System recommendation method based on Lanczos orthogonality
By applying a system recommendation method based on Lanczos orthogonality in community detection, using the Lanczos algorithm to project the Laplacian matrix and calculate the eigenvalues and eigenvectors, the difficulties in community structure detection are solved, and efficient and economical community detection and system recommendation are achieved.
Patent Information
- Application Number
- CN202510240124.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-02-28
- Publication Date
- 2025-06-27
AI Technical Summary
When detecting community structures in the network, the existing technology faces problems such as unknown number of communities, missing node attributes and edge attributes, and lack of interpretability in community embedding. Graph neural networks have high requirements for computer hardware and increase economic costs.
Using the system recommendation method based on Lanczos orthogonality, the community structure search problem is modeled in the network through the eigenvectors corresponding to the first k minimum eigenvalues of the undirected network Laplacian matrix. The high-dimensional Laplacian matrix is projected into a symmetric tridiagonal matrix on the low-dimensional Krylov subspace by using the Lanczos algorithm, its eigenvalue and eigenvector are calculated, and community structure detection is performed through the standard k-mean algorithm.
Effectively detect community structure in graph networks, improve system recommendation quality, reduce computing costs, improve computing efficiency, and provide interpretable community embeddings.
Smart Images
Figure CN120217128A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of data mining, and particularly to a system recommendation method based on Lanczos orthogonality. Background Art
[0002] Community structure plays a crucial role in graph-based recommendation systems because community members may have similar interests and preferences. By detecting the relationships between nodes (i.e., user-user, item-item, user-item), high-quality recommendations can be generated based on Lanczos orthogonality. For the scenario of community structure in the network, there are already some mature detection algorithms being widely used, such as methods of probability generation, matrix factorization, and deep learning for community structure detection. However, they face significant challenges such as unknown number of communities, missing node attributes and edge attributes, and lack of interpretability of community embedding.
[0003] A series of latest studies suggest fully capturing the structural information and context information of the graph through graph neural networks to better identify community structure. Graph neural networks (GNNs) can enhance the feature representation of nodes by incorporating complex structural information. But it has high requirements for computer hardware, which increases the economic cost, and there is a lack of good interpretability between modules of the model. In the case of limited computer hardware, the dimensionality reduction method has become an important method in the field of community discovery for recommendation systems due to its advantages of low requirements for computer hardware and high computational efficiency.
[0004] The Lanczos algorithm is an algorithm that transforms a symmetric matrix into a symmetric tridiagonal matrix through orthogonal similarity transformation, named after the 20th-century Hungarian mathematician Cornelius Lanczos. The main function of this algorithm is to transform the matrix, convert the original symmetric matrix into a "tridiagonal matrix" multiplied by an orthogonal matrix, and then process this "tridiagonal matrix", such as performing singular value decomposition. Finally, we can obtain an approximately equivalent matrix through this transformation, and the calculation speed of this process is fast. The finally obtained approximately equivalent matrix can replace the original matrix for certain operations to achieve the purpose of accelerating the calculation and obtaining approximate calculations. The Lanczos algorithm has significant advantages in solving linear equations, especially in dealing with the eigenvalue problem of large-scale sparse Hermitian matrices. Louvain algorithm It is a community discovery algorithm based on modularity. Its basic idea is that nodes in the network try to traverse the community labels of all neighbors and select the community label that maximizes the modularity increment. After maximizing the modularity, each community is regarded as a new node, and the process is repeated until the modularity no longer increases. Summary of the Invention
[0005] The object of the present invention is to provide a system recommendation method based on Lanczos orthogonality. Specifically, the eigenvectors corresponding to the first k smallest eigenvalues of the undirected network Laplacian matrix are used to model the problem of finding community structures in the network. This method first projects the Laplacian matrix in the high-dimensional Euclidean space into a symmetric tridiagonal matrix in the low-dimensional Krylov subspace through the Lanczos algorithm; then calculates the eigenvalues of the symmetric tridiagonal matrix according to the eigenvalue decomposition formula, and judges whether the eigenvalues have converged based on the eigenvalue convergence criterion; if the eigenvalues have converged, calculates the eigenvector corresponding to the converged eigenvalue of the symmetric tridiagonal matrix, and calculates the eigenvector of the high-dimensional Laplacian matrix accordingly, regards the converged eigenvalues of the low-dimensional symmetric tridiagonal matrix as the approximate eigenvalues of the Laplacian matrix, and saves the eigenpairs of the high-dimensional Laplacian matrix; then, de-orthogonalizes the Lanczos vectors and the eigenvectors of the Laplacian matrix to ensure that the Lanczos algorithm always maintains orthogonality during the iteration process; finally, arranges the saved converged eigenpairs in ascending order of eigenvalues, intercepts the first k eigenvectors as the representation learning of nodes, and uses the standard k-means algorithm for community structure detection.
