A method, system, device, and medium for ultra-large scale graph matrix factorization

Abstract

This application relates to a method, system, device, and medium for decomposing ultra-large-scale graph matrices. The method adaptively determines the upper limit of the spectral interval by extracting the degree distribution and community structure of the graph, and constructs a Chebyshev filter accordingly to filter the initial vector to amplify the target feature gap. Using the filtered vector as the starting block, block Lanczos iteration is performed to obtain Ritz values ​​and approximate eigenvectors. Convergence is determined based on the Ritz value gap and residuals. When stagnation occurs, the spectral interval is dynamically corrected based on the Ritz value distribution, and the filter is reconstructed. The approximate eigenvectors are then filtered again to restart the process, forming an estimation-filtering-convergence closed loop. Thus, the online spectral interval estimation, Chebyshev acceleration, and deep block Lanczos work together to suppress spectral radius overflow and eigenvector pseudo-alignment caused by extreme eigenvalues, overcoming convergence stagnation and achieving stable and efficient large-scale graph decomposition.

Description

Technical Field

[0001] This invention belongs to the field of computer science and numerical computing, and in particular relates to a method, system, device and medium for decomposing ultra-large-scale graph matrices. Background Technology

[0002] In the analysis of graph data such as ultra-large-scale social networks or financial networks, spectral clustering is a commonly used topological partitioning method. One of its core steps is to perform a low-rank approximate decomposition of the normalized Laplacian matrix of the graph to obtain the first few smallest non-zero eigenvalues ​​and their corresponding eigenvectors. For graphs containing hundreds of billions of nodes, such matrices have extremely high dimensionality and sparsity, and the degree distribution usually exhibits power-law characteristics. The community structure varies in strength, causing eigenvalues ​​to be highly clustered near zero, with extremely small spectral gaps, which brings fundamental difficulties to traditional numerical methods.

[0003] The classic Langios iteration and its block variant are commonly used methods for finding extreme eigenpairs of large-scale sparse matrices. They construct a set of orthogonal bases in Krylov subspaces, projecting the original matrix into a much smaller tridiagonal matrix, and then approximating the original matrix's eigenpairs with the eigenpairs of this tridiagonal matrix. However, in finite-precision computation environments, when the matrix contains eigenvalues ​​with extremely large absolute values ​​or tightly clustered eigenvalues, the Langios process is prone to numerical overflow. This results in spurious overlaps of originally disjoint eigenvalues ​​in the tridiagonal projection matrix, leading to a loss of basis vector orthogonality and pseudo-alignment of eigenvectors, severely distorting the final network partitioning result. Furthermore, when the spectral gap is extremely small, the convergence speed of the Langios iteration theoretically approaches infinity, but in practice, it exhibits prolonged convergence stagnation, requiring far more iterations than acceptable to obtain eigenpairs that meet the required accuracy.

[0004] To alleviate the aforementioned problems, researchers have introduced Chebyshev polynomial acceleration techniques. By applying polynomial filtering to the iterative vector, the components in the direction of interest are amplified while the components in other directions are suppressed, thereby artificially widening the effective gap between target and non-target eigenvalues ​​and improving convergence speed. However, the Chebyshev acceleration effect is strictly dependent on the accurate estimation of the matrix spectral interval, i.e., the interval containing the target eigenvalues. In ultra-large-scale heterogeneous graphs, the spectral interval itself is difficult to obtain accurately and inexpensively, usually requiring multiple additional Langios probes. This is not only costly, but also causes small estimation errors to be exponentially amplified in higher-order recursions, leading to the complete failure of the acceleration. Some existing improvement approaches, such as dynamically adjusting Chebyshev parameters or increasing the size of the Langios block, address the problems of inaccurate spectral estimation or convergence stagnation to some extent, but none of them incorporate the graph's topological prior information, spectral interval estimation, and iterative acceleration into a unified closed-loop framework for joint optimization. Therefore, when dealing with heterogeneous networks with hundreds of billions of nodes, it is difficult to simultaneously ensure numerical stability, convergence speed, and computational accuracy. Summary of the Invention

[0005] Therefore, it is necessary to provide a method, system, device, and medium for decomposing ultra-large-scale spectral matrices to address the aforementioned technical problems.

[0006] Firstly, this application provides a method for decomposing ultra-large-scale map matrices, including:

[0007] S1. Obtain the graph data to be decomposed, and construct the normalized Laplace matrix based on the adjacency matrix and degree matrix of the graph data.

[0008] S2. Extract the degree distribution features and community structure features of the graph data, and calculate the upper limit of the current spectral interval based on the degree distribution features and community structure features;

[0009] S3. Construct a Chebyshev polynomial filter based on the current upper limit of the spectral interval; wherein, the Chebyshev polynomial filter is configured to retain the components of the input vector that correspond to the eigenvalues ​​within the target eigenvalue interval of the normalized Laplacian matrix, and compress the components of the input vector that correspond to the eigenvalues ​​outside the target eigenvalue interval when filtering the input vector.

[0010] S4. Generate at least one initial vector, and filter the initial vector using a Chebyshev polynomial filter to obtain the starting block;

[0011] S5. Using the starting block as the initial block, perform block Ranchos iteration to generate a block tridiagonal matrix; perform eigenvalue solving on the block tridiagonal matrix to obtain a set of Ritz values ​​and a set of corresponding approximate eigenvectors; determine the convergence state of the approximate eigenvectors based on the gaps between the Ritz values ​​and the residuals between the approximate eigenvectors.

[0012] S6. When the convergence state is stagnant, the upper limit of the current spectral interval is adaptively adjusted based on the distribution of the Ritz values ​​to obtain the updated upper limit of the spectral interval. An updated Chebyshev polynomial filter is constructed based on the updated upper limit of the spectral interval. The updated Chebyshev polynomial filter is used to filter the approximate eigenvectors to obtain a new starting block. The new starting block is used as the starting block, and the process returns to step S5.

[0013] S7. When the convergence state is complete, output the final eigenvalues ​​and corresponding eigenvectors.

[0014] Secondly, this application also provides a large-scale graph matrix decomposition system for implementing the method described in the first aspect, the system comprising:

[0015] The Laplacian matrix construction module is used to obtain the graph data to be decomposed and construct a normalized Laplacian matrix based on the adjacency matrix and degree matrix of the graph data.

[0016] The spectral interval upper limit calculation module is used to extract the degree distribution features and community structure features of the graph data, and calculate the current spectral interval upper limit value based on the degree distribution features and community structure features.

[0017] The Chebyshev filter construction module is used to construct a Chebyshev polynomial filter based on the upper limit of the current spectral interval. The Chebyshev polynomial filter is configured to retain the components of the input vector that correspond to the eigenvalues ​​within the target eigenvalue interval of the normalized Laplacian matrix, and compress the components of the input vector that correspond to the eigenvalues ​​outside the target eigenvalue interval when filtering the input vector.

[0018] The starting block generation module is used to generate at least one initial vector, and to filter the initial vector using a Chebyshev polynomial filter to obtain the starting block;

[0019] The block Ranchos iteration and convergence determination module is used to perform block Ranchos iteration with the starting block as the initial block to generate a block tridiagonal matrix; perform eigenvalue solving on the block tridiagonal matrix to obtain a set of Ritz values ​​and a set of corresponding approximate eigenvectors; and determine the convergence state of the approximate eigenvectors based on the gaps between the Ritz values ​​and the residuals between the approximate eigenvectors.

[0020] The adaptive filtering restart module is used to adaptively adjust the upper limit of the current spectral interval based on the distribution of Ritz values ​​when the convergence state is stagnant, to obtain the updated upper limit of the spectral interval, and to construct an updated Chebyshev polynomial filter based on the updated upper limit of the spectral interval; the updated Chebyshev polynomial filter is used to filter the approximate eigenvectors to obtain a new starting block; the new starting block is used as the starting block, and the operation of the Lanchos iteration and convergence determination module is returned to be executed.

[0021] The decomposition result output module is used to output the final eigenvalues ​​and corresponding eigenvectors when the convergence status is complete.

[0022] Thirdly, this application also provides a computer device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement a method for decomposing a large-scale graph matrix as described in the first aspect.

[0023] Fourthly, this application also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements a method for decomposing a large-scale graph matrix as described in the first aspect.

[0024] The aforementioned method, system, device, and medium for ultra-large-scale graph matrix decomposition first adaptively determines the upper limit of the spectral interval based on the degree distribution and community structure characteristics of the graph data, and then constructs a Chebyshev multinomial filter. This filter is used to selectively scale the initial vector, retaining components within the target eigenvalue interval and compressing components outside the interval, thus artificially amplifying the gap between target features before the block Lanczos iteration. The filtered vector is then used as the starting block to perform block Lanczos iteration, obtaining Ritz values ​​and approximate eigenvectors. The convergence state is determined based on the gap between Ritz values ​​and the residual of the approximate eigenvectors. When convergence stall is detected, the upper limit of the spectral interval is dynamically corrected using the distribution of Ritz values, the filter is reconstructed, and the current approximate eigenvector is filtered again to generate a new starting block, forming a closed-loop iteration of "spectral estimation - filtering enhancement - convergence detection - adaptive restart", ultimately obtaining accurate feature pairs. This approach combines graph topological priors, online spectral interval estimation, Chebyshev acceleration, and block Lanczos depth to significantly suppress spectral radius overflow and eigenvector pseudo-alignment at extreme eigenvalues. It effectively overcomes convergence stagnation caused by eigenvalue aggregation, enabling stable and efficient spectral decomposition on ultra-large-scale heterogeneous graphs and improving the overall robustness and accuracy of low-rank approximation decomposition. Attached Figure Description

[0025] To more clearly illustrate the technical solutions in the embodiments or related technologies of this application, the accompanying drawings used in the description of the embodiments or related technologies will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0026] Figure 1 A flowchart illustrating a method for decomposing an ultra-large-scale spectral matrix according to the present invention;

[0027] Figure 2 This is a flowchart illustrating the process of determining the convergence state of an approximate eigenvector in an optional embodiment of the present invention.

