1 Introduction
With the development of computer vision applications, we have witnessed that hash technology has become an indispensable step in the processing of large data
wang2015learning bernabe2019efficient. In dealing with data analysis, organization, and storage, etc., there is an imminent need to use the effective hash code to process data clustering from big databases. Besides, most existed digital devices mainly based on binary code, which can effectively save computing time and storage space. In general, the similarity between the original data can be effectively preserved by encoding the original highdimensional data using a set of compact binary codes
dean2013fast , fuentes2019topic . These advantages make it obtained widely applied in the computer vision task, such as image clustering sudharshan2019multiple ahmed2019hash and multiview learning yang2017discrete etc.Nowadays, binary coding methods have been well investigated in many fields. Locality Sensitive Hashing (LSH) datar2004locality pioneered hash research by indexing similar data with hash codes and achieved largescale search in constant time. Commonly, the hashing method can be roughly divided into two major categories: supervised model and unsupervised model. The supervised hash code generates a discrete, efficient and compact hash code by using the label information of the data. For instance, Minimal Loss Hashing (MLH) norouzi2011minimal , Supervised Discrete Hashing (SDH) shen2015supervised , Supervised Discrete Hashing With Relaxation (SDHR) gui2016supervised and Fast Supervised Discrete Hashing (FSDH) gui2017fast . However, the problem of manually labeling largescale data are very expensive has not been considered. Thus, the unsupervised hash method is proposed to address this problem, which also obtained good performance in binary code learning. Unsupervised hash models include, but not limited to, Spectral Hashing (SH) weiss2009spectral , Iterative Quantization (ITQ) gong2012iterative , Discrete Graph Hashing (DGH) liu2014discrete , inductive Hashing on Manifolds shen2015hashing etc.. Because discrete hash codes reduce the quantization error, Discrete Hash (DGH) liu2014discrete and Supervised Discrete Hash (SDH) shen2015supervised have significant improvements in hash coding performance.
Up to now, most methods usually use a single view to learn binary code representation, which fails to explain the observed fact that the complementary features and diversity of multiple views. In many visual applicationswang2017effective , wu2018deep wang2015unsupervised wu2016exploiting , data is usually collected from datasets in various fields or from different feature extractorswu2018deep1 ; wang2016iterative , such as Histogram dalal2005histograms , Local Binary Patters (LBP) ojala2002multiresolution and Scale Invariant Feature Transform (SIFT) rublee2011orb etc.. Compared with singleview information, multiview data maybe include more potential comprehensive information. Therefore, multiview learning obtained more and more attention in many applications. Xia et al. xia2010multiview introduced a spectralembedding algorithm to explore the complementary information of different views, which have proved effective for image clustering and retrieval. Zhang et al. zhang2016flexible explicitly produced lowdimensional projections for different views, which could be applied to other data in the outofsample. Wang et al. wang2015robust effectively maintain wellencapsulated individual views while study subspace clustering for multiple views. Therefore, gathering information from multiple views and exploring the underlying structure of data is a key issue in data analysis. In addition, since the hash method could efficiently encode highdimensional data, a promising research field by adopting multiview binary code to improve clustering performance.
Recently, some efforts have been done to learn effective hash code from multiview data wu2018cycle . There are two types of research areas: crossview hashing and multiview hashing. Song et al. proposed a novel Imedia Hashing Method (IMH) method, which can explore the relevance among different media types from various data sets to achieve largescale retrieval intermedia. Besides, Zhu et al. zhu2013linear proposed Linear Crossmodal Hashing (LCMH) has obtained good performance in crossview retrieval tasks. Ding et al. ding2014collective by using the latent factor models from different modalities collective matrix decomposition. Composited Hashing with Multiple Information Sources (CHMIS) zhang2011composite is the first work in the multiview hash field. More recently, Multiview Alignment Hashing (MAH) liu2015multiview based on nonnegative matrix factorization can respect the distribution of data and reveal the hidden semantics representation. Then many multiview hash methods are proposed, such as Discrete Multiview Hashing (DMVH) yang2017discrete and Multiview Discrete Hashing (MvDH) shen2018multiview . Most of these related works of hash focus on mutual retrieval tasks between different views, which ignored the potential cluster structure and distribution of information in multiview data. Therefore, hash technology is of vital significance for multiview clustering and arouses attention from researchers in the computer vision region. Table 1 summarizes the current multiview hash methods from model learning paradigms, hash optimization strategies, and categories.