[0006] The technical solution for realizing the object of the present invention is: a system recommendation method based on Lanczos orthogonality, first models the problem of finding community structures in the network structure as the problem of solving the first k smallest eigenpairs of the network Laplacian matrix; secondly projects the Laplacian matrix in the high-dimensional Euclidean space into a symmetric tridiagonal matrix in the low-dimensional Krylov subspace; then, uses the eigenvalue convergence criterion to screen out the converged eigenpairs of the symmetric tridiagonal matrix, and calculates its corresponding eigenvector; then, calculates the eigenvector of the Laplacian matrix according to the eigenvector of the symmetric matrix; finally de-orthogonalizes the converged eigenvectors of the Laplacian matrix and the Lancozs vectors, and saves the converged eigenvectors of the Laplacian matrix, and uses the standard k-means algorithm for community structure detection. Specifically, it includes the following steps:
[0007] Step 1): For an undirected network graph G=(V, R), where V and R are the vertex set and edge set in the graph G, and the size of the vertex set is denoted as n;
[0008] Step 2): According to the adjacency matrix A of the undirected network, calculate the diagonal matrix where the first subscript represents the row of the matrix and the second subscript represents the column of the matrix, and calculate the Laplacian matrix L of the undirected network;
[0009] L = D - (A + A T )
[0010] Step 3): Given the Laplacian matrix $L$ and the initial vector $q$, construct the Krylov subspace $K$ of order $m$ m (L, q);
[0011] K m (L, q)=span{q, Lq, L 2 q, L 3 q,..., L m q}
[0012] Step 4): Project the Laplacian matrix $L$ on the high-dimensional Euclidean space onto the low-dimensional Krylov subspace $K$ of order $m$ m (L, q) by the symmetric tridiagonal matrix $H$,
[0013]
[0014] where the matrix $Q$ n×m has size $n\times m$, the matrix $Q$ n×(m+1) has size $n\times(m + 1)$, the matrix $H$ (m+1)×m has size $(m + 1)\times m$, the matrix $H$ m×m has size $m\times m$, and the column vector has elements equal to 1 only at the $m$-th index position and 0 at other index positions;
[0015] Step 5): Calculate the $i$-th eigenvalue $\lambda$ m×m of the matrix $H$ i , and the column vector is the eigenvector corresponding to the eigenvalue $\lambda$ i , where $1\leq i\leq m$;
[0016]
[0017] Step 6): According to the eigenvalue convergence criterion
[0018]
[0019] Assume that $\lambda$ i is the $i$-th true eigenvalue of the matrix $H$ m×m , $\gamma$ i is the approximate solution of the $i$-th eigenvalue of the matrix $H$ m×m , $\varepsilon$ is the error tolerance, set to $\varepsilon = 10$ -16 , $\|H$ m \|_2$ represents the spectral norm of the matrix $H$ m . When the product of the scalar $\beta$ m and the absolute value of the last element of the eigenvector is not greater than When the i-th true eigenvalue λ i starts to converge, the eigenvalue λ i at this time is acceptable, and its corresponding eigenvector is meaningful;
[0020] Step 7): According to the symmetric tridiagonal matrix H m×m the converged eigenvalue λ i and the corresponding eigenvector the eigenvector based on the Laplacian matrix
[0021]
[0022] Save the converged eigenpairs of the Laplacian
[0023] Step 8): During the iteration of the Lanczos algorithm, once an eigenpair converges, the columns of the matrix Q n×(m+1) will lose orthogonality. According to the orthogonality criterion
[0024]
[0025] When an eigenvalue converges, that is, when the denominator takes a very small positive real number that is almost close to zero, the inner product of the eigenvector and the Lanczos vector will be very large. At this time, the Lanczos vector needs to be de-orthogonalized with the converged eigenvector
[0026]
[0027] Step 9): For each eigenvalue in the matrix H m×m , its convergence is judged in turn. When convergence occurs, calculate the eigenvector of the low-dimensional tridiagonal symmetric matrix H m×m , then calculate the eigenvector of the high-dimensional Laplacian matrix, and save the converged eigenpair, and then de-orthogonalize it with the latest Lanczos vector; if the eigenvalue does not converge, there is no need to calculate the eigenvector of the matrix H m×m corresponding to this eigenvalue, let alone calculate the eigenvector of the high-dimensional Laplacian matrix;
