1 Introduction

Semi-supervised learning (SSL) [1, 2] is one of the branches of machine learning, which uses a small number of labeled data and a large number of unlabeled data to obtain high performance classifier, and lacking enough labeled data is the characteristic of most data sets in the real world.

There are many methods in SSL such as self-training methods [3, 4], graph-based method [5], generative model [6] and co-training methods [7, 8]. In co-training algorithm, two different classifiers on two different views are trained firstly, then unlabeled data are labeled through the complementary effects of two views. The process of standard co-training algorithm is shown as follows: 1) The two different classifiers are trained on two different views by labeled data; 2) The trained classifiers are used to classify unlabeled data; 3) Unlabeled data with high confidence are selected, and they are added to the other view respectively; 4)The classifiers, the labeled data set and the unlabeled data set are updated; 5) Repeat 3) and 4) until the iteration stop condition is satisfied.

Standard co-training algorithm requires two sufficient and redundant views, that is, the attributes can be naturally partitioned into two sets, each set is sufficient to learn a classifier with good performance, and they are both independent of each other [9]. But in practice, such conditions are difficult to be satisfied. Work [10] mentioned that the co-training algorithm had acceptable performance of using the randomly divided feature set when it learned from labeled data and unlabeled data. Work [11] proposed a heuristic division to split single view data sets into two views, and it could make co-training work reliably well. Tri-training [12] used the full view of three random data to label unlabeled data, and this method relaxed the condition that co-training algorithm requires two fully redundant independent views.

In co-training algorithm, the selection of unlabeled data has a great influence on the classifier performance. Zhang and Zhou [13] utilized data editing technique to estimate the confidence degree of unlabeled data, and the finding of Angluin and Laird [14] were adopted on the selected data to avoid generating noise points. Work [15] proposed a semi-supervised learning framework combining clustering and classification, which made use of the semi-supervised fuzzy c-means algorithm to find data spatial structure hidden by unlabeled data. However, Euclidean distance could not reflect the correlation between attributes, and it was difficult to choose appropriate parameters. Then Gong and Lv [16] used semi-supervised metric based fuzzy clustering (SMUC) to select unlabeled data. This method could select unlabeled data with implication information, but its calculation cost was too high. Thus, a co-training method combined with active learning and density peaks clustering (CTALDP) was proposed [17], which utilized density peak clustering [18] to select unlabeled data instead of semi-supervised clustering. The above methods all select data by high confidence criterion, the reason is that high confidence data are more likely to be predicted correctly [19].

Unfortunately, it is not always able to improve the classification performance with the high confidence unlabeled data [20]. Work [20] not only considered the high confidence of the data, but also added the nearest neighbor criterion in the selection of unlabeled data. Work [21] chose the data which were located at the margin of the current classifier and the confidence level above the preset threshold. Another problem in co-training algorithm is that the information of labeled data is not used in the process of selecting unlabeled data, but labeled data often provide more correct hidden information than unlabeled data, no matter how high confidence of selected unlabeled data.

In order to solve these problems above, a new co-training algorithm based on the entropy and multi-criteria (CTEMC) is proposed. Specifically, the view partitioning adopts entropy to get two views with the same amount of information, the selection of unlabeled data use multi-criteria to solve the problem that high confidence criterion is not always effective. Another advantage of multi-criteria is that it takes into account the difference between the two views of co-training, and the role of difference in co-training has been confirmed by many studies [22, 23]. Moreover, in CTEMC, labeled data are utilized to find more reliable unlabeled data. The major contributions of CTEMC are two-fold:

  1. (1)

    A novel co-training framework is defined, which uses two different criteria to select unlabeled data in two views and then they are added to the other view. The complementarity of co-training guarantees mutual promotion on two criteria.

  2. (2)

    In the two selection criteria, the effect of labeled data is fully considered, because the information of labeled data is more accurate than unlabeled data.

The rest of the paper is organized as follows. Section 2 describes related theory including entropy, K-means clustering algorithm and Naive Bayes algorithm. Section 3 introduces view partitioning, the two criteria of the proposed algorithm and presents the algorithm in detail. Experimental results on real data sets and further analysis are given in Section 4. Section 5 concludes this paper and indicates several issues for future work.