Methods  IMH  LCMH  CMFH  CHMIS  MAH  DMVH  MvDH  Ours 

Multiview cluster  ✘  ✘  ✘  ✘  ✘  ✘  ✘  
Discrete  ✘  ✘  ✘  ✘  ✘  
Unsupervised  ✘  ✘ 
In this paper, we introduce a novel frame for graphbased multiview binary code clustering. In order to learn an efficient binary code, our method attempts to efficiently learn discrete binary code and maintain manifold structure in Hamming space for multiview clustering tasks. To learn discriminated binary codes, the key design is to generate similar binary codes for similar data without destroying the inherent attributes of the original space, which can share information between multiple views as much as possible. By learning the cooperative work between hash codes and graph, clustering tasks and coding efficiency is significantly improved. Since direct optimization of binary codes is a difficult problem, an effective alternating iterative optimization strategy is developed to solve the hash coding. The construction process of GMBL has been shown as Fig.1. The main contributions of this paper are illustrated as follows:

We propose an innovative unsupervised hashing method to learn compact binary codes from multiview data. To preserve the original structure of input data, our proposed method combines hash codes learning and graph clustering through Locally Linear Embedding learning. Joint learning ensures that the generated hash code can improve performance for clustering.

Inspired in graph learning, the local similar structure of the original information is embedded into the Hamming space to learn compact hash codes. The viewspecific information is shared from multiple views by projecting different views of original features into the common subspace through local linear embedding.

In order to obtain an accurate clustering effect, we assign different weights to various views according to contribute for clustering. In addition, we introduce the alternating optimization algorithm with strict convergence proof in a new discrete optimization scheme to solve hash coding.
2 Related work
Most hashing algorithms are based on singleview data to generate binary codes. In this section, we first introduce theories and notations of multiview binary code learning. Then, we review a classical spectral embedding method by graph Laplacian matrix to preserve the data similarity structure. We will present how to learn binary code from multiple views, and then study complementary hash codes with similarity preservation in the next section.
2.1 Binary Code Learning
Assume we are given a dataset of examples with th views. The multiview matrix in th view can be represented as: .
Where is the dimension of th view. Unsupervised hashing to map the highdimensional data into the binary codes . Therefore, binary code generating is to learn a set of hash project functions to produce the corresponding set of bits binary code. For the th views sample of hash function as: , Where is a binary mapping. Such functions are usually constructed by combining dimension reduction and binary quantization. Since Hamming distance represents the similarity between binary codes, the hash objective function in th view can be constructed. Then, the binary code of the th view dataset can be written as:
(1) 
Where is the corresponding hash code of the whole dataset. In the process of binary code mapping, it is necessary to minimize the loss of data and the destruction of the original structure.
2.2 Singleview Graph Learning
The main purpose of similarity preservation is to preserve the geometry structure of manifold data by the local neighborhood of a point, which can be efficiently approximated by the point nearest neighbors. Generally, it has two steps, i.e., discovering similar neighbors and constructing weight matrix.
Let denote a feature of the samples and denotes
is a low dimensional vector mapped from
. Firstly, each sample is approximated to its nearest neighbor samples. Then minimizes the reconstruction error in original space are be used as follows:(2) 
Where if and are not neighbours. Local liner embedding assumes that such linear combinations should remain unchanged even if the manifold structure is mapped to a lower space. Then, used the lowdimensional representation minimizes the reconstruction error as follows:
(3) 
Where is a low dimensional matrix mapped from . Where is the graph Laplacian matrix and is the trace of matrix.
3 Graphbased Multiview Binary Learning
In this section, we first propose a novel clustering method called Graphbased Multiview Binary Learning(GMBL), which map the data to Hamming space and implement clustering tasks by efficient binary codes. Firstly, the anchor points of data are selected randomly, and the different views are mapped to the same dimension by nonlinear kernel mapping in section 3.1. Then, we propose a method of mapping hash codes, which can learn efficient binary codes with balanced binary code distribution in section 3.2. Furthermore, similarity preservation of different views means that similar data will be mapped to binary code by a short Hamming distance. To do this, our proposed method preserves the local similar structure of data through a similar matrix in section 3.3. Finally, an alternating iterative optimization strategy is applied to search for the optimal solution and the optimization process is illustrated in describe in section 3.4.
Suppose our multiview dataset can be represented , where contains all features matrix from the th view, is the corresponding feature dimension and is the total number of samples. The aim of our method is to learn hash code to represent multiview data, where is the binary code length. And some important formula symbols are summarized as follows in table 2.
Notation  Description 

Feature matrix of the th view data  
Encode each feature vector.of the th view  
Map matrix for features in the th view  
Collaborative binary code matrix  
The hash code representation to the th sample  
The anchor samples from the th view  
The weighting factor for the th view  
Set of all features Laplace matrix  
The spares relationships for the ith feature in the th view  
The dimension of features in th view  
thnearest points in with the th view 
3.1 Kernelization from Multiple Views
We normalize the data from each view to maintain the balance of the data. Since the dimensions of different views may be various, we demand to find an effective method to embed multiview data into a lowdimensional representation space.
In order to obtain lowdimensional representation, GMBL adopts nonlinear kernel mapping for each view. Inspired by zhang2018binary the simple nonlinear RBF kernel mapping method was used to encode each feature vector. GMBL adopt the above technique to explore various information for each view as follows:
(4) 
Where is the kernel width, and are the anchor points are randomly selected from the th view. In the algorithm, we choose the number of anchor points to mapping based on the size of the dataset. Besides, projecting data into the kernel space can avoid the problem of uneven dimensions. represents the dimensional nonlinear embedding of data features from the th view .
3.2 Common Discrete Binary Code representation
The features of different views are mapped into hash codes in Hamming space by the projection matrix. The representation of the hash code is ,where is the common binary code representation of the th sample from different views. is a sign operator function. is the projection matrix of the th view. GMBL combines different views to embed them simultaneously into a common Hamming subspace. The purpose of our method is learning an efficient projection matrix
, which to map all samples in the original space into binary code. Therefore, we construct a minimizing loss function as follows:
(5) 
Here is a binary code for the th samples. By optimizing the above formula, we can get an efficient binary code. It is important to note that learning equilibrium and stable binary codes by using regularized item constraints. In general, using the maximum entropy principle that the equation can be rewritten as:
(6)  
Where is a nonnegative normalized weighting vector assigned according to the contributes of different views. ¿1 is the weight parameter, which ensures all views have a special contribution to the final lowdimentional representation. The first item of the equation ensures to learn efficient binary code for multiview data. The last two terms of the equation are constraints for learning binary code. In this way, adding regularization on can ensure a balanced partition and reduce the redundancy of binary codes.
3.3 Graphbased binary code learning
This section introduces the method of similarity preservation for mapping data to binary codes. Due to the existence of similar underlying structures in different views, the structural features of the original data should also be considered when learning the binary code projection matrix. Keeping the similarity of data is one of the key problems of the hashing algorithms, which means that similar data should be mapped to binary codes with short Hamming distance. Based on this problem, we propose a method to construct a similarity matrix, which can not only preserve the local structure of the data but also preserve the similarity between the data. Then, we introduce the similarity preservation method to map data into binary codes.