[0028] Step 10): Let l be the number of iterations, and construct a low-dimensional m-order Krylov subspace
[0029] K m (L, q) = span{q, Lq, L 2 q, L 3 q,..., L m q}
[0030] where 2 ≤ m ≤ l. On these l - 1 low - dimensional Krylov subspaces, steps 4) to 8) are required for each, and the converged eigenpairs of the Laplacian matrix are appended and saved in the dictionary container;
[0031] Step 11): Sort the dictionary container in ascending order of eigenvalues, then intercept the first k eigenvectors, form an eigenmatrix as the representation learning of the nodes, and then perform community structure detection using the standard k - means algorithm;
[0032] The present invention transforms the adjacency matrix or incidence matrix into a network graph. The Laplacian matrix is applied in graph theory as a matrix representation of a graph. Given a graph G with n vertices, its Laplacian matrix L is defined as L = D - A; where D is the degree matrix of the graph and A is the adjacency matrix of the graph. In a directed graph, only one of the out - degree or in - degree needs to be considered. The Laplacian matrix is a positive semi - definite matrix; the number of times 0 appears in the eigenvalues is the number of connected regions of the graph; the smallest eigenvalue is 0 because the sum of each row of the Laplacian matrix is 0.
[0033] Beneficial effects: The present invention proposes a system recommendation method based on Lanczos orthogonality, which improves the recommendation quality of the system by using community structure attributes. Spectral clustering is a common method for community detection, which consists of three parts: (1) fully considering the attribute information of the data to construct a similarity matrix; (2) calculating the Laplacian matrix according to the similarity matrix; (3) calculating the eigenvectors of the Laplacian matrix and taking the first k eigenvectors (the first k eigenvalues) as the features of the nodes, that is, simplifying it to a problem of finding the community structure in the network by modeling with the eigenvectors corresponding to the first k smallest eigenvalues of the undirected network Laplacian matrix; during the iterative process of the Lanczos algorithm, it is noted that the columns of the matrix Q n×(m+1) will show a lack of orthogonality, and this lack of orthogonality is closely related to the convergence of the eigenvalues; at the same time, the eigenpairs at both ends of the Laplacian spectrum converge earlier than the eigenpairs inside the spectrum. These two important facts are ignored in most applications of the Lanczos algorithm. At the same time, in the applications related to the Lanczos algorithm, they only use the Lanczos algorithm to solve the eigenpairs of symmetric matrices, but ignore the above two important facts. The use of these two important facts can accelerate the operation of the algorithm and reduce the time cost.
[0034] The present invention designs a system recommendation method based on the orthogonality of the Lanczos algorithm. Driven by a real community detection data set and using the Lanczos algorithm as a model, it can accurately detect the community structure contained in the graph network, thereby improving the recommendation quality of the system. It has very important application value for expanding the application of the Lanczos algorithm. Brief Description of the Drawings
[0035] Figure 1 It is the overall framework of the LOCD algorithm. Specific Embodiments
[0036] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the scope of protection of the present invention.
[0037] In addition, in the description of the present invention, unless otherwise specified. In community detection, the node representations learned by dimensionality reduction methods (matrix factorization, graph embedding) do not have the guarantee of linear independence, resulting in redundant information in the learned node representations. Based on the spectral clustering algorithm, only the attribute information of the data is captured, and the attribute information contained in the algorithm itself is often ignored.