[0028] Figure 3 This is a schematic diagram of the structure of a large-scale map matrix decomposition system provided by the present invention. Detailed Implementation

[0029] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0030] This invention provides a method for ultra-large-scale graph matrix decomposition, applicable to spectral decomposition of large-scale graph data containing hundreds of billions of nodes and edges on a distributed computing cluster consisting of multiple general-purpose servers interconnected by a high-speed network. Graph data typically originates from social networks or financial networks and is stored in a distributed file system as an edge list, with each record containing a source node, a target node, and optional edge weights. The entire distributed system comprises a master node and multiple worker nodes. The master node is responsible for task scheduling, parameter broadcasting, and global scalar reduction. Each worker node holds a partition of the graph and its corresponding vector slice. They collaboratively perform local matrix operations and exchange boundary data over the network. To achieve a low-rank approximate decomposition of the normalized Laplace matrix, this invention integrates graph topological priors, Chebyshev polynomial filtering, block Langios iteration, and adaptive parameter correction into a closed-loop framework, fundamentally overcoming the spectral radius overflow and convergence stagnation that occur in traditional methods when dealing with extreme eigenvalues ​​and eigenvalue clusters. The technical solution of this invention is described in detail below with reference to different implementation methods, each based on the aforementioned distributed environment.

[0031] Example 1:

[0032] refer to Figure 1 The document presents a flowchart illustrating a method for decomposing a large-scale spectral matrix, which includes the following steps:

[0033] S1. Obtain the graph data to be decomposed, and construct the normalized Laplacian matrix based on the adjacency matrix and degree matrix of the graph data.

[0034] Specifically, the distributed computing framework reads the edge list of the graph data from external storage. For undirected graphs, each edge is stored in both directions or processed symmetrically in one step to ensure the symmetry of the adjacency matrix. Each worker node receives edges belonging to its partition according to the hash value of its node identifier and constructs the adjacency matrix locally. The logical representation of the adjacency matrix. At the node With nodes The value is equal to the weight of the edge if there is an edge between them, and zero if there is no edge between them, where the index is... and All values ​​are node numbers. Due to the extremely large size of the graph and the extremely sparse edges, the adjacency matrix is ​​stored in a compressed sparse row format, retaining only the list of neighbors and their corresponding weights for each node to save memory.

[0035] Construct the degree matrix The degree of each node needs to be calculated. Each worker node traverses its local adjacency list, accumulates the weights of the edges connected to each node, and calculates its local degree. Then, a global reduction operation (such as AllReduce summation) is used to aggregate the local degrees of each node, yielding the global degree of each node. ,in , representing the degree matrix The The diagonal elements equal the adjacency matrix. No. The sum of all elements in a row. Degree matrix. It is a diagonal matrix, but due to its extremely high dimension, it is not stored as a complete square matrix, but only as a matrix composed of all... This forms a degree vector. A complete copy of this degree vector is maintained on each worker node to ensure that subsequent scaling operations are fully localized.

[0036] The symmetric normalized Laplace matrix is ​​defined as:

[0037]

[0038] in, Represents the symmetric normalized Laplace matrix. It is the identity matrix. It is an adjacency matrix. It is a degree matrix. This means taking the square root of the reciprocal of each component of the degree vector, i.e., the diagonal elements are... In distributed implementations, dense networks are not explicitly generated. Instead, by retaining And degree vectors, and agree on the specific execution logic of matrix-vector multiplication to represent them equivalently. When it is necessary to calculate With a vector When multiplying, each worker node first uses its local [database name] to calculate the product. value pairs The corresponding components are scaled to obtain the intermediate vector. The components; then, sparse matrix-vector multiplication is performed using the local adjacency matrix to obtain The local part; then, again using Scale the results; finally, from the original vector Subtract the scaled result from the middle. Communication between boundary nodes is completed in the sparse matrix-vector multiplication stage; worker nodes only need to exchange vector elements on adjacent boundaries. The implicit matrix-vector multiplication thus formed is completely equivalent to... Multiplication avoids the overhead of storing dense matrices and can accommodate a scale of hundreds of billions of nodes. It is a symmetric positive semi-definite matrix, and all its eigenvalues ​​lie in the interval [0,2]. The eigenvector corresponding to its smallest eigenvalue 0 is... ,in Represents a vector that is all one. for The inverse matrix, with diagonal elements of .

[0039] S2. Extract the degree distribution features and community structure features of the graph data, and calculate the upper limit of the current spectral interval based on the degree distribution features and community structure features.

[0040] Specifically, in order to obtain the upper limit of the spectral interval that can reliably cover the target feature interval with a small computational cost, this embodiment adopts a method of fusion degree distribution statistics and coarse-grained community conductivity to avoid performing high-precision community detection or complex eigenvalue bound estimation.

[0041] The degree distribution feature is extracted by counting the degrees of all nodes. Each working node already possesses complete degree information locally, so the maximum likelihood estimation of the power-law exponent can be directly performed based on this information. Let the node degree sequence be... ,in The total number of nodes. Indicates the first The degree of each node. To eliminate the noise impact in low-degree regions, a minimum degree threshold needs to be selected. Only retain degrees not lower than Fit the nodes. The selection of can be determined by scanning multiple candidate values ​​and using the Kolmogorov-Smirnov statistic to evaluate the goodness of fit of the power law, or in practice, a lower quantile of the degree sequence can be used as a simple approximation. Let For degrees greater than or equal to The set of nodes, whose size is . ,in This represents the set of valid nodes participating in the fitting process. This represents the number of nodes in the set. Each worker node iterates through its local nodes, and for nodes belonging to this set... Node computation The sums are then accumulated, and a global sum is obtained through global reduction. Finally, the power-law exponent estimate of the degree distribution is obtained. It is given by the following formula:

[0042]

[0043] in, This represents the estimated power-law exponent. The smaller the value, the more skewed the degree distribution, the more a few nodes have extremely high degrees, and the stronger the heterogeneity of the network. Set of valid nodes Size; Indicates all belonging to Calculate the logarithmic ratio of each node and sum them; For nodes The degree; This is the minimum degree threshold.

[0044] The extraction of community structure features employs a lightweight label propagation algorithm. Initially, each node is assigned a unique label. Each node repeatedly updates its label with the most frequent label among its neighbors, stopping when the label distribution stabilizes or a preset number of iterations is reached. This process requires only local communication between adjacent nodes, making it highly efficient in a distributed environment. Ultimately, each node obtains a community ID, and the entire graph is divided into [number of nodes]. Each community is designated as ,in For the total number of communities, Indicates the first Each community. For each community Calculate its volume and boundary cut respectively.

[0045] The volume is defined as the sum of the degrees of all nodes within the community, expressed as:

[0046]

[0047] in, Indicates community volume, This represents the sum of the degrees of all nodes belonging to this community. For nodes The degree.

[0048] A boundary cut is defined as the sum of the weights of all edges connecting internal nodes to external nodes within a community, expressed as:

[0049]

[0050] in, Indicates community Boundary cut, This indicates that for all nodes that satisfy the condition... Within the community and nodes Sum of edge weights outside the community. For nodes With nodes Edge weights between them.

[0051] Then, the electrical conductivity of the community was calculated as follows:

[0052]

[0053] in, Indicates community The conductivity value; To demarcate the boundaries of this community; The volume of the community; This represents the total volume of the remaining nodes after removing this community. This indicates that the smaller of the two values ​​is taken. Conductivity measures community cohesion; a lower conductivity indicates a more tightly connected community and greater isolation from the outside world. The median conductivity of all communities is taken. As a representative value of the entire graph community structure, This represents the median of the set of conductivity values. Since the median conductivity value only requires sorting all the conductivity values ​​of all communities once and taking the middle value, the computational cost is minimal, and it can reflect the spectral gap level of typical communities within the map to a certain extent.

[0054] Next, the degree distribution characteristics and community structure characteristics are combined to calculate the upper limit of the current spectral interval. Introducing a power-law exponent. Related scaling factors The more sloping the degree distribution (i.e., The smaller the value (the smaller the value), the more diffuse the eigenvalue distribution tends to be. The actual upper limit of the target eigenvalue may be higher than the median conductance, therefore a larger scaling factor is needed to ensure the spectral interval can adequately cover the target eigenvalue. The scaling factor can be determined by a monotonically decreasing function, for example, taking:

[0055]

[0056] in, Let be the scaling factor, satisfying ; Let be the power-law exponent of the degree distribution. When When the denominator is close to 1, it is very small. Larger; when When it is large, Approaching 1.

[0057] Finally, the upper limit of the current spectral interval is given by the following formula:

[0058]

[0059] in, This represents the upper limit of the current spectral interval. This is the scaling factor determined based on the power-law exponent. This is the median of the community conductance. This value has extremely low computational complexity and incorporates both types of core prior information from the graph, providing a sufficiently large and reasonable upper bound for the construction of the initial filter.

