Network data graph topology joint inference method and system using stationary signals and non-stationary signals

Abstract

The present invention provides a method and system for jointly inferring the topology of network data graphs using stationary signals and non-stationary signals. The method includes mapping the observed signals into stationary signals and non-stationary signals; estimating the stationary signals and non-stationary signals to obtain their respective covariance matrices and obtaining an optimization problem; identifying the graph filters of the non-stationary signals through an iterative algorithm and performing eigen-decomposition to obtain the eigen-bases; relaxing the optimization problem into a convex optimization problem and solving to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator; adjusting parameters to control the sparsity, smoothness, and similarity of the learned graph structure to obtain the final graph structure. When the underlying graph structure has both stationary signals and non-stationary signals and these graphs are closely related, the present invention can jointly infer the topologies of these graphs, improve the quality of the learned graphs, and obtain more accurate results.

Description

Technical Field

[0001] The present invention relates to a method and system for jointly inferring the topology of a network data graph using stationary signals and non-stationary signals, and belongs to the field of graph signal processing. Background Art

[0002] Modern data analysis and processing tasks often involve large structured data sets, where the structure carries key information about the nature of the data. One can find many examples of such data sets in a wide variety of application domains, such as transportation networks, social networks, computer networks, and brain networks. Graph Signal Processing (GSP) is an emerging research field where graphs are used as mathematical tools to describe the structure of such data, providing a flexible way to represent the relationships between data entities. Among them, graph topology inference is an important problem because: firstly, the graph can capture the actual geometric structure of the structured data, which is essential for effective processing, analysis, and visualization; secondly, learning the relationships between data entities is beneficial to many application domains, such as understanding the functional connections between brain regions or the behavioral influence among a group of people; and thirdly, the inferred graph topology can help predict future data evolution.

[0003] In recent years, people have successfully used graphs to capture the irregular (non-Euclidean) structures that are typically inherent in contemporary data. In more and more cases, such as in statistics, machine learning, or signal processing, graphs are relied upon to capture the underlying irregular domains to solve a series of applications, such as communication, genetics, and brain networks. GSP has been able to generalize many of the initially envisioned tools for processing signals with regular time or space to signals on irregular domains represented by graphs, thus providing new insights and effective algorithms. Although graph-related methods are becoming increasingly popular, in many cases, the graph is unknown, and it is possible to learn the graph and infer its topology from a set of observed nodes under the basic assumption that there is a relationship between the attributes of the observed signals and the sought graph topology. This constitutes a prominent problem, commonly referred to as graph topology inference, also known as graph learning. Notable methods include correlation networks, partial correlation and (Gaussian) Markov random fields, sparse structural equation models, methods based on graph signal processing, and their non-linear generalizations.

[0004] However, current methods are usually limited to inferring a single network and assume that the underlying graph structure is either a stationary signal or a non-stationary signal. Based on this, a joint graph topology inference for stationary and non-stationary signals is proposed. Stationarity is a fundamental property that facilitates the analysis and processing of time-domain random signals. Although time-varying signals are inherently rich, in many practical scenarios, the information of interest exists in a more irregular graph domain. This lack of regularity hinders the generalization of the classical concept of stationarity to graph signals. Stationary signals can model stationary graph processes as the output of a linear graph filter applied to white noise input. However, many signals do not exist on such a regular structure, that is, non-stationary signals. For example, instead of having more than one sensor return a time signal, there are multiple sensors in a two-dimensional space, and each sensor only provides one value. In this case, the signal support is no longer regular, which is a non-stationary signal.

[0005] Currently, for the signals being observed, some are stationary and some are non-stationary on the graphs being sought, and these graphs are closely related. In such a case, there is a lack of a method for joint graph topology inference to improve the quality of the learned graphs.

[0006] The above problems should be considered and solved in the process of using joint graph topology inference for stationary and non-stationary signals. Summary of the Invention

[0007] The object of the present invention is to provide a method and system for joint graph topology inference of network data using stationary and non-stationary signals to solve the problem in the prior art that it is usually limited to inferring a single network and it is difficult to achieve joint graph topology inference when the underlying graph structure has both stationary and non-stationary signals.