[0038] Spectral clustering is a commonly used method for community detection. It consists of three parts: (1) constructing a similarity matrix by fully considering the attribute information of the data; (2) calculating the Laplacian matrix according to the similarity matrix; (3) calculating the eigenvectors of the Laplacian matrix and taking the first k eigenvectors (the first k eigenvalues) as the feature representations of the nodes for community structure detection.
[0039] Use the eigenvectors corresponding to the first k smallest eigenvalues of the Laplacian matrix of the undirected network to model the search for the community structure in the network: The eigenvalues can be used as approximate solutions of the eigenvalues of the Laplacian matrix. The eigenvectors of the Laplacian matrix are obtained by left multiplying the eigenvectors of the symmetric tridiagonal matrix by an orthogonal matrix. Continuously execute this step, add the newly converged eigenpairs to the dictionary container until the iteration is completed, and then arrange the eigenvectors in the dictionary container in ascending order of eigenvalues, and take the first k eigenvectors as the feature representations of the nodes for community detection.
[0040] Model social networks, citation networks, protein interaction networks, etc. as undirected networks and represent them using an adjacency matrix. The Laplacian matrix can be calculated through the adjacency matrix. The Lanczos algorithm only requires three parameters: a symmetric matrix, an initial vector, and the number of iterations. The symmetric matrix is the Laplacian matrix, the initial vector can be any non-zero vector, and the number of iterations is selected according to the data. The number of iterations required when the Lanczos algorithm converges is the final number of iterations. The Lanczos algorithm can transform the Laplacian matrix in a high-dimensional Euclidean space into a symmetric tridiagonal matrix in a low-dimensional Krylov subspace through a column orthogonal matrix, and calculate the eigenvalues and eigenvectors of the symmetric tridiagonal matrix.
[0041] The present invention proposes a system recommendation method based on Lanczos orthogonality, which models the problem of finding community structures in a network using the eigenvectors corresponding to the first k smallest eigenvalues of the Laplacian matrix of an undirected network; in combination with Figure 1 , including the following steps:
[0042] Step 1): For an undirected network graph G=(V, R), where V and R are the vertex set and edge set in graph G, and denote the size of the vertex set as n; for a directed network graph G=(V, R), ignore the direction of its edges and treat it as an undirected network graph, and proceed to step 2);
[0043] Step 2): According to the adjacency matrix A of the network, calculate the diagonal matrix scalar D ii represents the i-th main diagonal element of the diagonal matrix D, and the element A ij represents the element in the i-th row and j-th column of matrix A, and calculate the Laplacian matrix L of the undirected network;
[0044] L = D - (A + A T )
[0045] The Laplacian matrix is symmetric and positive semi-definite, and proceed to step 3);
[0046] Step 3): According to the Laplacian matrix L and the initial vector q, construct a Krylov subspace K of order m m (L, q);
[0047] K m (L, q) = span{q, Lq, L 2 q, L 3 q,..., L m q}
[0048] The Krylov subspace K of order m m(L, q) is a Krylov subspace K of order m + 1 m+1 a proper subspace of (L, q), go to step 4);
[0049] Step 4): Project the Laplacian matrix L on the high-dimensional Euclidean space into a low-dimensional Krylov subspace K of order m through the Lanczos algorithm m the symmetric tridiagonal matrix H of (L, q) (m+1)×m ,
[0050]
[0051] where the matrix Q n×m is of size n × m, the matrix Q n×(m+1) is of size n × (m + 1), the matrix H (m+1)×m is of size (m + 1) × m, the matrix H m×m is of size m × m, the column vector has elements of 1 only at the m-th index position and elements of 0 at other index positions. When the columns of Q n×(m+1) are orthogonal,
[0052]
[0053] the Laplacian matrix L on the high-dimensional Euclidean space is projected into the symmetric tridiagonal matrix H on the low-dimensional Krylov subspace under the mapping of the orthogonal matrix Q n×m ; Based on each step of iteration, it is m×m ; For the scalar
[0054]
[0055] go to step 5) go to step 5)
[0056] Step 5): Calculate the i-th eigenvalue λ m×m of the matrix H i , and the column vector is the eigenvector corresponding to the eigenvalue λ i , where 1 ≤ i ≤ m;
[0057]
[0058] The column vector is of size m × 1, go to step 6)
[0059] Step 6): According to the eigenvalue convergence criterion
[0060]