[0060] S3. Construct a Chebyshev polynomial filter based on the upper limit of the current spectral interval; wherein, the Chebyshev polynomial filter is configured to retain the components of the input vector that correspond to the eigenvalues ​​within the target eigenvalue interval of the normalized Laplacian matrix, and compress the components of the input vector that correspond to the eigenvalues ​​outside the target eigenvalue interval when filtering the input vector.

[0061] Specifically, the construction of a Chebyshev polynomial filter involves three levels: establishing a linear mapping, constructing the filter polynomial, and establishing recursive calculation rules.

[0062] First, the eigenvalue interval of the Laplace matrix needs to be mapped to the standard working interval [−1, 1] of the Chebyshev polynomial of the first kind. The Chebyshev polynomial of the first kind... exist Internal satisfaction ,in Let be the independent variable of the polynomial. The degree of the polynomial; while or Its absolute value increases exponentially with the order. This invention utilizes this property to target the desired range of eigenvalues. Mapped to [−1,1], while the target interval to be retained is... Mapping to the range less than -1. Define a linear mapping function. Its expression is:

[0063]

[0064] in, This represents an eigenvalue of the normalized Laplacian matrix. This represents the upper limit of the current spectral interval. Eigenvalues The corresponding value within the Chebyshev polynomial working interval after linear mapping. Substituting the endpoints, we can see that when hour, ;when hour, ;when hour, And because Hengyou ,in This represents the corresponding value of the zero eigenvalue after mapping. Thus, the region to be suppressed is mapped strictly linearly to [−1, 1], while the target retention region is mapped to... Its left endpoint is much smaller than -1.

[0065] Based on this mapping, the order is constructed as follows: The filtering polynomial, which is a ratio of Chebyshev polynomials, is expressed as:

[0066]

[0067] in, Indicates the order is The filtering polynomial in eigenvalues The value at; The order of the polynomial is preset. It is a Chebyshev polynomial of the first kind, satisfying , And for all There is a recursive relationship ,in It is the recurrence order; Eigenvalues The mapping value; The mapping value of the zero eigenvalue; The numerator represents the mapping value. place Chebyshev polynomial values; The denominator represents the mapping value. place The Chebyshev polynomial value. The key property of this filtering polynomial comes from the denominator. .because absolute value Follow It exhibits hyperbolic cosine exponential growth. For Located in the suppression zone In this situation, ,molecular ,therefore The signal is significantly attenuated; for Located in the target range In this situation, The numerator and denominator are of the same order, therefore near The signal is preserved with high fidelity. This selective scaling behavior achieves the purpose of "projective scaling," which relatively enhances the components of the vector in the target feature space while compressing the components in the non-target feature space.

[0068] The combination of the filter and the specific vector is accomplished through a three-term recursion, without the need for explicit computation of the matrix polynomial. The auxiliary scalar coefficients are defined as follows:

[0069]

[0070] in, and To be based on the upper limit of the current spectral interval The two scalar coefficients were calculated. This represents the upper limit of the current spectral interval.

[0071] For any input vector The recursive process is as follows: Let the initial vector... ,in The initial vector for the recursion; the first step is to calculate... ,in This is the vector obtained from the first step of the recursion. It is the identity matrix. This is the normalized Laplace matrix.

[0072] Subsequently for Execute in sequence:

[0073]

[0074] in, , , They represent the first Step, First Step, First The vector obtained by step recursion.

[0075] Finally, the filtered vector for:

[0076]

[0077] in, This represents the final output vector after filtering. For the first The vector obtained by step recursion; This is the scalar denominator value defined earlier. The denominator contains... The same three recursive formulas can be used to pre-calculate in the scalar domain: Let , ,in , These are the initial value and the first step value for the scalar recursion, respectively. The mapping value for zero eigenvalues; for arrive Iteration ,in , , They represent the first Step, First Step, First The scalar recursive value of the step; then That is, the first The scalar recursion result is calculated by the master node and broadcast to all worker nodes, allowing each node to perform the final division scaling locally. Throughout the recursion process, the core operation at each step is a matrix operation. The product of a vector is equivalent to an implicit Laplacian matrix-vector multiplication plus vector scaling, perfectly matching the communication and computation patterns of distributed graph computing engines.

[0078] To prevent the norm of vector components from continuously amplifying or even overflowing due to extreme eigenvalues ​​during higher-order recursion (i.e., the spectral radius overflow problem), this embodiment also introduces a simple dynamic norm control mechanism. At each order of the recursion... After completion, calculate the current vector. global norm ,in Representing vectors The Euclidean norm. If Exceeding the preset security threshold Then the scaling factor is calculated immediately. ,in, The predefined norm safety upper limit is used; and the current vector is simultaneously adjusted. Previous step vector And the vector in the previous step (when Apply the scaling factor (at the time), that is, multiply them all by . Because the three recurrence relations are homogeneous and linear, applying the same scaling factor to the three consecutive vectors preserves the equivalence of the recurrence. Subsequent vectors are essentially scaled by the same factor as the entire sequence, resulting in a final filtered vector. The direction remains completely unaffected, and subsequent orthogonalization operations will renormalize it, thus ensuring sufficient direction preservation. Through this mechanism, even when recursively applying higher-order Chebyshev polynomials, the numerical range of all vectors can be firmly controlled, fundamentally preventing spectral radius overflow.

[0079] The above steps fully define the Chebyshev polynomial filter, which includes the filtering polynomial. The three recursive calculation rules with dynamic norm control will be applied implicitly to the specified vector in subsequent steps.

[0080] S4. Generate at least one initial vector, and use a Chebyshev polynomial filter to filter the initial vector to obtain the starting block.

[0081] Specifically, the first step is to determine the number of target features that need to be solved. ,in The number of target feature vectors, for example, is set according to the number of clusters required by spectral clustering. The actual block size used is determined to ensure the Krylov subspace can reliably capture all target feature vectors. It should be slightly larger than Generally take ,in The column number of the starting block. The oversampling width can be a small positive integer. On the master node, generate... initial random vectors Each element of each vector is independently drawn from a standard normal distribution. These vectors are then distributed to worker nodes according to the partitioning of the graph nodes, meaning each worker node holds a segment of vector elements corresponding to its own partition node.

[0082] Before filtering these initial vectors, they can optionally be orthogonalized with respect to known trivial zero eigenvectors. Normalized Laplacian matrix The zero eigenvalue corresponds to the eigenvector. It is possible to subtract the value of the initial vector from the value of the initial vector beforehand. The projection in the direction is specifically done by calculating the projection coefficient. ,in Indicates the first The projection coefficients of the initial vectors onto the direction of the zero eigenvector. Representing vectors and The inner product, Representing vectors The norm square; and update Both the inner product and the norm are achieved through distributed reduction. This operation avoids wasting search resources in subsequent iterations towards zero features, which helps improve the convergence speed for non-trivial feature vectors.

[0083] Subsequently, each working node uses the recursive calculation rules established in S3 to perform filtering operations on each initial vector. For example, all work nodes collaborate to complete the task. Each matrix-vector multiplication and vector update requires swapping vector fragments between boundary nodes to ensure global consistency. Throughout the recursion, a dynamic norm control mechanism remains in effect to ensure that no intermediate vectors overflow. After completing all recursion and final scaling, each working node receives the filtered vector. Local sharding. The filtered vectors are combined column-wise to obtain a matrix. ,in This is a matrix composed of the filtered vectors.

[0084] Filtering operation makes The column vectors are greatly enhanced in the target feature space, but they are generally not orthogonal to each other. To ensure the numerical stability of subsequent block Lanchos iterations, they must be transformed into a set of orthonormal bases. This is achieved using distributed Cholesky QR decomposition: First, each working node computes... The local fragment of the column vector inner product is obtained by global reduction summation. Gram matrix ,in For Gram matrix, For matrix The transpose of this matrix results in a very small matrix that can be fully aggregated at the master node. The master node then... Performing Cholesky decomposition yields an upper triangular matrix. ,satisfy ,in It is an upper triangular matrix. Transpose it; then, the master node calculates... inverse matrix ,in Represents an upper triangular matrix The inverse matrix is ​​calculated and broadcast to all worker nodes. Each worker node calculates... This yields the orthogonal starting block, where... This is the starting block matrix. Each column has a norm of 1 and they are all pairwise orthogonal, providing a good initial basis for subsequent iterations.

[0085] S5. Using the starting block as the initial block, perform block Ranchos iteration to generate a block tridiagonal matrix; perform eigenvalue solving on the block tridiagonal matrix to obtain a set of Ritz values ​​and a set of corresponding approximate eigenvectors; determine the convergence state of the approximate eigenvectors based on the gaps between the Ritz values ​​and the residuals between the approximate eigenvectors.

[0086] Specifically, the goal of the block Ranchos iteration is to progressively generate a set of orthogonal bases such that the matrix The projection onto this basis is a block tridiagonal matrix, which is then used to extract approximate feature pairs. Let the starting block be... For the first Step (from) (Start), assuming the previous... Orthogonal basis blocks To generate the first Each block.

[0087] First, calculate the matrix product. ,in The result of matrix multiplication. For the first Each base block matrix. This product, in a distributed environment, is processed locally by each worker node. The aforementioned implicit Laplacian matrix-vector multiplication is performed on the partitioned data, and the boundary data is swapped to obtain the complete result. .

