The present application relates to the technical field of unsupervised clustering, and in particular to a multi-view clustering method and system based on matrix decomposition and multi-partition alignment.
Multi-view data refers to a large amount of data that describes the same batch of samples from different sources, or with different attributes. For example, an item can be represented by a picture and a short text description; a person can be identified from the face, voice, fingerprints, and pupils. Based on a large amount of unlabeled multi-view data, multi-view clustering has been developed and attracted great attention. Existing multi-view clustering algorithms can be further classified into four categories by model-based differences: co-training, multi-kernel learning, graph clustering, and subspace clustering. For the above four methods, the basic idea of early fusion can be used for view fusion. The main idea of early fusion is to fuse the feature representations or graph structures of a plurality of views into a common representation or one common graph structure. For example, graph-based clustering methods construct sample similarities under each view, and then fuse the graphs via a random walk strategy. The multi-kernel learning method fuses a plurality of base kernels through linear or nonlinear combinations to obtain an optimal clustering kernel. The purpose of subspace clustering is to find a suitable low-dimensional representation or structure for each view, and then fuse them into one common representation or structure containing rich information for clustering. Moreover, there is a way of late fusion in addition to early fusion of views. This method is to fuse the clustering results of the single views, and is also called decision-level fusion. Late fusion can be classified into ensemble learning and collaborative training. The input to the integrated clustering algorithm is the clustering results corresponding to a plurality of views. As in the work, the clustering result is obtained by defining the distance between the final clustering result and the input clustering result as a common loss function. The focus of collaborative training is how to obtain better clustering results in collaborative training. A plurality of clustering results are obtained by spectral embedding for each view, and the obtained clustering results are used to influence the original representations of other views. In addition, late fusion is applied to multi-kernel k-means clustering, which reduces the complexity and time cost of the algorithm.
NMF is widely used for clustering because of its ability to handle basic representations that capture different viewpoints. Some work reduces redundancy between different view representations by defining diversity. Furthermore, both cross-entropy cost function and neighbor information are introduced to guide the learning process. Although NMF can solve the high dimensional problem well, it seems to be useless in capturing the internal structure of data, so that the subsequent work achieves the purpose of retaining the local geometric structure of a data space by adding a graphic regularization item and a popular regularization item. In order to reduce the influence of outliers, the norm of manifold regularization must be introduced in the work. With the development of research, information extracted by single-layer NMF clustering often cannot meet the requirement of data information mining. In order to explore deeper latent information in data, a deep semi-NMF model is proposed in the prior art to explore complex hierarchical information with implicit low-level latent attributes. Under the influence of the deep semi-NMF, the model DMVC learns the common low-dimensional representation containing deep information through the guidance of the original data structure. Recently, a method of multi-view clustering by a deep NMF method has also been proposed to automatically learn the optimal weight of each view.
Current conventional existing NMF methods achieve a large increase in clustering performance by learning low-dimensional representations with rich information, however, they can still be improved with the following considerations. 1) The function of the original data is fully exerted to obtain more discrimination information; 2) the sharing between views and specific information between views is shared; 3) a strategy for fusion of multi-view information is improved.
For the defects of the prior art, an objective of the present application is to provide a multi-view clustering method and system based on matrix decomposition and multi-partition alignment.
In order to achieve the above objective, the present application uses the following technical solutions.
A multi-view clustering method based on matrix decomposition and multi-partition alignment includes:
Further, the constructing an objective function corresponding to the unified partition matrix in the step S4 is represented as:
Further, the optimizing the constructed objective function by using an alternating optimization method in the step S5 specifically includes:
represents a partition matrix after fusion;
min∥X(v)−ϕZi(v)Hi(v)∥F2
min∥X(v)−ΦHm(v)∥F2,s·t·Hi(v)≥0
min∥X(v)−ΦHm(v)∥F2−λtr(Hβ(v)Hm(v)TW(v)+G),s·t·Hm(v)≥−0
min−tr(W(v)TQ),s·t·W(v)W(v)T=Ik
min(α(v))2R(v),s·t·α(v)≥0,Σv=1Vα(v)=1
max fTβ,s·t·β≥0,β2
Further, the steps A1, A2, A3, A4, and A5 all further include: obtaining an optimized result through singular value decomposition (SVD).