[0061] Assume λ i is the matrix Hm×m The i-th true eigenvalue, γ i is for matrix H m×m The approximate solution of the i-th eigenvalue, ε is the error tolerance, set as ε = 10 -16 , ||H m ||2 represents the spectral norm of matrix H m When the product of the scalar β m and the absolute value of the last element of the eigenvector is not greater than , the i-th true eigenvalue λ i starts to converge. Therefore, the eigenvalue λ i at this time is acceptable, and its corresponding eigenvector is meaningful. Go to step 7)
[0062] Step 7): Based on the converged eigenvalue λ m×m of the symmetric tridiagonal matrix H i and the corresponding eigenvector The eigenvector of the Laplacian matrix
[0063]
[0064] The column vector has a size of n×1 and saves the converged eigenpairs of the Laplacian Go to step 8)
[0065] Step 8): During the iteration of the Lanczos algorithm, once an eigenpair converges, the columns of matrix Q n×(m+1) will lose orthogonality. According to the orthogonality criterion:
[0066]
[0067] When an eigenvalue converges, that is, when the denominator takes a very small positive real number that is almost close to zero, the inner product of the eigenvector and the Lanczos vector will be very large. At this time, the Lanczos vector needs to be de-orthogonalized with the converged eigenvector At the same time, the latest
[0068]
[0069] also needs to be de-orthogonalized with as Go to step 9) Go to step 9)
[0070] Step 9): For each eigenvalue in matrix H m×m , its convergence is judged in turn; when convergence occurs, the newly converged eigenvalue is compared with the eigenvalues already existing in the dictionary container. If the deviation between the two is less than the given threshold, there is no need to calculate the eigenvector corresponding to the newly converged eigenvalue. Only the converged eigenvector needs to be taken out and de - orthogonalized with the latest Lanczos vector; if the deviation between the newly converged eigenvalue and the eigenvalues already existing in the dictionary container is greater than the given threshold, it indicates that the newly converged eigenvalue has not been saved in the dictionary container. Calculate the eigenvector of the low - dimensional tridiagonal matrix H m×m , then calculate the eigenvector of the high - dimensional Laplacian matrix, save the converged eigenpair, and then perform de - orthogonalization with the latest Lanczos vector; if the eigenvalue does not converge, there is no need to calculate the eigenvector of matrix H m×m corresponding to this eigenvalue, let alone calculate the eigenvector of the high - dimensional Laplacian matrix;
[0071] Step 10): Let l be the number of iterations, construct a low - dimensional m - order Krylov subspace,
[0072] K m (L,q)=span{q,Lq,L 2 q,L 3 q,...,L m q}
[0073] where 2 ≤ m ≤ l. On these l - 1 low - dimensional Krylov subspaces, Steps 4) to 8) are required, and the converged eigenpairs of the Laplacian matrix are all appended and saved in the dictionary container;
[0074] Step 11): Sort the dictionary container in ascending order of eigenvalues, then intercept the first k eigenvectors to form a feature matrix for the representation learning of nodes, and then perform community structure detection using the standard k - means algorithm; the converged eigenpairs can not only be used to prevent the loss of orthogonality in the Lanczos algorithm, but also be used for the detection of hidden community structures in the network.
[0075] Figure 1Shows the overall framework of LOCD. During the iterative process of the Lanczos algorithm, first project the Laplacian matrix onto a symmetric tridiagonal matrix on a low-dimensional Krylov subspace, and calculate the eigenvalues of the symmetric tridiagonal matrix; according to the eigenvalue convergence criterion, judge whether the eigenvalues have converged; if the eigenvalues have converged, judge whether the converged eigenvalues already exist in the dictionary container. If they exist, there is no need to continue calculating the eigenvectors of the tridiagonal matrix and the Laplacian matrix. Just directly take out the existing eigenvectors and perform de-orthogonalization with the Lanczos vectors. If they do not exist, it is necessary to continue calculating the eigenvectors of the tridiagonal matrix and the Laplacian matrix, and store the newly converged eigenpairs in the container and perform de-orthogonalization with the Lanczos vectors; if the eigenvalues do not converge, there is no need to calculate the eigenvectors of the tridiagonal matrix and the Laplacian matrix, nor to perform de-orthogonalization with the Lanczos vectors; finally, sort all the saved converged eigenpairs in ascending order according to the eigenvalues, take the first k eigenvectors as the representation learning of the nodes, and use the k-means algorithm to detect the community structure.