[0088] Next, in order to eliminate the rounding errors accumulated in finite-precision calculations, it is necessary to... Perform two complete reorthogonalizations on all existing basis blocks. First orthogonalization: for Calculate the projection coefficient matrix (Summarized by reducing the local inner product of each node), where For the first Projection coefficient matrix, For the first Transpose the base block matrix, then update Second orthogonalization: Repeat the above process to further reduce the residual non-orthogonal components to machine accuracy. After two orthogonalizations, Already with They are strictly orthogonal in numerical terms.

[0089] Then, after orthogonalization Perform block Cholesky decomposition to extract new basic blocks and their relation matrices with the previous basic blocks. Specifically, calculate the local symmetric matrix. ,in For the first A diagonal block matrix, for The symmetric matrix, and the upper triangular matrix obtained by decomposition. and the new standard orthogonal basis blocks ,in For the first The nth diagonal block matrix is an upper triangular matrix, For the newly generated orthogonal basis blocks, the following relation holds:

[0090]

[0091] in, For the previous base block matrix, This is the transpose of the previous second-diagonal block matrix. It is conventionally defined as a zero matrix. If a new block is discovered during the decomposition process... If the norm of the vector is extremely small or close to singular, it indicates that the current basis vector block has become linearly dependent, and the expansion can be terminated early, with all the bases generated at this point being taken as the final Lanjosky.

[0092] Put all the diagonal pieces and the second diagonal piece Arranged in order, they form a block tridiagonal matrix. Its size is ,in It is a block tridiagonal matrix. For the number of Lanchos moves completed, The size is the block size.

[0093] When the tridiagonal matrix When the dimension reaches a preset upper limit, or when the basis vectors exhibit linear dependence, expansion stops. On the master node (or a specified computation node), for... Perform full spectrum decomposition. Because... Since the size is much smaller than the original matrix, it can be solved efficiently using the standard QR algorithm or divide-and-conquer algorithm. After decomposition, all Ritz values ​​are obtained and denoted in ascending order as follows: ,in Indicates the first Each Ritz value is calculated, and its corresponding Ritz vector is obtained. Each Ritz vector is then left-multiplied by the basis matrix. To recover the approximate eigenvectors of the original matrix. ,in For the first Approximate eigenvectors Let be the basis matrix composed of all the basis blocks. For the first Ritz vectors.

[0094] The evaluation of the convergence state is for the previous We perform this on each target feature pair. First, we calculate the residual norm of each approximate feature vector using the following formula:

[0095]

[0096] in, Indicates the first The residual norm of approximate eigenvectors, For the first Approximate eigenvectors For the first One liz value, Representing vectors Norm. This norm can be obtained by calculating the squared components of the residual vector locally at each working node, and then summing and taking the square root through global reduction.

[0097] If the maximum residual satisfies If the current approximate feature pair is sufficiently accurate, the convergence state is set to "convergence complete," and the process proceeds directly to S7 to output the result. Indicates the preceding The maximum value among the residual norms The number of target feature vectors. This is a pre-defined convergence tolerance.

[0098] If the maximum residual does not meet the tolerance, then the gap between the target eigenvalue and subsequent eigenvalues ​​is further examined. Take the first... The liz value and the first The difference between each riz value is taken as the target gap, denoted as: .in, For the target gap, and The first The and the first Each lizard value. (If the current subspace contains only...) One or fewer Ritz values ​​can be used to temporarily extend the Ranchos process by one more step to obtain Compare the target clearance with a preset minimum allowable clearance. :like If the target eigenvalue is too close to the subsequent eigenvalues, the Lanchos process will find it difficult to continue separating them, and it will be judged as "convergence stall"; otherwise, the gap is still acceptable, and the block Lanchos iteration will continue, repeating S5.

[0099] Minimum gap here Based on the graph size and experience, the idea is that when the separation between adjacent feature values ​​is too small, the Ritz vector generated by the iteration cannot effectively approximate the true feature direction, and a filter is needed to further enhance the separation.

[0100] S6. When the convergence state is stagnant, the upper limit of the current spectral interval is adaptively adjusted based on the distribution of the Ritz values ​​to obtain the updated upper limit of the spectral interval. An updated Chebyshev polynomial filter is constructed based on the updated upper limit of the spectral interval. The updated Chebyshev polynomial filter is used to filter the approximate eigenvectors to obtain a new starting block. The new starting block is used as the starting block, and the process returns to step S5.

[0101] Specifically, when S5 determines that convergence has stalled, it means that the suppression range of the currently used filter is no longer accurate enough, and the upper limit of the spectral range needs to be corrected using the obtained Ritz value distribution. This embodiment adopts a direct and effective midpoint adjustment scheme: selecting the first... The liz value and the first The midpoint of each riz value is used as the upper limit of the updated spectral interval, i.e.:

[0102]

[0103] in, This represents the upper limit of the updated spectral interval. and The first The and the first One liz value. If If it doesn't exist yet, a temporary Lanchos iteration can be added to obtain it. This midpoint value is located precisely between the target feature cluster and the subsequent interfering clusters; using it as a new spectral boundary allows the filter to suppress subsequent components more accurately.

[0104] Master node according to Recalculate coefficients , and polynomial denominator scalar The updated parameters are then broadcast to all working nodes. Subsequently, following the method in S3, the linear mapping, filtering polynomial, and recursive calculation rules with dynamic norm control are re-established to construct the updated Chebyshev polynomial filter.

[0105] Next, selective re-filtering is performed on the currently obtained approximate eigenvectors. Select from those with residual norm greater than the convergence tolerance and corresponding Ritz values ​​less than 1. Those vectors form the set of vectors to be purified. ,in This is a matrix composed of vectors to be purified. This ensures that vectors that have already converged are no longer disturbed, and only vectors that are still within the target interval and whose accuracy has not yet reached the target are subjected to enhancement processing.

[0106] Use the updated filter Repeat the same filtering recursive process as S4 for each column to obtain the filtered vector set. Then perform Cholesky QR orthogonalization again to generate new orthogonal start blocks. ,in This forms the new starting block matrix. Finally, the old base matrix is ​​discarded. and block tridiagonal matrix ,Will As a completely new starting block, return to S5 and restart the Lanchos iteration. Because the new starting block is re-filtered under a more accurate suppression range, the target feature components are further purified, and the subsequent Lanchos process will converge at a significantly faster speed, effectively breaking the stagnation phenomenon.

[0107] S7. When the convergence state is complete, output the final eigenvalues ​​and corresponding eigenvectors.

[0108] Specifically, when S5 determines that convergence is complete, the previous value is taken. The Ritz values ​​are used as the final eigenvalues, i.e. ,in Indicates the first The final eigenvalues, The corresponding Ritz value, and the corresponding approximate eigenvector. As the final feature vector output, Indicates the first These final eigenvectors meet the requirements for orthogonality and accuracy and can be directly used for subsequent spectral clustering tasks. For example, the eigenvectors can be row-normalized and input into the K-means algorithm to achieve network topology partitioning. The entire process of decomposing the ultra-large-scale graph matrix is ​​now complete.

[0109] The method described in Example 1 constructs a complete "estimation-filtering-convergence" closed loop. It utilizes graph topological priors to provide initial values ​​for the spectral interval, enhances feature separation through Chebyshev filtering, extracts feature pairs using block Langios iteration, and adaptively restarts based on the midpoint of the Ritz value when stalling. In particular, the dynamic norm control introduced in the filter construction ensures the numerical stability of the recursive process in a lightweight manner, enabling the method to operate stably on real-world graph data with hundreds of billions of nodes. Compared to traditional methods without filtering and adaptive adjustment, this represents a fundamental technological advancement.

[0110] The aforementioned method for decomposing ultra-large-scale graph matrices adaptively determines the upper limit of the spectral interval based on the degree distribution and community structure characteristics of the graph data. A Chebyshev multinomial filter is then constructed using this filter to selectively scale the initial vector, retaining components within the target eigenvalue interval and compressing components outside the interval. This artificially amplifies the gap between target features before the block Lanczos iteration. The filtered vector is then used as the starting block for block Lanczos iteration, yielding Ritz values ​​and approximate eigenvectors. The convergence state is determined based on the gap between Ritz values ​​and the residual of the approximate eigenvectors. When convergence stall is detected, the upper limit of the spectral interval is dynamically corrected using the distribution of Ritz values. The filter is reconstructed, and the current approximate eigenvector is filtered again to generate a new starting block, forming a closed-loop iteration of "spectral estimation—filter enhancement—convergence detection—adaptive restart," ultimately obtaining accurate feature pairs. This approach combines graph topological priors, online spectral interval estimation, Chebyshev acceleration, and block Lanczos depth to significantly suppress spectral radius overflow and eigenvector pseudo-alignment at extreme eigenvalues. It effectively overcomes convergence stagnation caused by eigenvalue aggregation, enabling stable and efficient spectral decomposition on ultra-large-scale heterogeneous graphs and improving the overall robustness and accuracy of low-rank approximation decomposition.

[0111] Example 2:

[0112] Extracting the degree distribution and community structure features from the graph data, and calculating the upper limit of the current spectral interval based on the degree distribution and community structure features, includes the following steps:

[0113] S11. Perform community detection on the graph data to obtain multiple communities. Calculate the conductivity value of each community based on the degree matrix and adjacency matrix to obtain a set of conductivity values.