2 Related theory

Co-training algorithm was proposed by Blum and Mitchell [24], which was inspired by the task of learning to classify web pages, and they wanted to use a large number of unlabeled data to improve the performance of the classifier. The co-training method is simple and effective, it has been extensively researched. In co-training, the two views of the data set may exist naturally, or they can be obtained by dividing a relatively large attribute set. A common variant of co-training algorithm is to generate different classifiers by using different learning algorithms. The decision tree can divide the data space into several equivalence classes, thus Goldman and Zhou [22] used two different decision tree algorithms to get classifiers. Hady and Schwenker [23] proposed the framework of Co-Training by Committee (CoBC), using different classifiers to learn from each other during the training process, and the difference between the classifiers effectively improves the performance of the collaborative training algorithm. Moreover, tri-training [12] is also one of the co-training variant algorithm. The above methods all want to use the difference to obtain a better-performing classifier.

Many researches on co-training are about the selection of unlabeled data [13, 15,16,17], however, the information of labeled data are rarely used. Obviously, labeled data can provide more correct information. It is also a waste if the information of labeled data are not used. Therefore, in our proposed method, different selection strategies are used to select unlabeled data on two views. This method makes use of the difference to provide more information to another view. At the same time, labeled data are considered in both clustering criterion and confidence criterion.

The rest of this section introduces some related theories which used in CTEMC, including entropy, K-means clustering algorithm and Naive Bayes algorithm. The entropy is used in view partitioning to estimate the information of each attribute, K-means clustering algorithm is used in clustering criterion to find hidden space structure of data set, and Naive Bayes algorithm is used in confidence criterion to calculate the posterior class probability of unlabeled data.

2.1 Entropy

The concept of Information is abstract, and it is difficult to evaluate in a quantitative form until Shannon [25] proposed mathematical formula named entropy. Generally, if the value of entropy is greater, the uncertainty of the event and the amount of information is greater.That is, the probability of an event occurring is 1 or 0, it means that the event will definitely occur (1) or will not occur (0). In this case, the certainty of the event is very high, the value of entropy is small. On the contrary, if the probability of events occur tends to be consistent, the uncertainty of the event is high, the value of entropy is larger.

Assume there are some events S = {s1,s2,...,sn}, where the probability of si is pi(0 ≤ pi ≤ 1), and p1 + p1 + ... + pn = 1. So the amount of information of each event si can be calculated as formula (1).

$$ I\left({s}_i\right)=-{\log}_2\left({p}_i\right) $$
(1)

The I(si) shows the occurrence of event si. It only focus on one event, but in most cases, the I(S) of the entire event set is usually showing more meaningful. Obviously, I(S) can be expressed as follows.

$$ I\left(\mathrm{S}\right)=I\left({s}_1\right)+I\left({s}_2\right)+\dots +I\left({s}_n\right) $$
(2)

The entropy is used to evaluate the average amount of information of the entire event set S, and the best average measure is the expectation of the random event si, that is, the calculation formula of entropy is as follows [24].

$$ I\left(\mathrm{S}\right)=-\sum \limits_{i=1}^n{p}_i{\log}_2\left({p}_i\right) $$
(3)

2.2 K-means clustering algorithm

K-means clustering algorithm [26] is a common algorithm, which has been widely used [27, 28] because of its simple idea and easy implementation. The result of clustering is that the similarity of data belonging to the same cluster is as high as possible, and the similarity between different clusters is as low as possible. K-means clustering is an unsupervised algorithm and it is often used in data mining. The characteristic of data can be found by k-means clustering, so that the data can be further processed. The division of original K-means algorithm often adopts Euclidean distance. The optimization objective is to minimize the sum of squares of the distance from each datum to each center of the cluster. Define D = {x1,x2,...,xm}, C = {C1,C2,...,Ck} is k-mans clustering center, squared error-sum can be expressed as [29]:

$$ E=\sum \limits_{i=1}^k\sum \limits_{x\in {C}_i}{\left\Vert x-{u}_i\right\Vert}_2^2 $$
(4)

The calculation formula of ui is as follows.

$$ {u}_i=\frac{1}{N_i}\sum \limits_{x\in {C}_i}x $$
(5)

Where Ni is the number of examples of cluster i. E describes how closely of examples in the cluster surround the cluster center, the smaller E is, the higher similarity of example in the cluster will be, the effect of clustering is better. Unfortunately, minimizing E is not easy, it is an NP-hard problem, and a greedy strategy is usually used to iteratively find an approximate solution. The iteration process of K-means clustering algorithm is carried out until the iteration stop condition is satisfied: 1) The cluster center is no longer changed or the objective function is minimized; 2) The number of iterations reaches the initial setting.