[0076] The parameters of the present invention corresponding to social networks (which can also be applied to citation networks, protein interaction networks) are as follows: Is the corresponding Lanczos vector, and the orthogonalization with Lanczos is:
[0077]
[0078] The k-means algorithm is a commonly used data classification algorithm. Just use the eigenvectors of the Laplacian matrix as the input of the k-matrix algorithm: the eigenvalues can be used as the approximate solutions of the eigenvectors of the Laplacian matrix, and the eigenvectors of the symmetric tridiagonal matrix multiplied by the orthogonal matrix on the left obtain the eigenvectors of the Laplacian matrix. Continuously execute this step, add the newly converged eigenpairs to the dictionary container until the iteration is completed, and then sort the eigenvectors in the dictionary container in ascending order according to the eigenvalues, and take the first k eigenvectors as the feature representation of the nodes for community detection.
[0079] As described above, it is only the specific implementation manner of the present invention. However, the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed by the present invention should be covered within the protection scope of the present invention. Therefore, the protection scope of the present invention should be subject to the protection scope of the claims.
Claims
1. A system recommendation method based on Lanczos orthogonality, characterized in that: Firstly, the community structure problem in the network structure is modeled as the problem of solving the first k minimum eigenpairs of the network Laplacian matrix; secondly, the Laplacian matrix on the high-dimensional Euclidean space is projected into a symmetric tridiagonal matrix on the low-dimensional Krylov subspace; then, the converged eigenpairs of the symmetric tridiagonal matrix are screened out using the eigenvalue convergence criterion, and its corresponding eigenvector is calculated; then, the eigenvector of the Laplacian matrix is calculated based on the eigenvector of the symmetric matrix; finally, the converged eigenvector of the Laplacian matrix is deorthogonalized with the Lancozs vector, and the converged eigenvector of the Laplacian is saved and used for community structure detection through the standard k-means algorithm; specifically, the following steps are included: Step 1): For an undirected network graph G = (V, R), V and R are the vertex set and edge set in the graph G, and the size of the vertex set is n; for a directed network graph G = (V, R), ignore the direction of its edges and treat it as an undirected network graph; Step 2): Based on the adjacency matrix A of the network, calculate the diagonal matrix Scalar D ii represents the i-th main diagonal element of the diagonal matrix D, element A ij Represents the element in the i-th row and j-th column of matrix A, and calculates the Laplacian matrix L of the undirected network; L=D-(A+A T ) Step 3): Given the Laplacian matrix L and the initial vector q, construct the m-order Krylov subspace K m (L,q); K m (L,q)=span{q,Lq,L 2 q,L 3 q,...,L m q} Step 4): Project the Laplacian matrix L on the high-dimensional Euclidean space into a low-dimensional m-order Krylov subspace K through the Lanczos algorithm m The symmetric tridiagonal matrix H of (L,q), Among them, the matrix Q n×m The size of the matrix Q is n×m. n×(m+1) The size of the matrix H is n×(m+1). (m+1)×m The size of the matrix H is (m+1)×m. m×m The size is m×m, column vector Only the element at the mth index position is 1, and the elements at other index positions are 0; Step 5): Calculate the matrix H m×m The i-th eigenvalue λ i , column vector is the eigenvalue λ i The corresponding eigenvector, where 1≤i≤m; Step 6): According to the eigenvalue convergence criterion Assume λ i is the matrix H m×m The i-th true eigenvalue, γ i is the matrix H m×m The approximate solution of the i-th eigenvalue, ε is the error tolerance, set to ε = 10 -16 ,||H m ||2 represents the matrix H m The spectral norm of m With the eigenvector The product of the absolute values of the last elements of When the i-th true eigenvalue λ i Convergence begins to occur, so the eigenvalue λ at this time i is acceptable, and its corresponding eigenvector It is meaningful; Step 7): Based on the symmetric tridiagonal matrix H m×m Converged eigenvalue λ i and the corresponding eigenvector Eigenvectors based on Laplacian matrix Save the