Further, the step A4 further includes:
H
m
(v)
=H
m
(v)⊗√{square root over (u(ZHW)/1(ZHW))}
u(ZHW)=2(α(v))2([ΦTX(v)]++[ΦTΦHm(v)]−)+λβ(v)[W(v)H]+
1(ZHW)=2(α(v))2([ΦTX(v)]−+[ΦTΦHm(v)]+)+λβ(v)[W(v)H]−
Further, the step A6 further includes:
α(v)=Σv=1VR(v)/R(v)
Further, the step A7 further includes:
β=f/√{square root over (Σf2)}
where f represents a set of traces of similarity matrices of different views.
Correspondingly, further provided is a multi-view clustering system based on matrix decomposition and multi-partition alignment, which includes:
Further, the constructing an objective function corresponding to the unified partition matrix in the construction module is represented as:
Further, the optimizing the constructed objective function by using an alternating optimization method in the optimization module specifically includes:
represents a partition matrix after fusion;
min∥X(v)−ϕZi(v)Hi(v)∥F2
min∥X(v)−ΦHi(v)∥F2,s·t·Hi(v)≥0
min∥X(v)−ΦHm(v)∥F2−λtr(Hβ(v)Hm(v)TW(v)+G),s·t·Hm(v)≥0
where Φ=Z1(v) Z2(v) . . . Zm(v) represents the multiplication of the first ith base matrices; G=Σo=1,o≈vVβ(o)Hm(o)TW(o) represents fusion of other basic partitions except for the partition matrix corresponding to the vth view;
min−tr(W(v)TQ),s·t·W(v)W(v)T=Ik
min(α(v))2R(v),s·t·α(v)≥0,Σv=1Vα(v)=1
max fTβ,s·t·β≥0,β2
Compared with the prior art, the present application provides a novel conventional clustering method based on deep matrix decomposition and partition alignment, which includes the optimization objective of a basic partition learning module and a multi-partition fusion module. A large number of ablation experiments can show that the multi-partition fusion module added in the present application is beneficial to better fusion of information between views and can acquire richer information along with the increase of the number of layers. The experimental results on the six common datasets demonstrate that the performance of the present application is superior to that of the existing methods.
The following describes the embodiments of the present application by specific examples, and other advantages and effects of the present application will be readily apparent to those skilled in the art from the disclosure of the present application. The present application can also be implemented or applied through other different specific embodiments, and various modifications or changes can be made to the details in this specification based on different viewpoints and applications without departing from the spirit of the present application. It should be noted that the following embodiments and features in the embodiments can be combined with each other without conflict.
The conventional clustering method based on matrix decomposition only considers the common information among the views and ignores the specific information of the views, which results in insufficient representation learning and the possibility of noise doping in early fusion, and thus leading to inaccurate learning of results; aiming at the above problem, the present application provides a multi-view clustering method and system based on matrix decomposition and multi-partition alignment, where a basic partition matrix of each view is obtained through deep semi-nonnegative matrix factorization, then the fused partition matrix is obtained by combining the column-selected matrices of these basic partition matrices, and the common partition matrix is approximated to the fused partition matrix. The optimization is optimized through basis partitioning matrix and late fusion process. Finally, k-means clustering is performed by using the public partition to realize the purpose of clustering.
This embodiment provides a multi-view clustering method based on matrix decomposition and multi-partition alignment, as shown in
This embodiment provides an unsupervised conventional clustering method based on matrix decomposition and late fusion, and as shown in
In the step S4, the obtained basic partition matrix of each view and the consistent fused partition matrix are unified, and an objective function corresponding to the unified partition matrix is constructed.
In order to reduce the possibility of noise affecting the result, reduce time and improve efficiency, partition level, namely decision-level fusion is adopted. The partition matrix Hi of different views and the consistent fused partition matrix H are learned. The objective function is represented as:
The above formula obtains the partition of each view through deep nonnegative matrix factorization, and in the subsequent steps, the partition of each view is column-selected to approach a unified partition matrix, and finally the unified partition matrix is used for clustering.
In the step S5, the constructed objective function is constructed by using an alternating optimization method to obtain an optimized unified partition matrix.
The optimization problem of the objective function is difficult to solve directly, so an iterative algorithm is provided to effectively solve the optimization problem.
The step specifically includes:
min−tr(HU),s·t·HHT=Ik
where tr( ) represents a trace;
represents the fused partition matrix; and U can be directly decomposed by SVD to obtain the optimized H.
min∥X(v)−ϕZi(v)Hi(v)∥F2
where ϕ=Z1(v)Z2(v) . . . Zi-1(v) represents the multiplication of the first i−1th base matrices; and φ can be directly decomposed by SVD to obtain the optimized Zi(v).
min∥X(v)−ΦHi(v)∥F2,s·t·Hi(v)≥0
where Φ=Zi(v)Z2(v) . . . Zi(v) represents the multiplication of the first i−1th base matrices; and Φ can be directly decomposed by SVD to obtain the optimized Hi(v).
min∥X(v)−ΦHm(v)∥F2−λtr(Hβ(v)Hm(v)TW(v)+G),s·t·Hm(v)≥0
The step further includes:
Hm(v)=Hm(v)⊗u(ZHW)/I(ZHW)
u(ZHW)=2(α(v))2([ΦTX(v)]++[ΦTΦHm(v)]−)+λβ(v)[W(v)H]+
1(ZHW)=2(α(v))2([ΦTX(v)]−+[ΦTΦHm(v)]+)+λβ(v)[W(v)H]−
min−tr(W(v)TQ),s·t·W(v)W(v)T=Ik
min(α(v))2R(v),s·t·α(v)≥0,Σv=1Vα(v)=1
α(v)=Σv=1VR(v)/R(v)
max fTβ,s·t·β≥0,β2
β=f/√{square root over (Σf2)}
In summary, the objective function value monotonically decreases as the above stepwise optimization is performed alternately. Meanwhile, the objective function has a lower bound. Thus, the above optimization process can ensure convergence. In addition, a multi-view clustering algorithm based on nonnegative matrix factorization and multi-partition alignment is proposed, which unifies the clustering process and fusion process in one framework. The learning of the consistent partition matrix is more suitable for clustering, so that the algorithm can achieve a better clustering effect.
The difference between this embodiment and the prior art is that:
(1) A multi-view clustering method based on deep semi-NMF and multi-partition alignment is provided. The basic partition learning and the late fusion stage are unified into a framework, enabling them to promote and guide each other to obtain the final common partition matrix for clustering.
(2) The feature matrix is first decomposed using a depth semi-NMF framework to obtain a base partition matrix of each view. Then the basic partition matrices are fused by adopting a late fusion mode, and finally the alignment of the fused basic partition matrix and the public partition matrix is maximized to obtain a public partition matrix.
(3) An alternating optimization algorithm is designed to solve the optimization problem and extensive experiments are performed on the six multi-view datasets.
This embodiment provides a novel conventional clustering method based on deep matrix decomposition and partition alignment, which includes the optimization objective of a basic partition learning module and a multi-partition fusion module. A large number of ablation experiments can show that the multi-partition fusion module added in this embodiment is beneficial to better fusion of information between views and can acquire richer information along with the increase of the number of layers.
Correspondingly, further provided is a multi-view clustering system based on matrix decomposition and multi-partition alignment, which includes:
Further, the constructing an objective function corresponding to the unified partition matrix in the construction module is represented as:
Further, the optimizing the constructed objective function by using an alternating optimization method in the optimization module specifically includes:
represents a partition matrix after fusion;
min∥X(v)−ϕZi(v)Hi(v)∥F2
min∥X(v)−ΦHi(v)∥F2,s·t·Hi(v)≥0
min∥X(v)−ΦHm(v)∥F2−λtr(Hβ(v)Hm(v)TW(v)+G),s·t·Hm(v)≥0
min−tr(W(v)TQ),s·t·W(v)W(v)T=Ik
min(α(v))2R(v),s·t·α(v)≥0,Σv=1Vα(v)=1
max fTβ,s·t·β≥0,β2
The difference between the multi-view clustering method based on matrix decomposition and multi-partition alignment provided by this embodiment and Embodiment 1 is as follows:
The image dataset may include face images, images during logistics transportation, medical images, and the like; and the non-image dataset includes a text dataset and the like.
This embodiment verifies this method by means of six data.
There are six datasets used in this embodiment, including three graph datasets and three non-graph datasets, and the statistics of the datasets are shown in Table 1.
This method is compared with 12 benchmark algorithms. The contrastive algorithm includes k-means used as input after view features are spliced, a kernel-based method MVKKM, a graph-based method GMC, two subspace-based PMSC and CSMCSC, two co-training methods Co-train and Co-reg, and five matrix decomposition-based models MultiNMF, MVCF, ScaMVC, DMVC, and AwDMVC.
Experiment Setting:
For this method and all contrastive methods, data preprocessing, i.e., normalization of all datasets, was performed first. The weighting coefficient γ was selected from [2-12, 2-11, . . . 24, 25]. This method considered that the cluster number k was the number of real classes of each dataset and the dimension of each layer in the decomposition process was related to the cluster number, therefore, two schemes were designed: a two-layer dimension p2=[11,k], and another layer dimension p3=[11,12,k]. The 11 in p2 was selected from [4k, 5k, 6k], and 11, 12 in p3 were selected from [8k, 10k, 12k] and [4k, 5k, 6k], respectively. This method repeated each experiment 50 times to avoid the effects of random initialization and to preserve the optimal results. All experiments were performed on a desktop computer configured as Intel i9-9900K CPU@ 3.60 GHz×16 and 64 GB memory.
Evaluation Indicators:
This method uses three evaluation indicators recognized in the field of conventional clustering algorithms: clustering accuracy (ACC), standard mutual information (NMI), and purity (PUR).
Experiment Results:
This method is compared with 12 benchmark algorithms on 6 standard datasets, and the result is shown in Table 2, where Table 2 is the comparison of this method with other deep clustering methods, and the best result is marked in bold. Table 3 shows the incremental values of the three different indicators over the second-best method on the six datasets. From these tables, the following conclusions are as follows: 1) Table 3 shows the increment values of the three different indicators over the second-best method on six datasets, and the increment values of ACC, NMI and Purity were 11.68%, 15.55% and 3.47% respectively in BBC data; the improvement values were 19.85%, 11.31%, and 17.46% on BBCSport data; for NMI in Retuers and HW, although the performance was reduced by 2.28% and 4.59% over that in the second round, the difference was small. In conclusion, this method outperforms these baseline algorithms on six benchmarks. 2) It was found that the best results were always obtained with this method, compared with DMVC and AwDMVC that also use the deep semi-NM/F framework strong baseline. This indicates that the late fusion strategy of this method is more efficient and robust for these datasets. 3) Compared with PMSC that performs graphic fusion firstly and then performs spectral clustering before later fusion, the method has more advantages. This further indicates that multi-layer semi-NMF can extract more latent useful information.
The experimental results on the six common datasets of this embodiment demonstrate that the performance of the present application is superior to that of the existing methods.
It should be noted that the foregoing are merely some embodiments of the present application and applied technical principles. Those skilled in the art may understand that the present application is not limited to specific embodiments described herein, and those skilled in the art may make various significant changes, readjustments, and replacements without departing from the protection scope of the present application. Therefore, although the present application is described in detail by using the foregoing embodiments, the present application is not limited to the foregoing embodiments, and may further include more other equivalent embodiments without departing from the concept of the present application. The scope of the present application is determined by the scope of the appended claims.
Number | Date | Country | Kind |
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202110705655.9 | Jun 2021 | CN | national |
202111326424.3 | Nov 2021 | CN | national |
This application is the national phase entry of International Application No. PCT/CN2022/098951, filed on Jun. 15, 2022, which is based upon and claims priority to Chinese Patent Application No. 202110705655.9, filed on Jun. 24, 2021; and Chinese Patent Application No. 202111326424.3, filed on Nov. 10, 2021, the entire contents of which are incorporated herein by reference.
Filing Document | Filing Date | Country | Kind |
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PCT/CN2022/098951 | 6/15/2022 | WO |