[0114] Specifically, unlike the lightweight label propagation algorithm used in Example 1, this example uses the Louvain algorithm based on modularity optimization for distributed community detection to obtain higher-quality community partitioning. The Louvain algorithm aims to maximize the modularity of the entire graph, where modularity measures the extent to which the internal connection density of a community exceeds that of a random graph model. In a distributed environment, the Louvain algorithm executes alternately in two phases: In the first phase, each worker node scans its local nodes in parallel, calculates the modularity gain for each node by moving from its current community into neighboring communities, and reassigns the node to the community with the largest and positive gain; in the second phase, the communities discovered in the previous phase are aggregated into supernodes, a new coarsened graph is constructed, and the first phase is recursively executed. This multi-scale process effectively reveals the inherent hierarchical community structure in the graph. Finally, the community label of each node is obtained on the original graph, dividing all nodes into... a community ,in For the total number of communities, Indicates the first The Louvain algorithm utilizes global modularity optimization more effectively than label propagation, resulting in higher-quality communities with denser internal connections and clearer boundaries between communities.

[0115] For each community Calculate its volume and cuts. The volume is defined as the sum of the degrees of all nodes within the community, expressed as: ,in Indicates community volume, This represents the sum of the degrees of all nodes belonging to this community. For nodes The degree of the boundary cut. The boundary cut is defined as the sum of the weights of all edges connecting internal nodes to external nodes of the community, expressed as: ,in Indicates community Boundary cut, This indicates that for all nodes that satisfy the condition... Within the community and nodes Sum of edge weights outside the community. For nodes With nodes The edge weights between them. The formula for calculating the community conductance is: ,in Indicates community The conductivity value; To demarcate the boundaries of this community; The volume of the community; The total volume of the remaining nodes after removing this community. The set of all nodes; This indicates that the smaller of the two values ​​is taken. This conductivity value precisely measures the degree of separation between the community and the rest of the graph—the smaller the conductivity, the more independent the community, and the tighter the upper bound of the corresponding eigenvalue. The conductivity values ​​of all communities form a set. ,in This is a set of conductivity values. Due to the use of modularity-optimized Louvain community detection, these conductivity values ​​can more accurately reflect the multi-scale spectral structure of the graph.

[0116] S12. Sort the set of conductance values ​​in ascending order, according to the preset number of target features. Select the first The conductance value; using the upper bound relationship between eigenvalues ​​and conductance in the multi-path Cheeger inequality, the upper bound of the eigenvalue is determined as the nth conductance value. Twice the conductivity value.

[0117] Specifically, for sets The elements in the array are sorted in ascending order of value to obtain an ordered sequence. ,in Indicates the first digit after ascending order. The minimum conductance value. The multi-way Cheeger inequality establishes a rigorous mathematical relationship between the eigenvalues ​​of the graph Laplace matrix and the community conductance. For the normalized Laplace matrix... The The smallest non-zero eigenvalues Satisfies the upper bound relation:

[0118]

[0119] in, Indicates the first 1 eigenvalue, After sorting in ascending order, the first The inequality shows that the first conductivity value. The eigenvalue will not exceed the th eigenvalue. Twice the conductivity value of the smallest community. Intuitively, the former Each eigenvalue corresponds to the one with the smallest conductance in the graph. Each community structure has an eigenvalue constrained by the corresponding community conductance. Therefore, the upper limit of the unexpanded baseline spectral interval can be taken as... Compared to Example 1, which used the median conductance of all communities. As a representative example, the estimation in this embodiment is directly aimed at the first... Each target feature value makes full use of conductivity sorting information, thus making it more compact—it is not unnecessarily inflated by the presence of a large number of high conductivity communities in the graph.

[0120] S13. The power-law exponent of the degree distribution of the statistical graph data. Based on the power-law exponent of the degree distribution, the safety expansion factor is determined according to the preset mapping relationship. The upper limit of the current spectral interval is calculated by combining the upper bound of the eigenvalue and the safety expansion factor. The formula for calculating the upper limit of the current spectral interval is as follows:

[0121]

[0122] in, This represents the upper limit of the current spectral interval. After sorting in ascending order, the first Each conductivity value, The preset number of target features; For security expansion factor and The mapping relationship is configured such that the value of the security extension factor increases as the degree distribution power-law exponent decreases.

[0123] Specifically, the power-law exponent of degree distribution The estimation can be performed using the same maximum likelihood method as in Example 1. Let represent the power-law exponent of the degree distribution. Although Cheeger's inequality provides a theoretical bound on the eigenvalues, in practical power-law networks, due to the extreme skewness of the degree distribution, the true eigenvalues ​​are often less than the theoretical bound. Some eigenvalues ​​may be slightly higher than this upper bound, or the eigenvalue distribution may be quite diffuse, resulting in insufficient spectral interval. Therefore, a safety expansion factor needs to be introduced. To appropriately amplify the spectral range. and The mapping relationship reflects the additional constraint of the degree distribution characteristics on the spectral distribution: when When the degree distribution is relatively small (highly skewed), the influence of a few supernodes on the spectral distribution is amplified, the eigenvalue distribution becomes more dispersed, and a larger eigenvalue distribution is required. To ensure coverage; when When the spectral density is relatively high (and the spectral distribution is relatively uniform), the spectral distribution is more regular. It can approach 1. An exemplary mapping function is: ,in For security expansion factor; The power-law exponent is the degree distribution. This is an empirical constant calibrated experimentally. When... When the denominator is close to 1, it is very small. Significantly greater than 1; when When it is much greater than 1, Approaching 1. Ultimately, the upper limit of the current spectral interval. That is, the upper limit of the benchmark. With security extension factor The product of.

[0124] Where, constant The specific value can be determined by pre-calibrating on a set of typical graph data with different power-law exponents. The calibration process can be as follows: select several representative real social network graphs or financial network graphs as a calibration graph set. The node size of these calibration graphs should be within the typical range of the expected application scenario, and each should have a different degree distribution power-law exponent. For each calibration graph, the benchmark upper bound is obtained using the Louvain community detection method in S11 and the Cheeger bound estimation method in S12 of this embodiment. Then, higher-order Langios iteration or approximate spectral decomposition methods are used to obtain the front of the graph. The largest of the eigenvalues ​​is used as the true reference value. For different candidates The value is used to calculate the corresponding security extension factor according to the mapping function described above. and the upper limit of the spectral interval And check if it meets the requirements. That is, whether the spectral interval can cover the true feature values. If This indicates that The given safety extension is insufficient, and the starting point of the filter suppression interval is too early, which may inadvertently exclude target feature components. From all candidate filters that can guarantee coverage... The smallest value is selected from the available values ​​to make the spectral interval as compact as possible while fully covering the target feature values. The final selected value is... The value can be fixed for decomposition tasks of similar power-law networks. When a calibration graph is lacking, a slightly larger conservative value can also be used during the first run of this method. The initial value is determined by the first round of decomposition, and adjustments are made based on the eigenvalue estimates obtained. The values ​​are used for subsequent decomposition of graphs of the same size. This calibration method fully utilizes the statistical commonality of the degree distribution in power-law graphs. Once calibration is complete, The value can be reused for the same type of graph data.

[0125] Implementation 2, through more accurate Louvain community detection, tight eigenvalue bounds based on ordinal conductance and Cheeger inequality, and a safety expansion factor adaptively bound to the power-law exponent, obtained an initial spectral interval that more closely approximates the true spectral distribution than Implementation 1. This improvement allows the Chebyshev filter to operate with near-optimal parameters from the outset, and the starting point of the suppression interval is closer to the true target eigenvalue boundary. The resulting technical benefits are: with the same filter recursion order, the contrast between the target eigencomponent and the suppressed eigencomponent is higher, the total number of steps required for subsequent block Langios iterations is reduced, and the overall decomposition efficiency is significantly improved.

[0126] Example 3:

[0127] Based on the upper limit of the current spectral interval, construct a Chebyshev polynomial filter, including the following steps:

[0128] S21. Based on the current upper limit of the spectral interval, establish a linear mapping that maps the eigenvalues ​​of the normalized Laplace matrix to the Chebyshev polynomial working interval, obtaining the mapping function; where the expression for the linear mapping is:

[0129]

[0130] in, For the eigenvalues ​​of the normalized Laplace matrix, This represents the upper limit of the current spectral interval. Eigenvalues The corresponding value in the working interval after linear mapping. This is the corresponding value of the zero eigenvalue in the working interval.

[0131] Specifically, this mapping is uniquely determined by two endpoint conditions: the mapping value must be equal to the left endpoint of the Chebyshev interval when the eigenvalue is located at the left endpoint of the suppression interval. When the eigenvalue lies at the maximum possible value of the eigenvalues ​​of the Laplace matrix, the mapped value is equal to the right endpoint of the Chebyshev interval, i.e. Solving these two conditions yields the linear function expression described above. This mapping is a strictly monotonically decreasing linear function, therefore a larger eigenvalue interval... The compression mapping is applied to the standard interval [−1,1], while the smaller eigenvalue interval is mapped to... Projected to And because This can be verified. This means that the target retention interval is mapped to a region much smaller than -1. This mapping design lays the foundation for the next step of utilizing the exponential growth property of Chebyshev polynomials in this region.

[0132] S22. Based on the mapping function and the preset polynomial order, construct a filter polynomial composed of Chebyshev polynomials of the first kind; wherein the numerator of the filter polynomial is the value of the mapping function at the eigenvalue variable. The first-order Chebyshev polynomial of order I, with the denominator being the value of the mapping function at the zero eigenvalue. The first-order Chebyshev polynomial of order I; where the expression for the filter polynomial is:

[0133]

[0134] in, The order of the polynomial is preset. for Chebyshev polynomial of the first kind.

[0135] Specifically, the polynomial order determines the trade-off between the attenuation intensity of the filter in the suppression range and the amount of recursive calculation. Let be a Chebyshev polynomial of the first kind, whose domain is all real numbers, satisfying , And for all There is a recursive relationship ,in Let be the independent variable of the polynomial. It is the recurrence order; Eigenvalues The mapping value; The mapping value of the zero eigenvalue; The numerator represents the mapping value. place Chebyshev polynomial values; The denominator represents the mapping value. place Chebyshev polynomial values.

[0136] Chebyshev polynomials of the first kind have the following key property: when the independent variable... When it is located in the closed interval [−1,1], The function value oscillates between 1 and -1; when hour, The parsing expression is Where cosh is the hyperbolic cosine function and arccosh is the inverse hyperbolic cosine function. For factors that alternate between positive and negative signs, when The term is positive when it is even, and positive when it is not. When the factor is odd, it is negative. This factor only determines the sign of the function value and does not affect the magnitude of its absolute value, while the hyperbolic cosine part... Follow It grows exponentially, therefore in hour It approaches infinity at an exponential rate; when It exhibits similar exponential growth properties.

[0137] It is this property that gives the filtering polynomial its power. Spectral selectivity. For ,have Therefore, molecules The denominator because Its absolute value is much greater than 1, and it increases with... Exponential growth. Therefore, within this range... The signal components are drastically compressed. For , The numerator and denominator grow exponentially of the same order, and their ratio is close to 1, allowing the signal components to be preserved with high fidelity. In particular, when... hour, Since the numerator and denominator are the same, therefore... This verifies that the gain is exactly 1 at the zero eigenvalue. Therefore, Mathematically, this achieves the selective preservation of the target feature value interval and the exponential compression of the suppression interval, which is equivalent to performing a spectral-aware projection scaling on the vector in the feature space.

[0138] S23. Using the three recursive relations of the first-kind Chebyshev polynomial, construct the recursive calculation rule for performing filtering on the input target vector from the filter polynomial, and obtain the Chebyshev polynomial filter including the filter polynomial and the recursive calculation rule.

[0139] Specifically, the filtering polynomial needs to be... Actual action on any input vector That is, calculation This can be achieved directly using the three-term recurrence relation of the Chebyshev polynomial without explicitly constructing a matrix polynomial. Therefore, a linear operator is defined. Two of the scalar coefficients are: and .in, and To be based on the upper limit of the current spectral interval The two scalar coefficients were calculated; This represents the upper limit of the current spectral interval. For the normalized Laplace matrix; It is the identity matrix. This can be verified. The action on a vector is equivalent to first applying the vector to itself. Then it is linearly scaled to the Chebyshev workspace.

[0140] Based on this operator, the filtering recursive process is defined as follows: Using the input vector... Set the initial vector as the starting point. ,in Use the initial vector for the recursion; then calculate the result of the first step of the recursion. ,in The vector obtained in the first step of the recursion; for Execute iteration: .in, , , They represent the first Step, First Step, First The vector obtained by step recursion, Let this be the current recurrence order. Let be the order of the polynomial.

[0141] After the recursion is complete, the filtered vector is .in, This represents the final output vector after filtering. For the first The vector obtained by step recursion; The scalar denominator value, i.e., the value mapped to the zero eigenvalue. Chebyshev polynomial values. Denominator scalar. It can be efficiently calculated in advance in the scalar field using the same recurrence relation: Let , ,in , These are the initial value and the first step value for the scalar recursion, respectively. The mapping value for zero eigenvalues; for arrive implement ,in , , They represent the first Step, First Step, First The scalar recursive value of the step; then That is, the first The scalar recursion result is obtained step by step. This scalar calculation only involves... Secondary scalar multiplication and subtraction can be performed by the master node and broadcast.

[0142] Example 4:

[0143] refer to Figure 2 Based on the gaps between Ritz values ​​and the residuals between approximate eigenvectors, the convergence state of the approximate eigenvectors is determined, including the following steps:

[0144] S31. Sort the Ritz values ​​in ascending order and calculate the th value after ascending order. The liz value and the first The difference between the ritz values ​​is used to obtain the target gap; where... The number of target features is preset.

[0145] Specifically, after each round of feature solving in the block Ranchos iteration, a set of Ritz values ​​is obtained. ,in Indicates the first Each riz value. First, sort them in ascending order of value to ensure... Then take the first... The and the first Calculate the difference between the two rits values. ,in For the target gap, and The first The and the first The ritz value represents the degree of separation between the “outer boundary” of the target eigenvalue cluster and the “inner boundary” of the subsequent interfering eigenvalue cluster.

[0146] S32, when When greater than 1, based on the first [previous] after ascending sort. For each ritz value, calculate the average value of the gap between adjacent ritz values ​​to obtain the reference gap; when When the value is 1, the target gap is used as the reference gap.

[0147] Specifically, when At this time, the target cluster contains at least two Ritz values, so the average value of the adjacent gaps within it can be calculated as a reference gap. The calculation formula is as follows:

[0148]

[0149] in, Indicates the reference gap, i.e., the front The average value of adjacent gaps within a ritz value; The number of target features; and This represents two adjacent ritz values. The average value. This reflects the typical density within the target cluster. Ideally, the eigenvalues ​​within the target cluster should be relatively uniformly distributed. It is a stable internal reference value. When At that time, the target cluster contained only one Ritz value, making it impossible to calculate the internal gaps. Therefore, the reference gap was directly set as the target gap. In this case, the role of the reference gap degenerates into a scaling factor.

[0150] S33. Based on the approximate eigenvectors and Ritz values, calculate the residual norm of each approximate eigenvector relative to the normalized Laplace matrix.

[0151] Specifically, the formula for calculating the residual norm is as follows: ,in Indicates the first The residual norm of approximate eigenvectors; For the normalized Laplace matrix; For the first Approximate eigenvectors; For the first One liz value; Representing vectors Norm. The residual norm is a direct indicator of the accuracy of the current approximate eigenvector. Its calculation method is the same as in Example 1, and it is completed in parallel by each working node.

[0152] S34. When the maximum residual norm is less than or equal to the preset convergence tolerance, the convergence state is determined to be complete; when the maximum residual norm is greater than the convergence tolerance, if... If the value is greater than 1 and the target gap is less than the product of the reference gap and a preset coefficient greater than 1, then the convergence state is determined to be convergence stagnation; when the maximum residual norm is greater than the convergence tolerance, if If the value is equal to 1, then the convergence state is determined to be convergence stagnation.

[0153] Specifically, the decision logic in this step provides three mutually exclusive paths for the convergence state evaluation. First, it checks the accuracy condition: if... Then, regardless of the gap condition, the approximate eigenvector is accurate enough, and convergence is considered complete; where, For the front The maximum value of the residual norm. To minimize convergence tolerance. If the accuracy requirement is not met, proceed to gap analysis: when At that time, the target gap With reference gap of Comparing by multiples, A pre-defined coefficient greater than 1. Inequality The physical meaning is that the outer boundary gap of the target eigenvalue cluster is not significantly larger than the typical internal gap; in other words, the first... The eigenvalue and the th eigenvalue The degree of separation of the eigenvalues ​​does not significantly exceed the degree of dense clustering within the target cluster. In this case, the Ranchos subspace struggles to effectively distinguish the eigenvalues. The eigenvector and the eigenvector The directional differences among the eigenvectors cause convergence to stall. If If the outer boundary gap is sufficiently large relative to the internal density, the subspace method can continue to effectively separate the elements and should not be considered stalled; iterative iteration can continue. When there is a lack of internal gap reference, the system is directly judged to be stagnant because the accuracy condition is not met, so as to ensure that the necessary adaptive adjustment can be triggered even when only a single feature vector is needed.

[0154] The adaptive convergence criterion in Example 4 fully utilizes the spectral distribution structure information contained in the Ritz value sequence. By dynamically comparing the outer boundary gap with the internal average gap, it can automatically adapt to the spectral density of different graphs. This is in contrast to Example 1, which relies solely on a fixed threshold. By making a judgment, this embodiment can effectively distinguish between two gaps with completely different properties: "naturally dense internal gaps" and "abnormally close external boundaries." This avoids the defects of misjudging and stagnating on dense graphs or missing judgment and stagnating on sparse graphs due to fixed thresholds.

[0155] Example 5:

[0156] The upper limit of the current spectral interval is adaptively adjusted based on the distribution of the Ritz values ​​to obtain the updated upper limit of the spectral interval. An updated Chebyshev polynomial filter is then constructed based on the updated upper limit of the spectral interval, including the following steps:

[0157] S41. Sort the Ritz values ​​in ascending order, when the preset number of target features is reached. When greater than 1, based on the first [previous] after ascending sort. The average gap between adjacent Ritz values ​​is calculated, and this average is multiplied by a preset threshold coefficient to obtain the adaptive gap threshold; when the preset number of target features... When the value is equal to 1, the gap between the first and second Ritz values ​​is multiplied by a preset threshold coefficient to obtain the adaptive gap threshold.

[0158] Specifically, if Before calculation The average value of the internal gap of each ritz value .in, Indicates the preceding The average value of the adjacent gaps within a given ritz value. and For two adjacent Ritz values, The number of target features. If Then, directly take the single gap between the first and second Ritz values. As the base value, where The interval between the first and second riz values. and These are the first and second Ritz values, respectively.

[0159] Introduce a preset threshold coefficient The adaptive gap threshold is defined as:

[0160]

[0161] in, For adaptive gap threshold, The preset threshold coefficient and , For reference gap (when) hour, Replace with ).Should This serves as a benchmark for judging whether subsequent Ritz value gaps are "significant"—only when a gap is greater than this adaptive threshold is it considered to constitute a natural target cluster boundary, where the filter's suppression starting point can be safely set.