eigenpairs for which the Laplacian has converged Step 8): During the Lanczos algorithm iteration, once an eigenvalue pair converges, the matrix Q n×(m+1) The columns of When an eigenvalue converges, that is, the denominator When we take a very small positive real number close to zero, the eigenvector With Lanczos vector The inner product of will be very large; at this time, the Lanczos vector Need to converge with the eigenvector Deorthogonalization occurs Step 9): For the matrix H m×m For each eigenvalue in , its convergence is judged in turn; Step 10): Let l be the number of iterations and construct a low-dimensional m-order Krylov subspace K m (L,q)=span{q,Lq,L 2 q,L 3 q,...,L m q} Where, 2≤m≤l; in these l-1 low-dimensional Krylov subspaces, steps 4) to 8) are all required, and the eigenpairs of the Laplacian matrix that have converged are added and saved in the dictionary container; Step 11): Sort the dictionary containers according to their eigenvalues from small to large, and then extract the eigenvectors of the first k items to form a feature matrix as the representation learning of the nodes, and then use the standard k-means algorithm to perform community structure detection.
2. The system recommendation method based on Lanczos orthogonality according to claim 1, characterized in that: In step 9), when convergence occurs in step 9-1), the newly converged eigenvalue is compared with the eigenvalue already existing in the dictionary container. If the deviation between the two is less than a given threshold, there is no need to calculate the eigenvector corresponding to the newly converged eigenvalue. Instead, the converged eigenvector only needs to be taken out and deorthogonalized with the latest Lanczos vector. Step 9-2) If the deviation between the newly converged eigenvalue and the existing eigenvalue in the dictionary container is greater than a given threshold, calculate the low-dimensional tri-symmetric matrix H m×m The corresponding eigenvector is then calculated, and the eigenvector of the high-dimensional Laplacian matrix is saved, and then deorthogonalized with the latest Lanczos vector; if the eigenvalue does not converge, the matrix H does not need to be calculated. m×m The eigenvalue corresponds to the eigenvector, not to mention the eigenvector of the high-dimensional Laplacian matrix.
3. The system recommendation method based on Lanczos orthogonality according to claim 1, characterized in that: The search for community structure in the network is modeled using the eigenvectors corresponding to the first k smallest eigenvalues of the Laplacian matrix of an undirected network: the eigenvalues can be used as approximate solutions to the eigenvalues of the Laplacian matrix, and the eigenvectors of the Laplacian matrix are obtained by multiplying the orthogonal matrix by the eigenvectors of the symmetric tridiagonal matrix on the left. This step is performed continuously, adding newly converged eigenpairs to the dictionary container until the iteration is completed, and then the eigenvectors in the dictionary container are arranged in ascending order of the eigenvalues, and the first k eigenvectors are taken as the feature representation of the node for community detection.
4. The system recommendation method based on Lanczos orthogonality according to claim 1, characterized in that: Social networks, citation networks, protein interaction networks, etc. are modeled as undirected networks and represented by adjacency matrices. The Laplacian matrix can be calculated through the adjacency matrix. The Lanczos algorithm only requires three parameters: symmetric matrix, initial vector, and number of iterations. The symmetric matrix is the Laplacian matrix. The initial vector can be any non-zero vector. The number of iterations is selected according to the data. The number of iterations required when the Lanczos algorithm converges is the final number of iterations. The Lanczos algorithm can transform the Laplacian matrix on the high-dimensional Euclidean space into a symmetric tridiagonal matrix of the low-dimensional Krylov subspace through a column orthogonal matrix, and calculate the eigenvalues and eigenvectors of the symmetric tridiagonal matrix.
5. The system recommendation method based on Lanczos orthogonality according to claim 1, characterized in that: The corresponding parameters in the social network are as follows: is the corresponding Lanczos vector, which is orthogonalized to Lanczos: