An efficient clustering method based on hypergraph signal processing

Abstract

The present invention discloses an efficient clustering method based on hypergraph signal processing, which belongs to the field of data cluster analysis technology. The method is as follows: data is input in the form of a hypergraph, an adjacency tensor is constructed according to the input hypergraph, and the number of clusters is given; an approximate value of a tensor factor is found by simultaneously diagonalizing a small number of random projections of the adjacency tensor, and the inverse projection of the approximate factor is subjected to another round of simultaneous diagonalization, thereby decomposing the adjacency tensor; a hypergraph Fourier basis and Fourier frequency coefficients are obtained according to the decomposition result, the Fourier basis is found, and a Fourier spectrum matrix is ​​constructed; the Fourier spectrum matrix is ​​clustered using a k-means algorithm, and finally the clustering result is output. The present invention adopts a hypergraph signal processing framework to solve the problem that the graph signal processing framework cannot represent the high-order interaction of the original signal, and at the same time adopts tensor decomposition based on matrix decomposition to solve the problem of low efficiency and imprecision of tensor decomposition in the traditional hypergraph signal processing framework, making the clustering task more efficient.

Description

Technical Field

[0001] The present invention belongs to the technical field of data cluster analysis, and in particular relates to an efficient clustering method based on hypergraph signal processing. Background Art

[0002] Graph signal processing technology is a powerful tool for modeling complex relationships in large datasets and is currently used in countless applications across diverse fields, including data science, communication networks, epidemiology, and sociology. Simple graphs can only model pairwise relationships between data, which hinders their application in modeling networks with higher-order relationships. A new hypergraph signal processing framework based on tensor representation addresses this issue, extending traditional graph signal processing to higher-order domains. However, due to the high-order nature of tensors, hypergraph signal processing frameworks require enormous computing power for analytical calculations.

[0003] In summary, current graph signal processing methods still have some problems when processing high-order tensor interactions. Therefore, how to solve the problem that the graph signal processing framework cannot represent the high-order interactions of the original signal, thereby improving the accuracy and efficiency of tensor decomposition in the hypergraph signal processing framework, is the technical problem that the present invention wants to solve. Summary of the Invention

[0004] The purpose of the present invention is to provide an efficient clustering method based on hypergraph signal processing to solve the problems raised in the above background technology.

[0005] The object of the present invention is achieved by: an efficient clustering method based on hypergraph signal processing, characterized in that the method comprises the following steps:

[0006] Step S1: Input the data in the form of a hypergraph, construct an adjacency tensor based on the input hypergraph, and give the number of clusters;

[0007] Step S2: Find an approximate value of the tensor factor by simultaneously diagonalizing a small number of random projections of the adjacent tensor, and perform another round of simultaneous diagonalization on the back projection of the approximate factor to decompose the adjacent tensor;

[0008] Step S3: Obtain the hypergraph Fourier basis and Fourier frequency coefficients according to the decomposition results, find the Fourier basis, and construct the Fourier spectrum matrix;

[0009] Step S4: Use the k-means algorithm to cluster the Fourier spectrum matrix and finally output the clustering results.

[0010] Preferably, the step S1 constructs an adjacency tensor based on the hypergraph and gives the number of clusters, specifically:

[0011] Select a hypergraph with N nodes in, is a node set, ε is a hyperedge set; the maximum number of nodes contained in its hyperedge is M, and its adjacency tensor is expressed as Among them, the adjacency tensor

[0012] For a hyperedge in a hypergraph,

[0013] Calculate the adjacency tensor A=c i / a i ,

[0014] Among them, c i =|e i |≤M,c i is the number of elements in the i-th hyperedge.