In many graphbased hash methods, a key step in similarity preservation is to build neighborhood graphs on the data. For the th view of each data point, we pick up all points set from to reconstruct . Where is one of nearest points. Thus, The optimization equation can be obtained as follows:
(7) 
By solving Eq.(7), we get
(8) 
Where , is a covariance matrix. Where are described the relationship between data points, which we can use to define the similar matrix as:
(9) 
Where denotes the th neighbor between and in , i.e.. In order to ensure the symmetry of matrix , we need to operate with . We consider setting weights for similar matrices from different views rather than simply accumulating similar matrices. i.e., where is a weight vector. Therefore, the similarity preservation part can be calculated as follows:
(10) 
Where is the hash code representation to the th sample, and is the length of the hash code. The last two constraints force the binary codes to be uncorrelated and balanced, respectively. Eq.(10) can be organized as:
(11) 
Where , is a diagonal matrix given by , and is the graph Laplcian matrix.
3.4 Overall Objective Function
In order to learn binary codes associated with the clustering task, we find that the binary code representation learning and the discrete similarity preserving is both crucial. At last, We combine similarity preservation with binary code learning into a common framework as follows:
where , and are regularization parameters to balance the effects of different terms. To optimize the complex discrete coding problems, an alternating optimization algorithm is proposed in the next section.
4 Optimization Algorithm
We have constructed a general framework named GMBL which can combine discrete hashing representation and structured binary clustering for multiview data. We apply an alternative iterative optimization strategy to optimize the proposed objective function. The problem is resolved to separate the problem into several, which are to update a variable while fixing the remaining variables until convergence. In order to fully understand the proposed GMBL method, we summarize in Algorithm 1.
Updating : When fixing other variables, we update the projection matrix by:
(12) 
It closedform solution can be obtained by setting partial derivative , whose optimal solution is , where .
Updating : We next move to update , the subproblem with respect to the are defined as follows:
(13) 
We design an effective algorithm that can maintain discrete constraints in the optimization process, and through this method we can obtain more efficient binary codes shen2015supervised ,shen2016fast . According to the DPLM algorithm, we can get as follows:
(14) 
Where is the gradient of . We update variable use to in each iteration.
Updating : According to the attributes of different views, Optimization of can be equivalent as the following optimization problem:
(15) 
Let then we can rewritten (15) as
(16) 
We can solve the constraint equation by Lagrange multiplier method, the Lagrange function of (16) is
By setting the partial derivative of with respect to and to zero, we can get
(17) 
Therefore, we can get as
(18) 
In order to obtain the local optimal solution, we update the three variables iteratively until the convergence.
5 Experimental Evaluation
In this section, extensive experiments are the command to evaluate the proposed binary clustering methods in clustering performance. All the experiments are conducted with Matlab 2018b using a standard Windows PC with an Intel 3.4 GHz CPU.
5.1 Experimental Settings
In this section, we describe the datasets and comparison methods. We evaluated the clustering performance of GMBL by comparing it with several classical hash methods in the multiview datasets. In addition, the effectiveness of GMBL algorithm is evaluated by comparing the realvalued multiview methods. In the end, we compared the singleview lowdimensional embedding in the framework with the original GMBL lowdimensional embedding to verify that our method can modify and supplement complementary information between different views.
5.1.1 Datasets
In most practical applications, images are generally represented by multiple features, which constitute multiple views of the experiment. Without loss of generality, we evaluated our image cluster method using five popular datasets. Some Image samples of datasets are presented in Fig.2. The details of utilizing the data information are listed as follows:
Caltech101^{1}^{1}1http://www.vision.caltech.edu/ImageDatasets/Caltech101/
contains 9144 images associated with 101 objects and a background category. It is a benchmark image dataset for image clustering and retrieval tasks. Each example is associated with a reciprocally class label. For this dataset, five publicly available features are engaged for experiments, i.e. 48dim Gabor feature, 928dim LBP feature, 512dim GIST feature, 254dim CENTRIST feature, 40dim wavelet moments and 1984dim HOG feature.
Caltech256^{2}^{2}2http://www.vision.caltech.edu/ImageDatasets/Caltech256/ contains 30,607 images of 256 object categories, each of which contains more than 80 images. We use a 729dim color histogram feature, 1024dim Gist feature and 1152dim HOG feature, which three different types of features.
NUSWIDEobj^{3}^{3}3http://lms.comp.nus.edu.sg/research/NUSWIDE.htm contains 30,000 images in 31 categories. The features of the dataset can be found on the contributor’s home page, including 65dim color histogram(CH), 226dim color moments(CM),74dim edge distribution(ED), 129dim wavelet texture(WT) and 145dim color correlation(CORR).
Coil100^{4}^{4}4http://www1.cs.columbia.edu/CAVE/software/softlib/coil100.php is the abbreviation of the Columbia object image library dataset, which consists of 7200 images in 100 object categories. Each category contains 72 images and all images are with size 32×32. Intensity, 220dim DSD, 512dim HOG and 768dim Gist features are extracted for representation.
CiteSeer^{5}^{5}5http://ligmembres.imag.fr/grimal/data.html
consists of 3,312 documents on scientific publications. These documents can be further classified into six categories: Agent, AI, DB, IR, ML and HCI. For our multiview learning clustering, we construct a 3703dimensional vector representing the keywords of text view and a 3279dimensional vector representing the reference relationship between another document. All the features of discretion are briefly summarized in table
3.Datasets  Caltech101  Caltech256  NUSWIDEobj  Coil100  CiteSeer 

Samples  9144  30608  30000  7200  3312 
Classes  102  175  31  100  6 
Views  6  3  5  3  2 
5.1.2 Compared Methods
We compared our approach with the following stateoftheart methods, including hash multiview and realvalue multiview methods for clustering. As for the hash method, we utilized seven famous singleview hash algorithms and two multiview hash clustering algorithms as comparing methods, including LSH gionis1999similarity , MFH song2013effective , SH weiss2009spectral , DSH jin2013density , SP xia2015sparse , SGH jiang2015scalable , BPH, ITQ gong2012iterative , BMVC zhang2018binary , HSIC zhang2018highly
. For singleview hash methods, we adopted the best result of each feature clustering. As for the realvalue multiview method, we adopted seven algorithms as comparing methods, including kmeans
jetsadalak2018algorithm , SC von2007tutorial , Coregularize kumar2011co , AMGL nie2016parameter , MulNMF liu2013multi , MLAN nie2017multi . It is noteworthy that the kmeans method concatenates multiview data into one vector as the evaluation result. The length of the hash code used in the experiment is 128bits. We use the source code from the author’s homepage for comparative experiments.5.1.3 Evaluation Metrics
To generally evaluate the performance for clustering, We report the experimental results using four most widely used evaluation metrics, including accuracy(ACC), normalized mutual information(NMI), Purity and Fscore
gao2015multi , cao2015diversity . For all algorithms, the higher value of metrics indicates better performance. For the hashing methods, five different bits coding length are used for all datasets.5.2 Hash Method Experimental Results and Analysis
In the section, we conducted experiments for hash clustering on 5 datasets (Include Caltech101, Caltech256, NUSWIDEobj, Coil100 and CitySeer) to prove the performance of our proposed method. We utilized all methods to project multiview features into five hash codes of different lengths and adopted the kmeans method to finish the task of image clustering. The results with different code lengths on four benchmark image datasets are reported in Figures 3, 4, 5 and 6. Table 4 shows the results when the hash code length is 128bits in text clustering from two views. We have the following observations:
For Caltech101 datasets, we adopted six views to complete the clustering task, which is the dataset with the most views in the experiment. We adopt a view with the best clustering performance is used to evaluate singleview hash methods experiments. It is clear that GMBL can achieve better performance than the other hash methods in different binary code lengths. Generally, the results of multiview algorithms are better than singleview hash ones. It shows that in GMBL, the result increases with the increase of the hash code length. GMBL can obtain better results compared with the multiview hash methods. Because GMBL can construct a similarity matrix to obtain the nearest neighbor relation of data, the optimal result can be obtained when the length of the hash code increases. It can be found from Fig.3 that when the hash code length of our method is short, the clustering result can’t obtain better performance. The reason may be the hash code length is short, which the nearest neighbor relationship of the data is not well preserved. In the algorithm of this paper, several parameters have a significant influence on the experiment in Eq.(3.4). Generally speaking, larger the values of and , the experimental results attempt to lower. It is possible that the restriction of large regular terms will restrict the efficient learning of hash codes. Through many experiments found that increasing the value of will reduce the clustering performance while increasing the value of the nearest neighbor parameter will improve the clustering result of GMBL. However, the running time will be affected by the number of nearest neighbors and our method selects nearest neighbors for clustering. If more strategies are adopted to select anchor points, clustering performance will be improved. Besides, when the number of anchor point is greater than 1000, the clustering performance is not significantly improved.
For the Caltech256 dataset, we randomly selected 175 categories of images as the experimental data with a total of 20222 images. It is explicit from Fig.4 that our approach obtains the perfect results in terms of ACC, NMI, and Purity among all the compared other hash methods in 128bits binary code. From Fig.4, we can observe that GMBL outperforms other hash methods when the code length is relatively large (i.e., greater than 32). As the length of the binary code increases, the performance of all algorithms improves. On the Caltech256 dataset, our model achieved 0.29 on NMI when the code length was 128bits while the secondhighest NMI was 0.26.
Fig.5 illustrates the experimental results on the Coil100 dataset. Our method outperforms all the other methods of NMI and Purity evaluation metrics. For the ACC, HSIC method obtains the best result and we can deduce that using individual information and shared information to capture the hidden correlations of multiple views is necessary. We can also find that BMVC and HSIC can achieve the best performance in the short hash code lengthIt can be observed from the Fig.5 that the results of roughly all singleview hash methods are significantly poorer than that of multiview hash methods under different hash code lengths.
As we all know, NUSWIDEobj is a widely used dataset that includes 30000 images from Flickr. The dataset consists of 31 categories and each image is marked by at least one label and multiple labels can be assigned to each image. The dataset was divided into 12072 test sets and 17928 train sets by the provider. In this algorithm, we use the test set to evaluate the clustering task. For multiple labels with the same sample, we will automatically specify the corresponding labels groundtruth after the clustering task of each algorithm is completed. In Fig.6, the ACC, NMI, and Purity of all hash algorithms under 128bits binary code are reported. Box diagram corroborates the advantages of our GMBL relative to its simulated alternatives.
Experiments demonstrate that the singleview hash method can also obtain satisfactory performance. Moreover, both SGH and ITQ get excellent performance. In general, multiview methods get better results than singleview methods. Generally speaking, Multiview methods exploit multiview information and achieve better results. For multiview data that holds more of the original data structures during the construction of the binary code that our method obtains a similarity matrix reflecting the local structure of the original data. Therefore, graphbased similarity matrix construction plays an important role in our method, which can effectively obtain the structural relationship between the initial input and adjacent data. GMBL has been verified can improve Largely clustering performance. We can notice that GMBL achieves much better results than BMVC on almost all datasets. The primary reason is that in the process of binary code, we can keep the local structure of data to further explore the internal relationship of data, so as to obtain better clustering results.
Methods  LSH  SP  MFH  ITQ  SH  BPH  DSH  BMVC  HSIC  GMBL 

ACC  0.1413  0.1887  0.1954  0.2017  0.2126  0.1730  0.2292  0.2343  0.2560  0.2766 
NMI  0.0105  0.0432  0.0079  0.0431  0.0026  0.0124  0.0353  0.0298  0.0298  0.0517 
Purity  0.2542  0.3222  0.2225  0.3206  0.2183  0.2292  0.2497  0.2844  0.2844  0.3273 
To demonstrate the robustness of the GMBL, we consider the clustering experiment in the text dataset. There are 3312 documents in CitySeer dataset, which are divided into six categories. We use keywords and references between documents as two views for clustering experiments. Table 4 compare ACC, NMI, and Purity when the length of the hash code is 128bits. We have the following observations: GMBL obtained a higher value in the three indexes of the clustering task, which consistently outperforms other methods by large margins in all situations. Compared with singleview hashing method, ACC, NMI and Purity were increased 24%49%, 17%90% and 3%33%; Compared with multiview hashing method, ACC, NMI and Purity were improved 7%16%, 42%60% and 13%17%; The results demonstrate that the proposed multiview algorithm is effective in utilizing the graphbased method.
In order to verify more clearly, Fig.7 demonstrate the clustering performance of the algorithm in GMBL method of single views, evaluated by ACC, NMI and Purity respectively. We can notice that our multiviews method get higher results than the GMBL methods of single views. Fig.7 illustrates the clustering results of GMBL algorithm from different views on the Caltech101 and Coil100 dataset. Multiview GMBL still obtains higher results than a single view of GMBL when the clustering result of a certain view is remarkably efficient. In particular, with 128bits, our multiview method exceeds the best of singleview GMBL methods by more than 24% and 67% in terms of ACC, NMI and Purity, respectively. Thus, multiview methods explore common cluster structure work better compared with singleview methods.
5.3 Comparison with stateofart multiview methods
In this section, we present the detailed clustering results of three datasets in tables 5, 6 and 7. In each table, the bold values illustrate the best clustering performance. These tables indicate that GMBL achieves excellent performance in four evaluation indexes of three datasets and is superior to other methods by Caltech101, clatech256 and NUSWIDEobj datasets. GMBL by learning discrete coding and realvalue representation of multiview clustering and has obtained encouraging results. In addition, even though the kmeans clustering method concatenates all multiple views into a vector, it can not achieve efficient clustering performance. Because kmeans clustering is essentially a singleview clustering method. In the experiment, the length of the hash code is set to 128bits when compared with the realvalued multiview clustering method.
Methods  kmeans  SC  Corec  Cores  AMGL  MulNMF  MLAN  BMVC  HSIC  GMBL 

ACC  0.1331  0.1365  0.2670  0.2425  0.1350  0.2018  0.1807  0.2930  0.2578  0.3070 
NMI  0.3056  0.3269  0.4691  0.4683  0.2645  0.4089  0.2686  0.4900  0.3511  0.4982 
Purity  0.2909  0.3187  0.4600  0.4694  0.1569  0.2300  0.3286  0.4907  0.3492  0.5008 
Fscore  0.1895  0.0955  0.2295  0.1867  0.0319  0.1705  0.0481  0.2466  0.2502  0.2586 
Table 5 shows that the clustering results on the Caltech101 dataset. GMBL outperforms all the other methods on ACC, NMI, Purity, Fscore and improves the baseline kmeans more than 50%. kmeans is the worst method because the views are directly concatenated together and more noise will be introduced. The Coregularization method expresses different views through coregularization spectrum clustering to pursue a better clustering index, which is suitable for experiments with fewer perspectives and takes a longer time. Compared with realvalue multiview methods, GMBL has been improved significantly. The main reason is that BMVC learns the binary code of the different views in Hamming space and improve calculation efficiency. However, the calculation of distance in Euclidean space by realvalue multiview method has low efficiency and high time cost. However, pursuing a similar matrix by graphbased clustering is timeconsuming, Compared with the hash multiview method, our calculation time is shorter.
Methods  kmeans  SC  Corec  Cores  AMGL  MulNMF  MLAN  BMVC  HSIC  GMBL 

ACC  0.1001  0.0924  0.1030  0.0738  0.0467  0.0713  0.0693  0.1028  0.0971  0.1049 
NMI  0.1184  0.2764  0.2856  0.2467  0.1070  0.2272  0.0794  0.2915  0.2503  0.2949 
Purity  0.1018  0.1339  0.1602  0.1070  0.0415  0.1119  0.0922  0.1428  0.1184  0.1475 
Fscore  0.0804  0.0628  0.0727  0.0415  0.0466  0.0458  0.0224  0.0781  0.0719  0.0878 
For Caltech256 dataset, we randomly selected 20222 samples as experimental data, which extracts three features and 175 categories of pictures. The clustering results with different methods can be found in table 6. GMBL method outperforms all the other methods on four evaluation metrics. Compared with the realvalued multiview method, the hash method has obvious advantages in large datasets and takes the least time in clustering. Compared with other multiview hash methods, the clustering performance of GMBL is improved. Therefore, it is very important to maintain the original spatial structure in the process of learning binary codes.
Methods  kmeans  SC  Corec  Cores  AMGL  MulNMF  MLAN  BMVC  HSIC  GMBL 

ACC  0.1459  0.1360  0.1521  0.1625  0.1281  0.1183  0.1554  0.1508  0.1621  0.1682 
NMI  0.1415  0.1289  0.1505  0.1604  0.1362  0.1029  0.1199  0.1527  0.1625  0.1649 
Purity  0.2576  0.2460  0.2816  0.2826  0.1484  0.1975  0.2604  0.2855  0.2790  0.2968 
Fscore  0.1105  0.0840  0.1038  0.1018  0.1125  0.1128  0.1136  0.1090  0.1190  0.1126 
We used the test set of the NUSWIDEobj dataset to complete the clustering task in table 7 methods. Since some images in the dataset have multiple labels, the most representative label was adopted as groundtruth in our comparative experiment. It can be found that our method is superior to other methods in three indicators. The adoption of the similarity matrix is more conducive to the hidden structure of mining data, but the use of the hash method only improves significantly in time, and the evaluation result does not improve significantly. In addition, GMBL takes a lower evaluation index Fscore than HSIC.
5.4 Visualization
In Fig.8, and are the visualization effects of binary codes and original data using tSNE maaten2008visualizing on the caltech101 dataset (we randomly selected 5 classes). The original data links all six features into a vector as input. In addition, and are the visualization effects of binary code and full connection raw data using tSNE in the Coil100 dataset (we randomly selected 10 classes). In Fig.8, different colors geometric figures belong to various categories and the clustering results are well when the same kinds are adjacent to each other. We observed that the visualization of binary codes was more discriminating than the original data because each category in the graph was more scattered in the visualizations.
All the experiments above can verify the excellent performance of our proposed GMBL. It can extend the Euclidean space measure method to binary code in Hamming space. Through the experiment results, GMBL is better than the realvalue multiview method exceeds in most situations. Compared with the hash method, the local structure of the data preserved by constructing the similarity matrix can effectively improve the clustering performance, which is stronger than the most hash algorithm.
5.5 Convergence analysis
We adopted the Alternating iterative optimization method to iteratively update all the pending parameter matrix in our optimization problem. Fig.9 indicates the objective function values on the Caltech101 dataset. We observe that the values of our objective function on the dataset decrease rapidly in each iteration and access to a point. It can be identified that the constructed function is monotone and convergent and has minimum values.
6 Conclusion
In this paper, we propose a discrete hash coding algorithm based on graph clustering, named GMBL. GMBL learns efficient binary code, which can fully explore the original information of multiview data and reduce the lack of information. With the Laplace similarity matrix, the proposed algorithm can be preserved in the local linear relationship of the original data and the multiview binary clustering task can be well optimized. In addition, since various views contribute differently to cluster tasks, we assign weights to different views of adaptively giving to their contributions. In order to optimize the binary code, we adopt the alternating iteration method to directly optimize it instead of the loan constraint. It can be found that different from the traditional realvalued multiview clustering method, the hashing clustering method can effectively reduce the experimental time. We evaluated our proposed framework on five multiview datasets for experimental examination. The experiment demonstrated the superiority of our proposed method.
7 Acknowledgements
This work was supported in part by the National Natural Science Foundation of China Grant 61370142 and Grant 61272368, by the Fundamental Research Funds for the Central Universities Grant 3132016352, by the Fundamental Research of Ministry of Transport of P. R. China Grant 2015329225300, by Chinese Postdoctoral Science Foundation 3620080307, by the Dalian Science and Technology Innovation Fund 2018J12GX037 and Dalian Leading talent Grant, by the Foundation of Liaoning Key Research and Development Program.
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