaudhury, & Deshmukh, 2016) and DAIMC (Hu & Chen, 2018) work on multi-view data. MIC (Shao et al., 2015) fills the missing instances in each view, thus its performance depends on the quality of the completion. GPMVC (Rai et al., 2016) extends PVC to multiple views and introduces view specific graph Laplacian regularization. Borrowing the idea of the weighted NMF, DAIMC (Hu & Chen, 2018) introduces a respective weight matrix for each incomplete view.

All the above methods do not state whether the clusters in each view are complete or not, but they require complete clusters implicitly. If we run these algorithms on data with partial clusters in the views, the multi-view interaction will inevitably cause negative effect. For example, given a two-view dataset with three clusters c1, c2, c3, and c3 misses in the 2nd view. Using existing partial multi-view clustering methods by ignoring whether there are clusters missing in the views, on one hand, the 2nd view will be partitioned into three clusters, which is far from the real two clusters in this view; on the other hand, knowledge of c1 and c2 in the 2nd view can be transferred to the forming of c3 in the 1st view, which disturbs the 1st view’s right clustering process. To our best knowledge, only one method CGC (Cheng et al., 2013) considers the partial clusters, but its formulation requires that each instance in one view is mapped to instances in another view at a certain probability, i.e., it requires probabilistic complete mapping among the instances.

In this paper, we propose a NMF based Multi-View clustering algorithm on data with Partial Instances and Clusters (MVPIC). In our setting, not only some instances miss in some views, but also some whole cluster members miss in some views. For the special case where instances are partial, MVPIC introduces an instance-view-indicator matrix to indicate whether an instance exists in a view and learns an approximated ground truth indicator matrix for the data objects. Then it maps the instances representing the same object to the same row of the approximated ground truth indicator matrix, and maps the instances representing different objects to different rows of the approximated ground truth indicator matrix. For the general case where both instances and clusters are partial, MVPIC further introduces a cluster-view-indicator matrix to indicate whether a cluster exists in a view. It uses the weighted product of the instance-view-indicator matrix, the approximated ground truth indicator matrix and the cluster-view-indicator matrix as the approximated indicator matrix of each view. In this way, MVPIC maps the instances representing the same object to the same vector, and in each view, MVPIC makes the elements of the vector 0 if the elements correspond to missing clusters. Since instances are in multiple views and the clusters vary with the views, then it minimizes the disagreements between the approximated indicator vectors of instances representing the same object. Experiments show that MVPIC performs well on data with partial instances and clusters, and outperforms existing algorithms on data with partial instances.

To sum up, the major contributions are highlighted as:

  • We propose the MVPIC algorithm to cluster multi-view data with both partial instances and clusters.

  • The proposed algorithm learns the consensus indicator matrix from multiple partial views simultaneously.

Section snippets

Multi-view clustering on complete data

Multi-view clustering starts up from this ideal case, many algorithms have been proposed and NMF (Lee & Seung, 2001) based algorithms have demonstrated their superiorities. They learn the consensus indicator matrix or minimizes the divergence of multiple indicator matrices. Examples are (Akata et al., 2011, Cai et al., 2013, Cheng et al., 2013, Liu et al., 2013, Zhao et al., 2017, Zong et al., 2017). Akata et al. (2011) and Cai et al. (2013) enforced a shared coefficient matrix among different

Review on non-negative matrix factorization

Non-negative matrix factorization (NMF) (Lee & Seung, 2001) decomposes the non-negative original matrix into several non-negative factor matrices. Denote XR+d×n as the originald-dimensional data matrix, where n is the number of instances. The basic form of NMF tries to find two non-negative matrices VR+n×r and UR+d×r whose product approximates to X. Using the Frobenius norm .F to define cost function, NMF tries to optimize the following objective function, minU,VXUVTF2s.t.U0,V0

Problem formulation

We are given a multi-view dataset with partial instances and clusters in the views, i.e., some instances and clusters may be missing in some views (Fig. 1). Whether an object has a representation in a view and whether a cluster exists in a view are known. The setting is realistic, e.g., for the example in Introduction, it is known that golf is in both Reuters sport and CNN sport, and cricket is only in Reuters sport and football is only in CNN sport. It is also known that a piece of news

Datasets

We experiment on six datasets which are typically used for testing multi-view clustering algorithms: Leaves (Mallah, Cope, & Orwell, 2013), ALOI,3 Football,4 Rugby4, CUB5 and Flower.6

  • Leaves contain the binary images of 1584 leaf samples of 99 classes. There are three views: Shape, Margin and

Conclusion

In this paper, we have studied the multi-view clustering problem on data with partial instances and clusters, which is more general in reality. We propose a NMF based algorithm. It introduces an instance-view-indicator matrix to indicate whether an instance exists in a view and introduces a cluster-view-indicator matrix to indicate whether a cluster exists in a view. When the instances are partial, it maps the instances representing the same object to the same vector, and maps the instance

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.

Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 61806034; No. 61876028; No. 61602081).

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