[0015] Preferably, in step S2, approximate values ​​of the tensor factors are found by simultaneously diagonalizing a small number of random projections of adjacent tensors, specifically:

[0016] Define the adjacency tensor A at the unit vector w l The projection on in Uniformly from the unit sphere Select from At the same time, diagonalization gives the factors and the inverse of the factor I is the identity matrix;

[0017] Define the adjacency tensor A in The projection on Depend on At the same time, diagonalization gives the factors and factor weights

[0018] Preferably, the decomposition of the adjacency tensor in step S2 is specifically as follows:

[0019] According to the decomposition form of the obtained adjacency tensor

[0020] Among them, A is an M-order tensor, M ≥ 3; λ r That is the Fourier frequency coefficient, f r is the Fourier basis.

[0021] Preferably, the step S3 obtains the hypergraph Fourier basis and Fourier frequency coefficients according to the decomposition result, finds the Fourier basis, and constructs the Fourier spectrum matrix, specifically:

[0022] According to the decomposition form of the obtained adjacency tensor

[0023] Among them, A is an M-order tensor, M ≥ 3; λ r That is the Fourier frequency coefficient, f r is the Fourier basis;

[0024] Find the first E corresponding Fourier coefficients λ i Fourier basis f≠0 i , the Fourier coefficient λ i ≠0 Fourier basis is merged into a Fourier spectrum matrix S∈R whose columns are Fourier basis N×E .

[0025] Preferably, in step S4, the k-means algorithm is used to cluster the Fourier spectrum matrix, specifically:

[0026] Step S4-1: In step S4, the k-means algorithm is used to cluster the Fourier spectrum matrix, specifically:

[0027] Step S4-1: Randomly select K rows from the Fourier spectrum matrix S as the initial cluster centers;

[0028] Step S4-2: Calculate the distance from each row of the Fourier spectrum matrix S to each cluster center, and assign each row to the cluster closest to it.

[0029] Step S4-3: After all rows are assigned, update the K cluster center positions. The cluster center is defined as the mean of all objects in the cluster in each dimension.

[0030] Step S4-4: Compare the K cluster centers calculated last time. If the cluster centers have changed, go to step S4-2; otherwise, go to step S4-5.

[0031] Step S4-5: If the cluster center no longer changes, stop and output the clustering results. If the i-th row is assigned to the j-th cluster, then the i-th node of the original data belongs to the j-th class.

[0032] Preferably, when A is a third-order tensor and M=3, A is defined as l The projection on in Uniformly from the unit sphere Select from At the same time, diagonalization gives the factors and the inverse of the factor

[0033] Define A in The projection on Depend on At the same time, diagonalization gives the factors and factor weights

[0034] The decomposition form of the adjacency tensor A is expressed as

[0035] Let A be a fourth-order tensor,

[0036] By projecting onto the matrix

[0037] Among them, w is Uniformly from the unit sphere The matrix selected in , u is To uniformly move from the unit sphere The matrix selected from ; determined by the projection of the first two components and This method is used to decompose adjacency tensors of any order.

[0038] Compared with the existing technology, the present invention has the following improvements and advantages: by simultaneously diagonalizing the random projections of adjacent tensors to find the approximate values ​​of tensor factors, complex steps are reduced and the efficiency of tensor decomposition is improved; the hypergraph signal processing framework is adopted to solve the problem that the graph signal processing framework cannot represent the high-order interactions of the original signal, and at the same time, the tensor decomposition based on matrix decomposition is adopted to solve the problems of low efficiency and inaccuracy of tensor decomposition in the traditional hypergraph signal processing framework, making the clustering task more efficient. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 1 is an overall flow chart of the method of the present invention. DETAILED DESCRIPTION

[0040] The present invention is further summarized below with reference to the accompanying drawings.

[0041] like Figure 1 As shown, an efficient clustering method based on hypergraph signal processing includes the following steps:

[0042] Step S1: Input the data in the form of a hypergraph, construct an adjacency tensor based on the input hypergraph, and give the number of clusters;

[0043] Construct an adjacency tensor based on the hypergraph and give the number of clusters, specifically:

[0044] Select a hypergraph with N nodes in, is a node set, ε is a hyperedge set; the maximum number of nodes contained in its hyperedge is M, and its adjacency tensor is expressed as Among them, the adjacency tensor

[0045] For a hyperedge in a hypergraph,

[0046] Calculate the adjacency tensor A=c i / a i ,

[0047] where c i =|e i |≤M,c i is the number of elements in the i-th hyperedge.

[0048] Step S2: Find an approximate value of the tensor factor by simultaneously diagonalizing a small number of random projections of the adjacent tensor, and perform another round of simultaneous diagonalization on the back projection of the approximate factor to decompose the adjacent tensor;

[0049] An approximation to the tensor factor is found by simultaneously diagonalizing a small number of random projections of adjacent tensors, specifically:

[0050] Define the adjacency tensor A at the unit vector w l The projection on in Uniformly from the unit sphere Select from At the same time, diagonalization gives the factors and the inverse of the factor I is the identity matrix;

[0051] Define the adjacency tensor A in The projection on Depend on At the same time, diagonalization gives the factors and factor weights

[0052] Decomposition of the adjacency tensor

[0053] Among them, A is an M-order tensor, M ≥ 3; λ r That is the Fourier frequency coefficient, f r is the Fourier basis;

[0054] Taking the third-order tensor as an example, that is, when M=3, define A in w l The projection on in Uniformly from the unit sphere Select from At the same time, diagonalization gives the factors and the inverse of the factor Define A in The projection on Depend on At the same time, diagonalization gives the factors and factor weights Finally, the decomposition form of the adjacency tensor A can be expressed as

[0055] A is a higher-order tensor, such as a fourth-order tensor Then we can first project it onto the matrix Then determine by the projection of the first two components and This method can be used to decompose adjacency tensors of any order.

[0056] Step S3: Obtain the hypergraph Fourier basis and Fourier frequency coefficients according to the decomposition results, find the Fourier basis, and construct the Fourier spectrum matrix;

[0057] According to the decomposition results, we get the hypergraph Fourier basis and Fourier frequency coefficients, find the Fourier basis, and construct the Fourier spectrum matrix, specifically:

[0058] According to the decomposition form of the obtained adjacency tensor

[0059] Among them, λ r That is the Fourier frequency coefficient, f r is the Fourier basis;

[0060] Find the first E corresponding Fourier coefficients λ i Fourier basis f≠0 i , the Fourier coefficient λ i ≠0 Fourier basis is merged into a Fourier spectrum matrix S∈R whose columns are Fourier basis N×E .

[0061] Step S4: Use the k-means algorithm to cluster the Fourier spectrum matrix and finally output the clustering results.

[0062] The k-means algorithm is used to cluster the Fourier spectrum matrix, specifically:

[0063] Step S4-1: Randomly select K rows from the Fourier spectrum matrix S as the initial cluster centers;

[0064] Step S4-2: Calculate the distance from each row of the Fourier spectrum matrix S to each cluster center, and assign each row to the cluster closest to it.

[0065] Step S4-3: After all rows are assigned, update the K cluster center positions. The cluster center is defined as the mean of all objects in the cluster in each dimension.

[0066] Step S4-4: Compare the K cluster centers calculated last time. If the cluster centers have changed, go to step S4-2; otherwise, go to step S4-5.

[0067] Step S4-5: If the cluster center no longer changes, stop and output the clustering results. If the i-th row is assigned to the j-th cluster, then the i-th node of the original data belongs to the j-th class.

[0068] In order to prove the effect of the inventive method, the present invention was subjected to the following experiments:

[0069] In the Matlab environment, you need to rely on the MATLAB Tensor Toolbox 2.5 and Tensorlab 2.02 toolkits, input the adjacent-order tensor, decompose the tensor, obtain the results, and compare them with other methods.

[0070] This experimental scheme analyzes the performance of a third-order adjacency tensor decomposition, as shown in Table 1:

[0071] Table 1

[0072] Tensor decomposition algorithm Error coefficient Time coefficient Nonlinear Least Squares (NLS) 0.154390 167.800000 Lath algorithm 0.235587 12.000000 Alternating Least Squares (ALS) 0.200180 404.800000 Tensor decomposition algorithm based on matrix decomposition (this method) 0.143216 128.800000

[0073] As shown in Table 1, this method can effectively reduce the error generated when using the traditional decomposition algorithm, and at the same time, the running time is also at a relatively good level.

[0074] The foregoing is merely an embodiment of the present invention and is not intended to limit the present invention. It will be apparent to those skilled in the art that various modifications and variations of the present invention are possible. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention are intended to be included within the scope of the claims of the present invention.

Claims

1. An efficient clustering method based on hypergraph signal processing, characterized by: The method comprises the following steps: Step S1: Input the data in the form of a hypergraph, construct an adjacency tensor based on the input hypergraph, and give the number of clusters; Construct an adjacency tensor based on the hypergraph and give the number of clusters, specifically: Select a hypergraph with N nodes ,in, is a node set, is a hyperedge set; the maximum number of nodes contained in its hyperedge is M, and its adjacency tensor is expressed as ; Among them, the adjacency tensor ; For a hyperedge in a hypergraph, , Compute the adjacency tensor , , ; in, is the number of elements in the i-th hyperedge; Step S2: Find an approximate value of the tensor factor by simultaneously diagonalizing a small number of random projections of the adjacent tensor, and perform another round of simultaneous diagonalization on the back projection of the approximate factor to decompose the adjacent tensor; An approximation to the tensor factor is found by simultaneously diagonalizing a small number of random projections of adjacent tensors, specifically: Define the adjacency tensor A in terms of unit vectors The projection on ,in Uniformly from the unit sphere Select from At the same time, diagonalization gives the factors and the inverse of the factor ; is the identity matrix; Define the adjacency tensor A in The projection on ,Depend on At the same time, diagonalization gives the factors and factor weights ; Decompose the adjacency tensor as follows: Decomposition of the adjacency tensor ; in, for rank tensor, ; is the Fourier frequency coefficient, is the Fourier basis; Step S3: Obtain the hypergraph Fourier basis and Fourier frequency coefficients according to the decomposition results, find the Fourier basis, and construct the Fourier spectrum matrix; According to the decomposition results, we get the hypergraph Fourier basis and Fourier frequency coefficients, find the Fourier basis, and construct the Fourier spectrum matrix, specifically: According to the decomposition form of the obtained adjacency tensor ; in, for rank tensor, ; is the Fourier frequency coefficient, is the Fourier basis; Find the first E corresponding Fourier coefficients Fourier basis , the Fourier coefficients The Fourier basis is merged into a Fourier spectrum matrix whose columns are the Fourier basis ; Step S4: cluster the Fourier spectrum matrix using the k-means algorithm and finally output the clustering results; The k-means algorithm is used to cluster the Fourier spectrum matrix, specifically: Step S4-1: Randomly select K rows from the Fourier spectrum matrix S as the initial cluster centers; Step S4-2: Calculate the distance from each row of the Fourier spectrum matrix S to each cluster center, and assign each row to the cluster closest to it. Step S4-3: After all rows are assigned, update the K cluster center positions. The cluster center is defined as the mean of all objects in the cluster in each dimension. Step S4-4: Compare the K cluster centers calculated last time. If the cluster centers have changed, go to step S4-2; otherwise, go to step S4-5. Step S4-5: If the cluster center no longer changes, stop and output the clustering results. If the i-th row is assigned to the j-th cluster, then the i-th node of the original data belongs to the j-th class.

2. The efficient clustering method based on hypergraph signal processing according to claim 1, characterized in that: described When it is a third-order tensor, M=3, define A in The projection on ,in Uniformly from the unit sphere Select from At the same time, diagonalization gives the factors and the inverse of the factor ; Define A in The projection on ,Depend on At the same time, diagonalization gives the factors and factor weights ; The decomposition form of the adjacency tensor A is expressed as ; Let A be a fourth-order tensor, ; By projecting onto the matrix ; in, It is for Uniformly from the unit sphere The matrix selected in , It is for Uniformly from the unit sphere The matrix selected from ; determined by the projection of the first two components and , this method is used to decompose the adjacency tensor of any order.

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