CN108734187A - A kind of multiple view spectral clustering based on tensor singular value decomposition - Google Patents
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Abstract
The present invention provides a kind of multiple view spectral clustering based on tensor singular value decomposition, three rank tensors of the algorithmIndicate the probability transfer matrix of all viewdatas.Since there is tensor the low-rank in lateral, longitudinal, vertical three directions, the present invention to characterize low-rank of the tensor in each dimension using the multiple order (multi-rank) based on tensor singular value decomposition (Tensor-SVD).Because Tensor-SVD decomposition is generated based on tube convolution, it can not only be than other tensor resolution modes and the method based on two-dimensional structure relationship modeling can more give full expression to the correlation on space structure, and can quickly be calculated by Fourier transformation, improve computational efficiency.Therefore, modeled based on Tensor-SVD tensor resolutions, can more science, quickly, efficiently, and the experimental results showed that can effectively improve multiple view cluster effect.
Description
Technical Field
The invention relates to the field of data mining, in particular to a multi-view spectral clustering algorithm based on tensor singular value decomposition.
Background
The multi-view clustering problem improves clustering performance mainly by integrating useful complementary information in multiple views. Currently relevant algorithmic studies can be broadly divided into three categories: based on a multi-graph fusion algorithm, a collaborative training algorithm and a subspace learning algorithm.
First, based on a multi-graph fusion algorithm. The idea of this type of approach is to construct a graph structure for each view separately and then fuse these graph structures. The 2007 work by professor Dengyong Zhou and professor christopher j.c. burges, microsoft, research, showed that defining a random walk for each graph structure, and then defining a markov mixture model for all random walks to integrate information from each view can achieve good results. Professor Abhishek Kumar, university of Maryland, USA, and professor Piyush Rai, university of Utah, 2011 first proposed that information be integrated using collaborative regularized clustering assumptions between views based on spectral clustering. In 2014, the panyan assistant professor and the like at the university of middle mountains in China propose to fuse the probability transition matrixes of all views based on low-rank and sparse assumptions, so that a probability transition matrix with higher accuracy is extracted as the input of spectral clustering, and a clustering result with higher accuracy is also obtained.
Second, based on a co-training algorithm. The method is characterized in that training learning is carried out on different views respectively, but the information learned from other views is used for constraint in the training process, and multiple times of iterative training are carried out until the clustering results of multiple views tend to be consistent. For example, research algorithms of professor Steffen Bickel and professor tobias scheffer, university of berlin hough, germany, indicate that partial information is exchanged to realize cooperative training every time different views are clustered once. 2011 professor Abhishek Kumar and Hal Daum' e III, university of Maryland, USA, during each iteration, use the spectral representation of each view to restrict updating the adjacency matrix of other views until the adjacency matrix of all views tends to be stable.
And thirdly, based on a subspace learning algorithm. This type of approach is based on the assumption that: for multiple different views (eigenrepresentations), they can all map to the same potential low-dimensional subspace. The core of the multi-view clustering problem is then to map to this common potential low-dimensional subspace and then perform clustering. For example, the Classic Correlation Analysis (CCA) method proposed by professor KamalikaChaudhuri, university of california, usa, 2009 to project multi-view high-dimensional data into a low-dimensional subspace, and the professor Mathieu Salzmann, university of california, berkeley, 2010, to decompose all views into a vertical potential subspace, has also received attention.
At present, the existing method of multi-view clustering based on multi-core learning uses a matrix to represent each view data, and only uses a two-dimensional relation structure to perform modeling and solving to obtain a fusion view, neglects the information of the whole space structure between the views, thus resulting in insufficient utilization of effective information and poor clustering effect.
Disclosure of Invention
The invention provides a multi-view spectral clustering algorithm based on tensor singular value decomposition, which establishes a more accurate and scientific model and improves the multi-view clustering effect.
In order to achieve the technical effects, the technical scheme of the invention is as follows:
a Tensor singular value decomposition (Tensor-SVD) based multi-view spectral clustering algorithm comprises the following steps:
s1: expressing each view by a Gaussian kernel to obtain respective probability transition matrix;
s2: by a tensorRepresenting the probability transition matrix of all views, representing the probability transition matrix of one view in the front part of each tensor, and obtaining a probability transition matrix L by modeling and solving by using a data distribution rule, whereinWhere n represents the total number of samples and m represents the total number of views;
s3: and taking the probability transition matrix L as the key input of a Markov chain-based spectral clustering algorithm, and calculating to obtain a spectral clustering output result.
Further, the specific process of step S2 is:
s21: analysis tensorBecause the data of each view is interfered by noise in the actual acquisition process,inevitably containing noise, provided thatWhereinA tensor that represents the composition of the probability transfer matrix near the real,ε represents the noise tensor;
s22: tensorThe low rank performance is reflected by 3 dimensions of (1), and the following two aspects are mainly expressed:
1) if a group of objects can be clustered into a plurality of clusters, the objects belonging to the same cluster are similar, the object difference between different clusters is larger, and the tensor composed of the probability transfer matrixes of all the viewsEach of the front pieces of (b) characterizes a similarity between the set of objects, thenEach front piece of (1) has a correlation with the row or column vectors belonging to the same cluster, and the rank of the row or column vector group is relatively small compared with the dimension, so thatLow rank in both the transverse and longitudinal directions;
2) data descriptions, namely feature sets, obtained by observing the same group of objects from different angles have certain differences, but all the data descriptions and feature sets represent the internal relations among the group of objects, namely the internal relations presented by all the objects are similar, and tensors composed of probability transfer matrixes of all the viewsEach front piece of (A) represents the similarity between the objects in the group, and is the data representing the internal relationship of the objects, so thatEach front piece of (a) is similar, showing a vertical low rank property;
the low rank nature of the tensor multi-rank (tensor multi-rank) tensor defined based on tensor singular value decomposition is employed herein. The convex envelope of Tensor multi-rank is the Tensor Nuclear Norm (Tensor-Nuclear-Norm). Tensor singular value decomposition is generated based on pipe fiber (tube fiber) convolution, and not only can the correlation on a space structure be fully expressed compared with other tensor decomposition modes, but also fast calculation can be carried out through Fourier transformation, and the calculation efficiency is improved;
s23: tensorEach front piece is a probability transition matrix, each element must be greater than or equal to zero, and each row of the front piece is a probability distribution with a probability sum of 1, thus a tensorThe requirements are satisfied:
wherein e is a column vector with element values all being 1;
s24: assuming that the interference of the noise is random by a small amount, then the noise tensor ε is sparse and characterized by L1-norm;
s25: a model can be established by using the analysis results of S21-S24, and an optimization target is obtained:
where, λ is a compromise factor,nirepresenting the dimensions of the tensor εThe size of the degree;
s26: obtaining a low-rank tensor after solving the optimization target in the S25Are summed and averaged to obtain a probability transition matrix L, i.e.
Further, the optimization target obtained in step S25 needs to be solved by iamm optimization, and the specific process is as follows:
s251: the optimization target is subjected to convex relaxation processing, and the convex envelope of Tensor multiple rank (multi-rank) is obtained by Tensor Nuclear Norm (TNN):
s252: introducing an auxiliary variableThe optimization objective is converted into:
s253: input tensorParameter(s)
S254: the following variables are initialized:
ε＝0，μ＝10-4，ρ＝1.1，μmax＝105，η＝10-8；
s256: performing an iterative process, wherein in each iteration, the variables are updatedε，mu, calculating the value of the optimization target in S252, and stopping iteration if the value of the optimization target in S252 is less than a threshold eta;
s257: output variableThe value of ε is the solution to the optimization goal of S25.
Further, in step S256, the variables are updated by using different mathematical formulasε，Mu, the specific process is as follows:
s2561: constructing an augmented Lagrangian function:
wherein,is the Lagrange multiplier, mu>0 is a penalty parameter;
s2562: variables ofThe update formula of (2) is:
the optimization problem is equivalent to: fixing other variables in S2561, solving forThe sub-problems of (1):
is equivalent to:
its closed form solution is:
wherein,the singular values of the tensor of (a) are decomposed into:shrinkage operatorEach of the front sheets of (a) is a diagonal array, and the tube fiber tube on the ith diagonal is:
is thatFast fourier transform of (a).
S2563: the updated formula for the variable ε is:
the optimization problem is equivalent to: fixing the other variables in S2561, solving a sub-problem on epsilon:
is equivalent to:
its closed form is solved:wherein Sθ(·)＝max(·-θ,0)+min(·+θ,0)；
S2564: variables ofThe update formula of (2) is:
the optimization problem is equivalent to: fixing other variables in S2561, solving forThe sub-problems of (1):
is equivalent to:
order toThe problem then translates into:
a pair of questionsQuestion (I)Carrying out optimization solution to obtain an optimized variableSolutions, i.e. variables, ofThe solution of (1);
s2565: variables ofThe update formula of (2) is:
s2566: variables ofThe update formula of (2) is:
s2567: the updated formula for the variable μ is:
μk←min(μmax,ρμk-1)。
further, step S2564 pairs of sub-problemsThe process of carrying out the optimization solution is as follows:
s25641: to pairThe elements in (a) are ordered and are relabeled as: u. of1≥u2≥…≥un；
S25642: for each element uiComputingA value of (d);
s25643: from all viFinding out subscript i corresponding to the maximum value;
s25644: order toThen output
Compared with the prior art, the technical scheme of the invention has the beneficial effects that:
the invention is based on tensorRepresenting all view data, Tensor multi-rank token Tensor defined based on Tensor-SVD Tensor decompositionLow rank property of (1). Because the Tensor-SVD decomposition is generated based on tube convolution, the correlation on a space structure can be more fully expressed than other Tensor decomposition modes, and fast calculation can be carried out through Fourier transformation, so that the calculation efficiency is improved. Therefore, modeling is carried out based on Tensor-SVD Tensor decomposition, the method is more scientific, faster and more efficient, and the effect of multi-view clustering is improved.
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FIG. 1 is a flow chart of the present invention;
FIG. 2 shows a size n1×n2×n3tensors-SVD diagram.
Detailed Description
The drawings are for illustrative purposes only and are not to be construed as limiting the patent;
for the purpose of better illustrating the embodiments, certain features of the drawings may be omitted, enlarged or reduced, and do not represent the size of an actual product;
it will be understood by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
The technical solution of the present invention is further described below with reference to the accompanying drawings and examples.
Example 1
As shown in fig. 1, a multi-view spectral clustering algorithm based on tensor singular value decomposition is characterized by comprising the following steps:
s1: expressing each view by a Gaussian kernel to obtain respective probability transition matrix;
s2: by a tensorRepresenting the probability transition matrix of all views, representing the probability transition matrix of one view in the front part of each tensor, and obtaining a probability transition matrix L by modeling and solving by using a data distribution rule, whereinWhere n represents the total number of samples and m represents the total number of views;
s3: and taking the probability transition matrix L as the key input of a Markov chain-based spectral clustering algorithm, and calculating to obtain a spectral clustering output result.
The specific process of step S2 is:
s21: analysis tensorBecause the data of each view is interfered by noise in the actual acquisition process,inevitably containing noise, provided thatWhereinRepresenting a tensor composed of a probability transfer matrix close to a real, wherein epsilon represents a noise tensor;
s22: tensorThe low rank performance is reflected by 3 dimensions of (1), and the following two aspects are mainly expressed:
1) if a group of objects can be clustered into a plurality of clusters, the objects belonging to the same cluster are similar, the object difference between different clusters is larger, and the tensor composed of the probability transfer matrixes of all the viewsEach of the front pieces of (b) characterizes a similarity between the set of objects, thenEach front piece of (1) has a correlation with the row or column vectors belonging to the same cluster, and the rank of the row or column vector group is relatively small compared with the dimension, so thatLow rank in both the transverse and longitudinal directions;
2) from different angles to the same group of objectsThe data descriptions, i.e. feature sets, obtained by line observation have certain differences, but they all characterize the internal relations among the group of objects, i.e. the internal relations they present are all similar, and the tensor is composed of probability transition matrices of all viewsEach front piece of (A) represents the similarity between the objects in the group, and is the data representing the internal relationship of the objects, so thatEach front piece of (a) is similar, showing a vertical low rank property;
in the invention, Tensor multiple rank (tensermelti-rank) defined based on Tensor singular value decomposition (Tensor-SVD) is adopted to represent TensorThe convex envelope of Tensor multi-rank is a Tensor Nuclear Norm (Tensor-Nuclear-Norm),
tensor singular value decomposition is generated based on pipe fiber (tube fiber) convolution, and not only can the correlation on a space structure be fully expressed compared with other tensor decomposition modes, but also fast calculation can be carried out through Fourier transformation, and the calculation efficiency is improved;
s23: tensorEach front piece is a probability transition matrix, each element must be greater than or equal to zero, and each row of the front piece is a probability distribution with a probability sum of 1, thus a tensorThe requirements are satisfied:
wherein e is a column vector with element values all being 1;
s24: assuming that the interference of the noise is random by a small amount, then the noise tensor ε is sparse and characterized by L1-norm;
s25: a model can be established by using the analysis results of S21-S24, and an optimization target is obtained:
where, λ is a compromise factor,nirepresenting the size of each dimension of the tensor epsilon;
s26: obtaining a low-rank tensor after solving the optimization target in the S25Are summed and averaged to obtain a probability transition matrix L, i.e.
The optimization target obtained in step S25 needs to be solved by iamm optimization, and the specific process is as follows:
s251: convex relaxation processing is carried out on the optimization target, and the convex envelope of tensor multiple ranks can be obtained by tensor nuclear norm:
s252: introducing an auxiliary variableThe optimization objective can be translated into:
s253: inputting: tensorParameter(s)
S254: some variables are initialized:
ε＝0，μ＝10-4，ρ＝1.1，μmax＝105，η＝10-8；
s256: an iterative process is performed. In each iteration, the variables are updated using different mathematical formulasε，μ, and calculates the value of the optimization target in S252. If the optimization goal is in S252if the target value is less than the threshold η, the iteration is stopped.
S257: output variableThe value of ε is the solution to the optimization goal of S25.
Updating variables using different mathematical formulas in step S256ε，Mu, the specific process is as follows:
s2561: constructing an augmented Lagrangian function:
wherein,is the Lagrange multiplier, mu>0 is a penalty parameter;
s2562: variables ofThe update formula of (2) is:
the optimization problem is equivalent to: fixing other variables in S2561, solving forThe sub-problems of (1):
is equivalent to:
the problem has a closed form solution:
wherein,the tensor-SVD of (A) is decomposed into:as shown in fig. 2, the shrink operatorEach front panel of (a) is a diagonal array, and the tube on the ith diagonal is:
is thatFast fourier transform of (a).
S2563: the updated formula for the variable ε is:
the optimization problem is equivalent to: fixing the other variables in S2561, solving a sub-problem on epsilon:
is equivalent to:
the problem has a closed form solution:wherein Sθ(· max (· - θ,0) + min (· + θ,0) S2564: variables ofThe update formula of (2) is:
the optimization problem is equivalent to: fixing other variables in S2561, solving forThe sub-problems of (1):
is equivalent to:
order toThe problem then translates into:
pair problemCarrying out optimization solution to obtain optimized variablesSolutions, i.e. variables, ofThe solution of (1).
S2565: variables ofThe update formula of (2) is:
s2566: variables ofThe update formula of (2) is:
s2567: the updated formula for the variable μ is:
μk←min(μmax,ρμk-1)。
step S2564 sub-problemThe process of carrying out the optimization solution is as follows:
s25641: inputting: vector quantity
S25642: for theThe elements in (a) are ordered and are relabeled as: u. of1≥u2≥…≥un；
S25643: for each element uiComputingA value of (d);
s25644: from all viFinding out subscript i corresponding to the maximum value;
s25645: order toThen output
In order to verify the clustering performance of the multi-view spectral clustering algorithm based on Tensor singular value decomposition, a large number of experiments are carried out on 2 kinds of classical multi-view data sets, and the advantages of the multi-view clustering algorithm based on Tensor-SVD decomposition are analyzed in detail. In the following, the present section will be described in detail in terms of experimental data sets, comparison algorithms, evaluation indexes, experimental result display and analysis, and the like.
Experimental comparative analysis was performed using classical two multi-view clustered data sets:
cora data set: the data set consisted of scientific papers where 2708 paper documents were selected, including two views, the first view being the textual content of the documents (after extraction of stems, removal of stop words, words with a frequency of less than 10, the vocabulary consisted of 1433 different words) and the second attempt consisting of 5429 reference links between documents. The documents contain genetic algorithms, learning theory, neural networks, probabilistic learning methods, reinforcement learning, learning rules, and case-based reasoning 7 categories. The similarity between documents in each view is calculated by cosine similarity, i.e.Wherein | | | xiI represents the vector xiThe euclidean norm of (a).
Handwritten digital dataset (hand digit dataset): the data set is a classical data set on the UCI machine learning repeater, a set of 2000 handwritten digital pictures (200 different handwritten pictures per number between 0-9). The data set has two views, a first view being characterized by pixel values of the digital picture and a second view being characterized by fourier coefficients of the digital shape. The similarity between the digital samples in each view is calculated by a gaussian kernel, i.e.:wherein | | | xi-xjI represents the sample point xiAnd xjAnd σ represents the mean of the euclidean distances between all sample points.
The information of the specific samples, view characteristics, categories, etc. of the 2 data sets is summarized and counted in table 1 below:
TABLE 1 four data set information Table
Because the multi-view clustering algorithm based on tensor singular value decomposition provided by the invention belongs to multi-view clustering research based on multi-core learning, 2 single-view clustering and 5 typical multi-view clustering algorithms based on multi-core learning are selected to carry out comparison experiments in this chapter, and the method specifically comprises the following steps:
single View (Single View): and (3) calculating each View by using a Markov chain-based spectral clustering method, and selecting the Best result (Best Single View) and the Worst result (Worst Single View) from all the Single-View clustering results.
Feature ligation (FC): and directly splicing the characteristics of all the views into a group of characteristics, and clustering by using spectral clustering based on a Markov chain.
Nuclear Mean (Kernel Mean, KM): and constructing a kernel matrix based on a Gaussian kernel for each view, then taking the mean value of all the kernel matrices as a fusion view, clustering by using spectral clustering based on a Markov chain, and outputting a final result.
Collaborative regularized Spectral Clustering (Co-regularized Spectral Clustering, Co-Reg): the method establishes a model by minimizing the cooperative rule of the inconsistency among a plurality of views, obtains a fused probability transition matrix by optimization solution, and takes the fused probability transition matrix as the key input of the Markov chain-based spectral clustering so as to obtain the final clustering result.
Robust Multiview Spectral Clustering (RMSC): the method includes the steps that a probability transfer matrix based on random walk of a Markov chain is constructed for each view, then low-rank and sparse decomposition is conducted on all the probability transfer matrices, and a common low-rank probability transfer matrix is obtained through optimization. And finally, the obtained target probability transition matrix is used as the key input of the spectral clustering based on the Markov chain, so that the final clustering result is obtained.
Tensor expansion based multi-view spectral clustering algorithm (TM): the multi-dimensional constraint relation of multi-view data is analyzed by expanding tensors on all dimensions, key structure information is stored by using the ideas of low-rank matrix representation and sparse representation for reference, so that a solution model based on tensor expansion is established, and then an IALM optimization algorithm is used for solving. And finally, the obtained target probability transition matrix is used as the key input of the spectral clustering based on the Markov chain, so that the final clustering result is obtained.
The experimental result evaluation indexes selected in the invention mainly comprise: f-score (F-score), Precision (Precision), Recall (Recall), Normalized Mutual Information (Normalized Mutual Information), average entropy (AverageEntropy), Adjusted Lande Index (Adjusted Range Index).
The higher the index values of F-score, Precision, Recall, NMI and Adj-RI are, the better the clustering effect is; the smaller the control index value is, the better the clustering effect is.
The experimental environment is based on a Microsoft Windows 8 system, the CPU is Intel (R) Core (TM) i5-44603.20GHz, the memory is 16GB, and the IDE for algorithm implementation is MatlabR2014 b.
On each data set, 20 different initial runs were performed, and the various indices in the following test results were the mean of the 20 runs (4 decimal places retained after rounding), wherein the standard deviation of each index is shown in the following table brackets (4 decimal places retained after rounding). Next, the experimental results of analyzing each data set are as follows:
cora data set test results and analysis
Firstly, comparing the clustering effects of the BSV and the WSV of the two single views, the Precision, NMI and Adj-RI indexes have large difference values, which shows that the observation angles of the two views are very different, and directly comparing the data descriptions (feature sets) of the two views, the potential consistent structures of the two views are not easy to find, and better clustering performance can be obtained only by mutually supplementing the data descriptions.
Second, compare two simple fusion algorithms, FC and KM. The FC directly splices the two feature sets, is simple and rough, does not deeply utilize complementary information of multiple views, and can see that the effect of the FC is almost the same as that of a single-view BSV. The KM is based on fusion of core matrixes, and the core matrixes represent similarity of objects and reveal intrinsic relation characteristics among the objects, so that the KM is more reasonable and effective than an FC algorithm in simple fusion based on the intrinsic relation of the objects.
Table 2 Cora data set experimental results table
| Method | F-score | Precision | Recall | NMI | Adj-RI | Entropy |
| WSV | 0.2933(0.0022) | 0.1804(0.0011) | 0.2837(0.0117) | 0.0851(0.0020) | 0.0026(0.0025) | 2.4946(0.0039) |
| BSV | 0.2771(0.0128) | 0.3069(0.0178) | 0.2527(0.0095) | 0.1733(0.0173) | 0.1373(0.0173) | 2.1719(0.0486) |
| FC | 0.2882(0.0151) | 0.3077(0.0196) | 0.2711(0.0130) | 0.1956(0.0108) | 0.1444(0.0201) | 2.1169(0.0297) |
| KM | 0.3208(0.0169) | 0.3451(0.0230) | 0.3000(0.0137) | 0.2482(0.0050) | 0.1848(0.0225) | 1.9741(0.0116) |
| Co-Reg | 0.2463(0.0128) | 0.2702(0.0185) | 0.2265(0.0088) | 0.1359(0.0104) | 0.0988(0.0188) | 2.2742(0.0297) |
| RMSC | 0.2727(0.0098) | 0.3021(0.0107) | 0.2486(0.0094) | 0.1739(0.0173) | 0.1323(0.0115) | 2.1702(0.0463) |
| TM | 0.4426(0.0236) | 0.4666(0.0144) | 0.4217(0.0346) | 0.4333(0.0166) | 0.3286(0.0247) | 1.4830(0.0395) |
| T-SVD | 0.5087(0.0258) | 0.5160(0.0399) | 0.5036(0.0292) | 0.4650(0.0109) | 0.4028(0.0337) | 1.4162(0.0341) |
Thirdly, comparing two representative multi-view clustering algorithms based on multi-core learning, namely Co-reg and RMSC, which are probability transition matrixes based on matrix representation of each view, modeling is carried out by utilizing a two-dimensional relational structure, namely, a common consistent relation meeting a certain assumption in a plurality of views is extracted, the observation angle difference of the two views in the data set is large, and the completely consistent relational structure part is few, so that the two methods are not good at the data structure distribution, obviously feel arduous and have almost the same effect as single-view BSV.
For two multi-view clustering algorithms TM and T-SVD based on tensor decomposition, index values of six clustering evaluation indexes are superior to those of the other six comparison algorithms, and the specific expression is that the improved proportion (calculation mode is [ optimal algorithm index value-suboptimal algorithm index value ]/suboptimal algorithm index value) of TM in F-fraction, accuracy, recall rate, normalized mutual information and adjusted Lande index five indexes compared with the second-best Kernel Mean algorithm (Kernel Mean) is respectively 37.97%, 35.21%, 40.57%, 74.58%, 77.81% and T-SVD is respectively 58.57%, 49.52%, 67.87%, 87.35% and 117.97%, which shows that the two algorithms have higher accuracy and are closer to the real result in clustering result than other comparison algorithms, and the average entropy index is also reduced by 24.88% relative to the second-best Kernel Mean algorithm, 28.26%, indicating that they perform better than other algorithms in stability.
Two multi-view clustering algorithms TM and T-SVD based on tensor decomposition greatly improve the clustering effect of the Cora data set, and the method based on tensor modeling is better at mining the data distribution structure with large observation visual angle difference compared with the existing algorithm. Because the tensor model not only considers the two-dimensional relationship of each view, but also performs the overall (stereo) structural analysis of multiple views, more complementary relationships between multiple views can be captured.
Finally, from above
As can be found in Table 2, the index values of the six clustering evaluation indexes of the T-SVD algorithm are better than that of the TM algorithm, specifically, the index values of the F-score, the accuracy, the recall ratio, the normalized mutual information and the adjusted Lande index are respectively improved by 14.93%, 10.59%, 19.42%, 7.32% and 22.58% and are relatively reduced by 4.5% in the average entropy index, which shows that the T-SVD algorithm can dig more multi-view spatial structure information on the data set than the TM algorithm, and verifies that the T-SVD method can utilize the spatial structure information more deeply.
UCI digits experimental results and analysis
Firstly, the clustering effects of the two single-view BSVs and the WSVs are compared, the difference is not large under six indexes, and the clustering precision of the two single-view BSVs and the WSVs reaches about six-fold. And then, a relatively simple and effective shallow multi-view fusion algorithm KM is found that 0.1262, 0.1191, 0.1346, 0.1239 and 0.14 are respectively improved on six indexes relative to BSV, and the ratio of the improved values to the original index values of about six is relatively small, which shows that the potential consistent structure directly reflected by two views in the UCI digits data set has a large ratio and relatively less information needs to be mutually supplemented.
TABLE 3 UCI digits data set Experimental results
| Method | F-score | Precision | Recall | NMI | Adj-RI | Entropy |
| BSV | 0.6030(0.0307) | 0.5914(0.0316) | 0.6152(0.0313) | 0.6502(0.0208) | 0.5582(0.0342) | 1.1712(0.0686) |
| WSV | 0.5921(0.0196) | 0.5882(0.0194) | 0.5960(0.0200) | 0.6471(0.0129) | 0.5467(0.0218) | 1.1753(0.0428) |
| FC | 0.5762(0.0392) | 0.5677(0.0414) | 0.5851(0.0373) | 0.6292(0.0280) | 0.5285(0.0439) | 1.2385(0.0945) |
| KM | 0.7292(0.0479) | 0.7105(0.0613) | 0.7498(0.0358) | 0.7741(0.029) | 0.6982(0.0539) | 0.7664(0.1047) |
| Co-Reg | 0.6599(0.0335) | 0.6489(0.0363) | 0.6715(0.0312) | 0.7029(0.0242) | 0.6216(0.0375) | 0.9954(0.0821) |
| RMSC | 0.7394(0.0551) | 0.7300(0.0630) | 0.7493(0.0470) | 0.7713(0.0383) | 0.7101(0.0617) | 0.7668(0.1330) |
| TM | 0.7717(0.0291) | 0.7642(0.0392) | 0.7796(0.0182) | 0.7963(0.0150) | 0.7461(0.0329) | 0.6823(0.0579) |
| T-SVD | 0.7847(0.0026) | 0.7827(0.0026) | 0.7868(0.0026) | 0.8048(0.0015) | 0.7609(0.0028) | 0.6498(0.0051) |
The data distribution rule is typical and common, and is a data distribution structure which is good for processing by existing algorithms. On the upper part
Co-reg and RMSC in the table are two representative multi-view clustering algorithms based on multi-core learning, each probability transition matrix is represented based on a matrix, and a two-dimensional relation structure is used for modeling, namely a common consistent relation meeting a certain assumption in a plurality of views is extracted.
Secondly, comparing the experimental effects of two tensor decomposition-based multi-view clustering algorithms TM and T-SVD, the index values of six clustering evaluation indexes are superior to those of the other six comparison algorithms, specifically, the improvement ratios of robust multi-view spectral clustering (RMSC) which are superior to the second one in five indexes of F-fraction, accuracy, recall ratio, normalized mutual information and adjusted Lande index are respectively 4.37%, 4.68%, 4.04%, 3.24% and 5.07%, and the improvement ratios of T-SVD are respectively 6.13%, 7.22%, 5.00%, 4.34% and 7.15%, which shows that the accuracy of the two algorithms is further improved in clustering results compared with other comparison algorithms, and in addition, the average entropy index is also respectively reduced by 11.02% and 15.26% compared with the second one kernel mean algorithm, which shows that the two algorithms are better in stability compared with other algorithms.
On the basis of the data distribution structure which is good in the existing algorithm, the two tensor decomposition-based multi-view clustering algorithms TM and T-SVD provided by the invention can also improve some clustering performance, which shows that the overall (three-dimensional) structure analysis of a plurality of views can really obtain more complementary relations among the multi-views. This also verifies the superiority and validity of the two tensor models of the present invention.
Finally, from above
In the table, the T-SVD is found to be superior to TM in index values of six clustering evaluation indexes, specifically, 1.68%, 2.42%, 0.92%, 1.07% and 1.98% are respectively improved in five indexes of F-score, accuracy, recall ratio, normalized mutual information and adjusted Lande index, and 4.76% is relatively reduced in average entropy index, which shows that the T-SVD algorithm can dig more multi-view spatial structure information on the data set than the TM algorithm, thereby improving the clustering effect, and verifying that the T-SVD method can utilize the spatial structure information more deeply.
The same or similar reference numerals correspond to the same or similar parts;
the positional relationships depicted in the drawings are for illustrative purposes only and are not to be construed as limiting the present patent;
it should be understood that the above-described embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. And are neither required nor exhaustive of all embodiments. Any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention should be included in the protection scope of the claims of the present invention.
Claims (5)
1. A multi-view spectral clustering algorithm based on tensor singular value decomposition is characterized by comprising the following steps:
s1: expressing each view by a Gaussian kernel to obtain respective probability transition matrix;
s2: by a tensorRepresenting the probability transition matrix of all views, the front part of each tensor represents the probability transition matrix of one view, and the probability transition matrix is established by using the data distribution ruleSolving the model to obtain a probability transition matrix L, whereinWhere n represents the total number of samples and m represents the total number of views;
s3: and taking the probability transition matrix L as the key input of a Markov chain-based spectral clustering algorithm, and calculating to obtain a spectral clustering output result.
2. The multi-view spectral clustering algorithm based on tensor singular value decomposition as claimed in claim 1, wherein the specific process of step S2 is:
s21: analysis tensorBecause the data of each view is interfered by noise in the actual acquisition process,inevitably containing noise, provided thatWhereinRepresenting a tensor composed of a probability transfer matrix close to a real, wherein epsilon represents a noise tensor;
s22: tensorThe low rank performance is reflected by 3 dimensions of (1), and the following two aspects are mainly expressed:
1) if a group of objects can be clustered into a plurality of clusters, the objects belonging to the same cluster are similar, the object difference between different clusters is larger, and the tensor composed of the probability transfer matrixes of all the viewsEach of the front pieces of (b) characterizes a similarity between the set of objects, thenEach front piece of (1) has a correlation with the row or column vectors belonging to the same cluster, and the rank of the row or column vector group is relatively small compared with the dimension, so thatLow rank in both the transverse and longitudinal directions;
2) data descriptions, namely feature sets, obtained by observing the same group of objects from different angles have certain differences, but all the data descriptions and feature sets represent the internal relations among the group of objects, namely the internal relations presented by all the objects are similar, and tensors composed of probability transfer matrixes of all the viewsEach front piece of (A) represents the similarity between the objects in the group, and is the data representing the internal relationship of the objects, so thatEach front piece of (a) is similar, showing a vertical low rank property;
s23: tensorEach front piece is a probability transition matrix, each element must be greater than or equal to zero, and each row of the front piece is a probability distribution with a probability sum of 1, thus a tensorThe requirements are satisfied:
wherein e is a column vector with element values all being 1;
s24: assuming that the interference of the noise is random by a small amount, then the noise tensor ε is sparse and characterized by L1-norm;
s25: a model can be established by using the analysis results of S21-S24, and an optimization target is obtained:
where, λ is a compromise factor,nirepresenting the size of each dimension of the tensor epsilon;
s26: obtaining a low-rank tensor after solving the optimization target in the S25Are summed and averaged to obtain a probability transition matrix L, i.e.
3. The multi-view spectral clustering algorithm based on tensor singular value decomposition as claimed in claim 1, wherein the optimization objective obtained in step S25 needs to be solved by IALM optimization, and the specific process is as follows:
s251: convex relaxation processing is carried out on the optimization target, and the convex envelope of tensor multiple ranks is obtained by tensor nuclear norm:
s252: introducing an auxiliary variableThe optimization objective is converted into:
s253: input tensorParameter(s)
S254: the following variables are initialized:
ε＝0，μ＝10-4，ρ＝1.1，μmax＝105,η＝10-8；
s256: performing an iterative process, wherein in each iteration, the variables are updatedε，mu, calculating the value of the optimization target in S252, and stopping iteration if the value of the optimization target in S252 is less than a threshold eta;
s257: output variableThe value of ε is the solution to the optimization goal of S25.
4. The multi-view spectral clustering algorithm based on tensor singular value decomposition as claimed in claim 1, wherein different mathematical formulas are used to update variables in step S256ε，Mu, the specific process is as follows:
s2561: constructing an augmented Lagrangian function:
wherein,is the Lagrange multiplier, mu>0 is a penalty parameter;
s2562: variables ofThe update formula of (2) is:
the optimization problem is equivalent to: fixing other variables in S2561, solving forThe sub-problems of (1):
is equivalent to:
its closed form solution is:
wherein,the singular values of the tensor of (a) are decomposed into:shrinkage operatorEach of the front panels of (a) is a diagonal array, and the tube fibers at the ith diagonal are:
is thatFast fourier transform of (a).
S2563: the updated formula for the variable ε is:
the optimization problem is equivalent to: fixing the other variables in S2561, solving a sub-problem on epsilon:
is equivalent to:
its closed form is solved:wherein Sθ(·)＝max(·-θ,0)+min(·+θ,0)；
S2564: variables ofThe update formula of (2) is:
the optimization problem is equivalent to: fixing other variables in S2561, solving forThe sub-problems of (1):
is equivalent to:
order toThe problem then translates into:
pair problemCarrying out optimization solution to obtain an optimized variableSolutions, i.e. variables, ofThe solution of (1);
s2565: variables ofThe update formula of (2) is:
s2566: variables ofThe update formula of (2) is:
s2567: the updated formula for the variable μ is:
μk←min(μmax,ρμk-1)。
5. the multi-view spectral clustering algorithm based on tensor singular value decomposition as claimed in claim 3, wherein step S2564 is to solve the sub-problemThe process of carrying out the optimization solution is as follows:
s25641: to pairThe elements in (a) are ordered and are relabeled as: u. of1≥u2≥…≥un；
S25642: for each element uiComputingA value of (d);
s25643: from all viFinding out subscript i corresponding to the maximum value;
s25644: order toThen output
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