[0162] S42, from the first in ascending order Starting with a Ritz value, the Ritz value sequence is scanned backward, and the gaps between adjacent Ritz values ​​are calculated to obtain the gap sequence.

[0163] Specifically, this step involves examining each case in turn. ,in Given the Ritz values ​​in ascending order, we obtain a gap sequence composed of these differences.

[0164] S43. Based on the gap sequence, determine whether there is a gap greater than the adaptive gap threshold.

[0165] Specifically, this judgment operation only requires comparing from front to back until the first gap that meets the conditions is found or all available gaps have been scanned.

[0166] S44. When a gap satisfying the condition exists, select the adjacent Ritz value pair corresponding to the first gap satisfying the condition, and set the upper limit of the updated spectral interval as the midpoint of the adjacent Ritz value pair; update the Chebyshev polynomial filter according to the upper limit of the updated spectral interval to obtain the updated Chebyshev polynomial filter; wherein, the condition is that there is a gap in the gap sequence that is greater than the adaptive gap threshold; when there is an adjacent Ritz value satisfying the condition... and When updated, the formula for calculating the upper limit of the spectral interval is:

[0167]

[0168] in, This represents the upper limit of the updated spectral interval. and To meet The adjacent Ritz values, The index of the smaller of the adjacent Ritz values ​​and ; This is an adaptive gap threshold.

[0169] Specifically, this midpoint is precisely located in the center of the natural gap between the target feature cluster and subsequent interfering clusters, enabling it to retain the target component and suppress interfering components with the highest balance. Then, the linear mapping, filtering polynomial, and recursive rules are re-established according to the method in S3, thereby constructing the updated Chebyshev polynomial filter.

[0170] S45. When there is no gap that meets the conditions, maintain the updated upper limit of the spectral interval as the current upper limit of the spectral interval, and increase the order of the Chebyshev polynomial filter to obtain the updated filter order; update the Chebyshev polynomial filter according to the updated upper limit of the spectral interval and the updated filter order to obtain the updated Chebyshev polynomial filter.

[0171] Specifically, if scanning backwards to the end of the sequence (or all Ritz values ​​provided by the subspace) does not find any Ritz values ​​greater than 1, then... The gaps indicate that within the currently resolved spectral range, there is no obvious natural separating band between the target feature cluster and subsequent eigenvalues. Forcibly shifting the spectral boundaries at this point could potentially cut off the interior of the target cluster, leading to the inaccurate suppression of useful signals. Therefore, maintaining... Unchanged, among which Instead of using the upper limit of the current spectral interval, the separation is improved by increasing the filter strength. Specifically, the filter order is increased, for example, by taking... ,in The updated filter order. The original order remains unchanged. As the order increases, although the starting position of the suppression interval remains the same, according to the exponential growth characteristic of Chebyshev polynomials, the decay coefficient within the suppression interval will increase exponentially. This allows for the forced suppression of interfering feature values ​​that closely follow the target feature, artificially creating an effective gap. Then, using... and As new parameters, all construction steps in S3 are re-executed to obtain the updated Chebyshev polynomial filter.

[0172] The dual adaptive strategy provided in Example 5 treats "spectral boundary shifting" and "filter strength enhancement" as two complementary control dimensions, and automatically selects the optimal control method based on real-time evaluation of the significance of Ritz value gaps. When significant natural gaps exist, adjusting the spectral boundaries is prioritized, a "go with the flow" strategy that is low-cost and effective. When natural gaps are missing, increasing the filter order is used, a "counter-current pressure" strategy that, although computationally increased, can forcefully open the convergence path. The two methods work together under a unified gap threshold criterion, which is far more robust and efficient than using a single method. This adaptive mechanism is not a simple combination of existing technologies, but is based on a deep understanding of the dynamic characteristics of the "estimation-filtering-convergence" closed loop: it recognizes the coupling relationship between spectral interval estimation accuracy, filter suppression strength, and Langios convergence behavior, and dynamically balances this coupled system through real-time sensing feedback. When faced with the highly irregular spectral distribution of heterogeneous networks with hundreds of billions of nodes, the method in this embodiment can flexibly make decisions based on the actual spectral structure, always maintaining a highly efficient convergence driving force, and achieving a stable acceleration effect that cannot be achieved by using fixed parameters or a single method. This is a significant breakthrough in existing spectral decomposition acceleration technology.

[0173] The aforementioned method for decomposing ultra-large-scale graph matrices adaptively determines the upper limit of the spectral interval based on the degree distribution and community structure characteristics of the graph data. A Chebyshev multinomial filter is then constructed using this filter to selectively scale the initial vector, retaining components within the target eigenvalue interval and compressing components outside the interval. This artificially amplifies the gap between target features before the block Lanczos iteration. The filtered vector is then used as the starting block for block Lanczos iteration, yielding Ritz values ​​and approximate eigenvectors. The convergence state is determined based on the gap between Ritz values ​​and the residual of the approximate eigenvectors. When convergence stall is detected, the upper limit of the spectral interval is dynamically corrected using the distribution of Ritz values. The filter is reconstructed, and the current approximate eigenvector is filtered again to generate a new starting block, forming a closed-loop iteration of "spectral estimation—filter enhancement—convergence detection—adaptive restart," ultimately obtaining accurate feature pairs. This approach combines graph topological priors, online spectral interval estimation, Chebyshev acceleration, and block Lanczos depth to significantly suppress spectral radius overflow and eigenvector pseudo-alignment at extreme eigenvalues. It effectively overcomes convergence stagnation caused by eigenvalue aggregation, enabling stable and efficient spectral decomposition on ultra-large-scale heterogeneous graphs and improving the overall robustness and accuracy of low-rank approximation decomposition.

[0174] It should be understood that although the steps in the flowcharts of the embodiments described above are shown sequentially according to the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless explicitly stated herein, there is no strict order restriction on the execution of these steps, and they can be executed in other orders. Moreover, at least some steps in the flowcharts of the embodiments described above may include multiple steps or multiple stages. These steps or stages are not necessarily completed at the same time, but can be executed at different times. The execution order of these steps or stages is not necessarily sequential, but can be performed alternately or in turn with other steps or at least some of the steps or stages of other steps.

[0175] Based on the same inventive concept, this application also provides a system for implementing the ultra-large-scale spectral matrix decomposition method described above. The solution provided by this system is similar to the implementation scheme described in the above method; therefore, the specific limitations of one or more ultra-large-scale spectral matrix decomposition system embodiments provided below can be found in the limitations of the ultra-large-scale spectral matrix decomposition method described above, and will not be repeated here.

[0176] In one exemplary embodiment, such as Figure 3 As shown, a large-scale graph matrix decomposition system 30 is provided to implement the methods in the above-described method embodiments. The system includes:

[0177] The Laplacian matrix construction module 31 is used to obtain the graph data to be decomposed and construct a normalized Laplacian matrix based on the adjacency matrix and degree matrix of the graph data.

[0178] The upper limit calculation module 32 for the spectral interval is used to extract the degree distribution features and community structure features of the graph data, and calculate the upper limit value of the current spectral interval based on the degree distribution features and community structure features.

[0179] Chebyshev filter construction module 33 is used to construct a Chebyshev polynomial filter based on the upper limit of the current spectral interval; wherein, the Chebyshev polynomial filter is configured to retain the components of the input vector that correspond to the eigenvalues ​​within the target eigenvalue interval of the normalized Laplacian matrix, and compress the components of the input vector that correspond to the eigenvalues ​​outside the target eigenvalue interval when filtering the input vector.

[0180] The starting block generation module 34 is used to generate at least one initial vector, and to filter the initial vector using a Chebyshev polynomial filter to obtain the starting block.

[0181] The block Ranchos iteration and convergence determination module 35 is used to perform block Ranchos iteration with the starting block as the initial block to generate a block tridiagonal matrix; perform eigenvalue solving on the block tridiagonal matrix to obtain a set of Ritz values ​​and a set of corresponding approximate eigenvectors; and determine the convergence state of the approximate eigenvectors based on the gap between the Ritz values ​​and the residual between the approximate eigenvectors.

[0182] The adaptive filtering restart module 36 is used to adaptively adjust the upper limit of the current spectral interval based on the distribution of Ritz values ​​when the convergence state is stagnant, to obtain the updated upper limit of the spectral interval, and to construct an updated Chebyshev polynomial filter based on the updated upper limit of the spectral interval; to filter the approximate eigenvectors using the updated Chebyshev polynomial filter to obtain a new starting block; and to use the new starting block as the starting block and return to the operation of the Lanchos iteration and convergence determination module.

[0183] The decomposition result output module 37 is used to output the final eigenvalues ​​and corresponding eigenvectors when the convergence state is complete.

[0184] Embodiments of this application also provide a computer device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps in the aforementioned method embodiments.

[0185] Embodiments of this application also provide a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps in the above-described method embodiments.

[0186] For the device embodiments, since they basically correspond to the method embodiments, the relevant parts can be referred to in the description of the method embodiments. The device embodiments described above are merely illustrative. The components described as separate parts may or may not be physically separate, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this disclosure according to actual needs. Those skilled in the art can understand and implement this without creative effort.

[0187] The above-described embodiments are merely illustrative of several implementation methods of the embodiments of this application, and their descriptions are relatively specific and detailed. However, they should not be construed as limiting the scope of the patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the embodiments of this application, and these modifications and improvements all fall within the protection scope of the embodiments of this application.

Claims

1. A method for decomposing a large-scale map matrix, characterized in that, The method includes: S1. Obtain the graph data to be decomposed, and construct a normalized Laplace matrix based on the adjacency matrix and degree matrix of the graph data. S2. Extract the degree distribution features and community structure features of the graph data, and calculate the upper limit of the current spectral interval based on the degree distribution features and the community structure features; S3. Construct a Chebyshev polynomial filter based on the upper limit of the current spectral interval; wherein, the Chebyshev polynomial filter is configured to retain the components of the input vector that correspond to the eigenvalues ​​within the target eigenvalue interval of the normalized Laplacian matrix, and compress the components of the input vector that correspond to the eigenvalues ​​outside the target eigenvalue interval when filtering the input vector. S4. Generate at least one initial vector, and filter the initial vector using the Chebyshev polynomial filter to obtain the starting block; S5. Using the starting block as the initial block, perform block Ranchos iteration to generate a block tridiagonal matrix; perform eigenvalue solving on the block tridiagonal matrix to obtain a set of Ritz values ​​and a set of corresponding approximate eigenvectors; determine the convergence state of the approximate eigenvectors based on the gaps between the Ritz values ​​and the residuals between the approximate eigenvectors. S6. When the convergence state is stagnant, the current upper limit of the spectral interval is adaptively adjusted based on the distribution of the Ritz values ​​to obtain an updated upper limit of the spectral interval. An updated Chebyshev polynomial filter is constructed based on the updated upper limit of the spectral interval. The updated Chebyshev polynomial filter is used to filter the approximate feature vector to obtain a new starting block. The new starting block is used as the starting block, and the process returns to step S5. S7. When the convergence state is complete, output the final feature value and the corresponding feature vector.

2. The method according to claim 1, characterized in that, The step of extracting the degree distribution features and community structure features of the graph data, and calculating the upper limit of the current spectral interval based on the degree distribution features and the community structure features, includes: S11. Perform community detection on the graph data to obtain multiple communities, calculate the conductivity value of each community based on the degree matrix and the adjacency matrix, and obtain a set of conductivity values; S12. Arrange the set of conductivity values ​​in ascending order, according to the preset number of target features. Select the first The conductance value; using the upper bound relationship between eigenvalues ​​and conductance in the multi-path Cheeger inequality, the upper bound of the eigenvalue is determined as the first... Twice the conductivity value; S13. Calculate the power-law exponent of the degree distribution of the graph data; determine the safety expansion factor according to the preset mapping relationship based on the power-law exponent of the degree distribution; and calculate the upper limit of the current spectral interval by combining the upper bound of the eigenvalue and the safety expansion factor; wherein, the formula for calculating the upper limit of the current spectral interval is: in, This represents the upper limit of the current spectral interval. After sorting in ascending order, the first Each conductivity value, The number of the preset target features; For the security extension factor and The mapping relationship is configured such that the value of the security extension factor increases as the power-law exponent of the degree distribution decreases.

3. The method according to claim 1, characterized in that, The step of constructing a Chebyshev polynomial filter based on the upper limit of the current spectral interval includes: S21. Based on the current upper limit of the spectral interval, establish a linear mapping that maps the eigenvalues ​​of the normalized Laplace matrix to the Chebyshev polynomial working interval, obtaining the mapping function; wherein, the expression of the linear mapping is: in, Let be the eigenvalue variables of the normalized Laplace matrix. The upper limit of the current spectral interval is... Eigenvalues The corresponding value in the working interval after the linear mapping. The corresponding value of the zero eigenvalue in the working interval; S22. Based on the mapping function and the preset polynomial order, construct a filter polynomial composed of Chebyshev polynomials of the first kind; wherein, the numerator of the filter polynomial is the value of the mapping function at the eigenvalue variable. The first-order Chebyshev polynomial of order, with the denominator being the value of the mapping function at the zero eigenvalue. A Chebyshev polynomial of the first kind; wherein, the expression of the filtering polynomial is: in, The order of the polynomial is preset. for Chebyshev polynomials of the first kind of order; S23. Using the three-term recursive relation of the first-kind Chebyshev polynomial, construct a recursive calculation rule for performing filtering on the input target vector from the filtering polynomial, and obtain the Chebyshev polynomial filter including the filtering polynomial and the recursive calculation rule.

4. The method according to claim 1, characterized in that, Determining the convergence state of the approximate eigenvector based on the gap between the Ritz values ​​and the residual between the approximate eigenvectors includes: S31. Sort the Ritz values ​​in ascending order, and calculate the first value after the ascending order. The liz value and the first The difference between the ritz values ​​is used to obtain the target gap; where... The preset number of target features; S32, when When greater than 1, based on the first [previous] after ascending sort. For each ritz value, calculate the average value of the gap between adjacent ritz values ​​to obtain the reference gap; when When the value is equal to 1, the target gap is used as the reference gap; S33. Based on the approximate eigenvectors and the Ritz values, calculate the residual norm of each approximate eigenvector relative to the normalized Laplace matrix; S34. When the largest residual norm is less than or equal to a preset convergence tolerance, the convergence state is determined to be convergence complete; when the largest residual norm is greater than the convergence tolerance, if If the target gap is greater than 1 and less than the product of the reference gap and a preset coefficient greater than 1, then the convergence state is determined to be convergence stagnation; when the largest residual norm is greater than the convergence tolerance, if If the value is equal to 1, then the convergence state is determined to be a convergence stagnation.

5. The method according to claim 1, characterized in that, The adaptive adjustment of the current upper limit of the spectral interval based on the distribution of the Ritz value to obtain the updated upper limit of the spectral interval, and the construction of the updated Chebyshev polynomial filter based on the updated upper limit of the spectral interval, includes: S41. Sort the Ritz values ​​in ascending order, when the preset number of target features... When greater than 1, based on the first [previous] after ascending sort. The average gap between adjacent Ritz values ​​is calculated, and this average is multiplied by a preset threshold coefficient to obtain an adaptive gap threshold; when the preset number of target features... When the value is equal to 1, the gap between the first and second Ritz values ​​is multiplied by a preset threshold coefficient to obtain the adaptive gap threshold. S42, from the first in ascending order Starting with a Ritz value, the Ritz value sequence is scanned backward, and the gaps between adjacent Ritz values ​​are calculated to obtain the gap sequence; S43. Based on the gap sequence, determine whether there is a gap greater than the adaptive gap threshold; S44. When a gap satisfying the condition exists, select the first adjacent Ritz value pair corresponding to the gap satisfying the condition, and set the upper limit of the updated spectral interval as the midpoint of the adjacent Ritz value pair; update the Chebyshev polynomial filter according to the upper limit of the updated spectral interval to obtain the updated Chebyshev polynomial filter; wherein, the condition is that there is a gap in the gap sequence that is greater than the adaptive gap threshold; when there is an adjacent Ritz value satisfying the condition... and When the updated upper limit of the spectral interval is calculated, the formula is as follows: in, This represents the upper limit of the updated spectral interval. and To meet The adjacent Ritz values, The index of the smaller of the adjacent Ritz values ​​and ; The adaptive gap threshold; S45. When there is no gap that satisfies the condition, the updated upper limit of the spectral interval is maintained at the current upper limit of the spectral interval, and the order of the Chebyshev polynomial filter is increased to obtain the updated filter order; the Chebyshev polynomial filter is updated according to the updated upper limit of the spectral interval and the updated filter order to obtain the updated Chebyshev polynomial filter.

6. A large-scale graph matrix decomposition system for implementing the method according to any one of claims 1 to 5, characterized in that, The system includes: The Laplacian matrix construction module is used to obtain the graph data to be decomposed and construct a normalized Laplacian matrix based on the adjacency matrix and degree matrix of the graph data. The spectral interval upper limit calculation module is used to extract the degree distribution features and community structure features of the graph data, and calculate the current spectral interval upper limit value based on the degree distribution features and community structure features; A Chebyshev filter construction module is used to construct a Chebyshev polynomial filter based on the current upper limit of the spectral interval; wherein, the Chebyshev polynomial filter is configured to retain the components of the input vector that correspond to the eigenvalues ​​within the target eigenvalue interval of the normalized Laplacian matrix, and compress the components of the input vector that correspond to the eigenvalues ​​outside the target eigenvalue interval when filtering the input vector. A start block generation module is used to generate at least one initial vector, and to filter the initial vector using the Chebyshev polynomial filter to obtain a start block; The block Rankius iteration and convergence determination module is used to perform block Rankius iteration with the starting block as the initial block to generate a block tridiagonal matrix; perform eigenvalue solving on the block tridiagonal matrix to obtain a set of Ritz values ​​and a set of corresponding approximate eigenvectors; and determine the convergence state of the approximate eigenvectors based on the gap between the Ritz values ​​and the residual between the approximate eigenvectors. The adaptive filtering restart module is used to adaptively adjust the current upper limit of the spectral interval based on the distribution of the Ritz values ​​when the convergence state is stagnant, to obtain an updated upper limit of the spectral interval, and to construct an updated Chebyshev polynomial filter based on the updated upper limit of the spectral interval; to filter the approximate feature vector using the updated Chebyshev polynomial filter to obtain a new starting block; and to use the new starting block as the starting block and return to execute the operation of the block Lanchos iteration and convergence determination module. The decomposition result output module is used to output the final feature value and the corresponding feature vector when the convergence state is convergence complete.

7. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the method of any one of claims 1 to 5.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method of any one of claims 1 to 5.

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