2.3 Naive Bayes algorithm

Naive Bayes is a classification algorithm based on probability theory [30], and its result represents the uncertainty of all conditions and implicates the degree of confidence in the possibility of different labels. Hence, many researchers used it to estimate the confidence of data [31, 32]. Naive Bayes requires relative independence between the attributes of the data set in theory. Its classification ability is stable and has a small error rate under this condition. This assumption, however, is often invalid in practical applications. So assuming that the attributes are mutually independent to make Naive Bayes algorithm simplified. Although this simplification method reduces the classification effect of the naive Bayes classification algorithm, it greatly simplifies its complexity in actual application scenarios.

Let x = {a1,a2,...,am} denotes a test data, and a1,a2,...,am are the attributes of x, m is the number of the attributes of x. Let y = {c1,c2,...,ck} denotes the label set, and k denotes the number of the labels. Then the probability P(aj|ci), j = 1,2,...,m, i = 1,2,...,k are calculated. Assuming that the attributes are independent of each other, the probability of x can be calculated as [33]:

$$ P\left({c}_i\left|x\right.\right)\propto P\left(x\left|{c}_i\right.\right)P\left({c}_i\right)=P\left({c}_i\right)\prod \limits_{j=1}^mP\left({a}_j\left|{c}_i\right.\right),i=1,\cdots, k $$
(6)

3 A co-training method based on entropy and multi-criteria

The proposed method consists of two parts: the first part is view segmentation, and another part is multi-criteria strategy. In view partitioning, the entropy is used to obtain two views with same amount of information, more description is given in Section 3.1. In multi-criteria strategy, the structure of standard co-training algorithm is changed, and it has two different criteria for selecting unlabeled data in two views, instead of only one criterion in standard co-training algorithm. This change aims to solve the problem that high confidence strategy is not always effective, it can also play the complementary role of co-training through the difference between the two selection criteria, hence more useful unlabeled data can be found. In the multi-criteria, each view has its own strategy to select unlabeled data, and its framework can be described as Fig. 1. View 1 uses clustering criterion (criterion 1) to select unlabeled data, view 2 uses confidence criterion (criterion 2) to select unlabeled data. The selected unlabeled data are added to the opposite view respectively to achieve the complementary effect of co-training method.

Fig. 1
figure 1

The framework of multi-criteria co-training

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Labeled data can provide more correct information than unlabeled data, so labeled data are emphasized in both clustering criterion and confidence criterion. Figure 2 is an example of the role of labeled data. Red circle represents labeled data, and blue rhombus represents unlabeled data. In this figure, the confidence of unlabeled data ‘b’ is 0.988, and the confidence of unlabeled data ‘a’ is 0.985. From a numerical point of view, the confidence of point ‘a’ is lower than the confidence of point ‘b’, but the distance between instance ‘b’ and labeled data is much higher than instance ‘a’ to labeled data, then we think unlabeled data ‘a’ would be a better choice than unlabeled data ‘b’.

Fig. 2
figure 2

The example of the role of labeled data

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Specific clustering criterion and confidence criterion are given in 3.2 and 3.3, the whole methods are described in 3.4.

3.1 View partitioning

A simple way to divide a single view into two views is using entropy [34]. The value of entropy is a mathematical representation of the amount of information, thus it can reflect the importance of features.

Let D = {D1,D2,...,Dm} denotes a data set, and D1,D2,...,Dm are the features of D, m is the number of the attributes of D. In order to get two views with the same amount of information, firstly, the entropy of each feature based on the whole data set is calculated, then they are sorted by their entropy. Finally, odd position features are assigned to view 1 and even position features are assigned to view 2. The entropy calculation formula in view partitioning can be written as follows.

$$ Info\left({D}_j\right)=-\sum \limits_{i=1}^k{p}_i{\log}_2\left({p}_i\right) $$
(7)

Where Dj (j = 1,...,m) is the data set which need to calculate the entropy, pi is the probability that Dj belongs to class i, and k is the number of classes.

3.2 Clustering criterion

K-means clustering algorithm has many advantages, but one of the disadvantage is that the clustering result may be not optimal on account of the randomly selected initial clustering centers.

In the real world, the data in a class consist of many subsets [35]. N. Piroonsup et al. [36] proposed that if the data in a class are clustered into different sub-classes, and the clustering results are more consistent with the data distribution. Therefore, the initial clustering center in the proposed clustering criterion is defined as labeled data. Thus it can avoid the bad effort brought by randomly initial clustering centers. Another superiority is fully considered the information of labeled data. The detailed steps for selecting unlabeled data using clustering criterion are shown in Algorithm 1. At Line 2, the process of clustering criterion will stop when cluster centers are not changed anymore.

figure c

3.3 Confidence criterion

Confidence criterion is a common criterion to select the unlabeled data in co-training algorithm. Nevertheless, researchers often ignore the role of labeled data in confidence criterion. The proposed confidence criterion in this paper uses naive Bayes to obtain the posterior class probability of each datum firstly, those data with high posterior probabilities can be seen high confidence data, then the distance between the data with high confidence and labeled data is calculated. Finally, the high confidence unlabeled data which are closer to labeled data are selected. The selected unlabeled data have two characteristics: 1) Compared with other high-confidence data, the distance between the selected data and labeled data is relatively small; 2) High confidence. Algorithm 2 shows the detailed process of confidence criterion. At Line 2, the confidence of unlabeled data is calculated by Naive Bayes algorithm.

figure d

3.4 CTEMC

Before the iteration process of CTEMC, view partitioning by entropy is firstly carried out. Secondly, unlabeled data with high ambiguity are selected and they are added into the labeled data set by active learning, this step is the prior work in [17]. Thirdly, unlabeled data are selected according to criterion 1 and they are added into view 2. Similarly, criterion 2 are used to select unlabeled data, then they are added into view 1. Finally, the same data with different labels are found and they are relabeled by the weighted K-nearest neighbor algorithm. The progress will stop when all unlabeled data have been labeled. Algorithm 3 describes the CTEMC in detail.

figure e

At Lines 1–5, the unlabeled data with high ambiguity are defined and selected. The data set is divided into two parts by entropy at Lines 7–8. At Line 11, R1 and R2 are selected by clustering criterion and confidence criterion, they are described in Algorithm 1 and Algorithm 2.

4 Experiments

In order to illustrate the effectiveness of the proposed algorithm, 10 UCI data sets and one artificial data set are selected. Those data sets are different in size, the number of attributes (Attribute) and the number of classes(Class). The further description is shown in Table 1. The base classifiers, comparison algorithms and their parameters involved in the experiment are given in Table 2. Naive Bayes and K-nearest neighbor algorithms are selected for analyzing the effect of the algorithms under different base classifiers. More descriptions are shown as follows.

  1. (1)

    K-nearest neighbor(KNN) algorithm is based on instance, the class label of the unlabeled data can be determined by the class label of the K labeled data closest to this unlabeled data. The KNN algorithm does not have a specific target equation, usually, it uses the distance between the data to measure its similarity.

  2. (2)

    Naive Bayes algorithm is based on Bayes’ theorem and has a solid mathematical foundation. The probabilities that the example belongs to each category are calculated first, and then the category with the largest probability is selected as the class of the example. The naive Bayes algorithm can obtain the class of the example and the probability that the example belongs to each category, so it can not only be used for classification, but also can be used to measure the confidence of the data.

Table 1 The description of UCI data sets
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Table 2 The comparison algorithms and parameters
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At the experimental stage, the 10-fold cross-validation is adopted. The training set is divided into labeled data and unlabeled data by random way, that is, part of the data in training set are selected randomly to be the labeled data, and the label of the rest of data are removed to obtain unlabeled data.

The algorithm is evaluated by accuracy, and the computation of it can be described as formula (8), where Tcorrect is the number of data which are predicted correctly in testing set and T is the number of testing set.

$$ accuracy=\frac{1}{10}\sum \limits_{i=1}^{10}\frac{T_{correct}}{T}\times 100\% $$
(8)

4.1 Feature partitioning result

To illustrate the effectiveness of view partitioning based on entropy, a comparative experiment is conducted on 15 UCI data sets. And the another two view partitioning methods are view complementarity and random view. If a data set consists of eight attributes, view complementarity means that view 1 consists of the first four attributes and view 2 is comprised of the remaining attributes. Random view, that is, the data set is divided into two views randomly. The proportion of labeled data is 10% and the experimental results are shown in Table 3.

Table 3 Accuracy of different view partitioning methods when the proportion of labeled data is 10%
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As shown in Table 3, the accuracy of entropy is higher than view complementary and random view in most cases, the average accuracy of entropy also achieves the highest. It proves the superiority of the entropy to some extent. The performance of view complementary and random view is poor, due to the considerable randomness in feature information.

In the subsequent experiments, CTEMC adopt entropy in view partitioning.

4.2 The result of using the information of labeled data

For analyzing the validity of labeled data, our experiment is carried out on the artificial data set, the percentage of labeled data is 10% and the contrast algorithms are CTSMUC and CTALDP. Theses different methods are utilized to choose unlabeled data which need to be labeled. The results are shown in Fig. 3. Figure 3a is the distribution of artificial data set, it contains 2 attributes and three classes of data, the ‘◇’ represents data of class 1, the ‘⭐’ are data of class 2, and the ‘×’ represents data of class 3. Figure 3b, c and d are schematic diagram of unlabeled data are selected by CTSMUC, CTALDP and CTEMC, respectively. The ‘○’, ‘*’ and ‘□’ represent unlabeled data, labeled data and selected unlabeled data.

Fig. 3
figure 3

Schematic diagram of unlabeled data are selected by different methods. a Distribution of artificial data set. b Unlabeled data are selected by CTSMUC. c Unlabeled data are selected by CTALDP. d Unlabeled data are selected by propose method

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From Fig. 3, the selected unlabeled data by SMUC are possibly the edge points when the data sets have not obvious boundary line between clusters, and the selected unlabeled data by CTALDP will be clustered in the part with high density.

Figure 3d shows the result of the information of labeled data. As shown in Fig. 3d, if a cluster contains more labeled data than other clusters, the method of labeled data will select more unlabeled data in this cluster, and these data also conform to the distribution of data set.

4.3 The complementarity of co-training algorithm

For the sake of demonstrating that multi-criteria can make use of the complementarity of co-training, we compare multi-criteria with confidence criterion and clustering criterion in this section. Figure 4 shows the results with base classifier Naive Bayes, and results with base classifier KNN are shown in Fig. 5.

Fig. 4
figure 4

Result of multi-criteria and single criterion with the base classifier Naive Bayes

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Fig. 5
figure 5

Result of multi-criteria and single criterion with the base classifier KNN

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As shown in Figs. 4 and 5, in most cases, the accuracy of multi-criteria is higher than single clustering criterion or single confidence criterion, which reflects the effectiveness of multi-criteria for selecting unlabeled data.

4.4 The performance of CTEMC

In order to prove the effectiveness of the proposed algorithm, CTEMC is compared with SCT, CTSFCM, CTSMUC and CTALDP.

Tables 4 and 5 are the experimental results of the 5 algorithms on 15 data sets, Naive Bayes is used as the base classifier of co-training algorithm in Table 4 and KNN is used as the base classifier in Table 5, the percentage of labeled data is 8%.

Table 4 Accuracy of 5 algorithms with the base classifier Naive Bayes when the proportion of labeled data is 8%
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Table 5 Accuracy of 5 algorithms with the base classifier KNN when the proportion of labeled data is 8%
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As shown in Tables 4 and 5, under most circumstances, the accuracy of CTEMC is higher than the contrast algorithms, no matter the base classifier is Naive Bayes or KNN. In detail, 4/5 of the highest accuracy come from CTEMC in Tables 4 and 5. It also has the highest average accuracy on 15 data sets in both Tables 4 and 5, that means the performance of CTEMC is better. In some cases, the accuracy of CTEMC is not the highest in Tables 4 and 5, which may be caused by different characteristics from different base classifiers. On data sets IONO and IRIS, the performance of CTEMC is worse than the contrast algorithms when the base classifier is Naive Bayes, but it is a opposite situation when the base classifier is KNN. Also, on data sets BA and SEED, the accuracy of CTEMC is lower than the contrast algorithms when the base classifier is KNN, it is, however, completely opposite when the base classifier is Naive Bayes. On data set BCW, the algorithm CTSMUC has the highest accuracy, which maybe because the distribution of the BCW is more suitable for SMUC clustering.

In general, the accuracy of CTEMC shows the superiority from Tables 4 and 5, which means the classifier Naive Bayes and KNN can be trained better by CTEMC.

4.5 Effect of the ratio of labeled data

The further discussion about the influence of percentage of labeled data in our algorithm is given in this part. Figure 6 shows the accuracy of 5 algorithms (CTEMC and its contrast algorithms SCT, CTSMUC, CTALDP, CTSFCM) on 9 data sets when the percentage of labeled data are 8%, 11%, 14%, 17% and 20%, respectively.

Fig. 6
figure 6

Relationship chart of classification accuracy and labeled data ratio of 5 algorithms. a PIMA. b ECOLI. c ABAL. d CB. e IONO. f BA. g IRIS. h SEED. i WINE

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The results are presented in Fig. 6. Observing Fig. 6 we can find that the accuracy of CTEMC is higher than comparison algorithms on PIMA, ECOLI, CB, BA, IRIS, SEED and WINE data sets. It means CTEMC has a better performance in improving accuracy. Compared to contrast algorithm, the accuracy of CTEMC is relative poor on IONO data set, and the reason is that K-means clustering algorithm is not consistent with the distribution of IONO. On ABAL data set, the performance of CTEMC is lower than CTSMUC, but the accuracy of CTEMC is better than SCT, CTALDP, CTSFCM and CTEMC also get a better performance than contrast algorithms when the percentage of labeled data is relatively high.

In general, Fig. 6 shows CTEMC has a better performance under most circumstances.

4.6 Time complexity and time consumption

Finally, The performance of the algorithm is analyzed in terms of time complexity and time consumption. Table 6 is the time complexity and of 5 algorithms, where n is the number of data set and m is the number of selected unlabeled data. The results of time consumption on 6 data sets are shown as Figs. 7 and 8, and the base classifier is Naive Bayes and KNN, respectively. Like Section 4.4 and Section 4.5, the contrast algorithms are SCT, CTSMUC, CTALDP, CTSFCM and CTEMC, the parameters of those algorithms are shown in Table 2, and the percentage of labeled data is 8%.

Table 6 Time complexity of 5 algorithms
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Fig. 7
figure 7

Result of time consumption with the base classifier Naive Bayes

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Fig. 8
figure 8

Result of time consumption with the base classifier KNN

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As can be seen from Table 6, the time complexity of SCT, CTALDP, CTSFCM and CTEMC are equal, and CTSMUC has the highest time complexity. This is because the time-consuming Mahalanobis distance is adopted in SMUC when calculating the distance between data. This phenomenon can also be seen from Figs. 7 and 8, the time consumption of CTEMC (blue column in the histogram), CTSFCM (yellow column), CTALDP (gray column), and SCT (black column) tend to be equal, while the time consumption of CTSMUC (orange column) is much higher than 4 other algorithms. In general, in terms of time consumption and time complexity, the algorithm proposed in this paper does not have a poor performance.

5 Conclusions

In this paper, a new co-training framework based on entropy and multi-criteria is introduced. Firstly, entropy is utilized in view partitioning to get two view sets with the same amount of information. Then, the multi-criteria is put forward to solve the problem that high confidence criterion is not always effective, and it can also take advantage of the complementary of co-training algorithm. Multi-criteria consists of clustering criterion and confidence criterion, and unlabeled data are selected by clustering criterion on view 1 and are added to view 2, unlabeled data are chosen by confidence criterion on the view 2 and are added to the view 1. Besides, the usage of labeled data is strengthened in CTEMC to find more reliable unlabeled data. In detail, the initial clustering center is defined as labeled data in the clustering criterion, and unlabeled data with higher confidence and smaller distance from labeled data are selected in the confidence criterion. In order to verify the effectiveness of the CTEMC, we conduct our experiments on 15 UCI data sets and one artificial data set. The experimental results show that the accuracy of multi-criteria selection of unlabeled data is higher than single criterion, and unlabeled data which are selected by labeled data are also more representative. Compared to SCT, CTSMUC, CTALDP and CTSFCM, CTEMC has superiority in improving accuracy.

In the future, we will discuss the feasibility of other methods in multi-criteria. Additionally, it is very important to enhance the theoretical analysis of our approach. Furthermore, more effective view partitioning methods are also worthy of researching.