[0008] The technical solution of the present invention is as follows:

[0009] A method for joint graph topology inference of network data using stationary and non-stationary signals, comprising the following steps:

[0010] S1. Map the observed signals into stationary signals and non-stationary signals, where the stationary signals are the output of a linear graph filter with white noise input, and the non-stationary signals are the output of a linear graph filter with non-white noise input;

[0011] S2. Estimate the stationary signals and non-stationary signals, respectively obtain their covariance matrices, and obtain an optimization problem;

[0012] S3. Identify the graph filter of the non-stationary signals through an iterative algorithm and perform eigen-decomposition to obtain eigen-bases;

[0013] S4. Use the spectral template to relax the optimization problem obtained in step S2 to a convex optimization problem, and solve to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph;

[0014] S5. Adjust the parameters to control the sparsity, smoothness, and similarity of the learned graph structure adjacency matrix, and obtain the final graph structure.

[0015] Further, in step S1, map the observation signal to a stationary signal and a non-stationary signal. Specifically,

[0016] The observation signal is two graphs defined on a common set of N nodes, each graph having P1 and P2 observation values respectively. The observation signal satisfies a linear diffusion process, that is,

[0017] y1 = Hx1, y2 = Hx2

[0018] where y1 is the output of the white noise x1 input to the linear graph filter, that is, the stationary signal, y2 is the output of the non-white noise x2 input to the linear graph filter, that is, the non-stationary signal, and the observation signal is random noise subject to a Gaussian distribution; H is the graph filter: where h l is the coefficient of the graph filter, S is the graph shift operator, and L is the filter order.

[0019] Further, in step S2, estimate the stationary signal and the non-stationary signal to obtain their respective covariance matrices. Specifically,

[0020] S21. According to the expression of the graph filter, the signal diffusion process is Since the graph filter H is a polynomial of the symmetric graph shift operator S, the graph filter H is also symmetric. Thus, the covariance matrix of the output signal is

[0021] C y := E[yy T = E[Hx(Hx) T = HE[xx T H

[0022] where E is to take the expectation of the matrix, x is the observation signal, y is the output signal, T is the transpose of the matrix, and h l is the coefficient of the graph filter;

[0023] S22. According to the definition formula of the graph shift operator, the graph filter can also be expressed as

[0024]

[0025] where h lare the coefficients of the graph filter, V := [v1,..., v N is the eigenvector matrix;

[0026] S23. Since the observed signal is white noise, i.e., where, is the covariance matrix of the white noise x1, and I N is the N - order identity matrix, so the covariance matrix of the stationary signal y1 is

[0027]

[0028] where, h l are the coefficients of the graph filter;

[0029] It can be seen from this that the shift operator S of the stationary signal, the graph filter H, and the covariance matrix of the stationary signal y1 have the same eigenvector matrix; for P observed values the estimated value of the covariance matrix of the non - stationary signal y2 is

[0030] S24. Stationarity means that the graph shift operator S1 of the stationary signal and the covariance matrix of the stationary signal can be simultaneously diagonalized. For diagonalizable matrices, this is equivalent to commutativity, that is, it is necessary to satisfy Since the exact value of the covariance matrix cannot be known in practice and only the estimated value can be obtained, the equality constraint is relaxed to a convex constraint on the difference, i.e., to adapt to the difference between the sample and the exact value. Based on this, the optimization problem proposed is:

[0031]

[0032]

[0033]

[0034] d(S1 - S2) ≤ ε3

[0035] where, S1 is the graph shift operator of the stationary signal, S2 is the graph shift operator of the non - stationary signal, Λ2 is the eigenvalue matrix of the non - stationary signal, is the estimated value of the output signal covariance matrix of the stationary signal, ε1, ε2, ε3 are constant parameters, is the estimated value of the eigenvector matrix of the non - stationary signal, T is the transpose of the matrix, and d is the distance function.

[0036] Furthermore, in step S3, the graph filter of the non - stationary signal is identified through an iterative algorithm, and the eigen - decomposition is performed to obtain the eigen - basis. Specifically,

[0037] S31. Stop after multiple iterations or when the error between two consecutive iterations is below δ. The iteration formula is:

[0038]

[0039]

[0040] where the gradient of the graph filter is is the iterative estimate of the graph filter, H k , H k+1 are graph filters, η is the iteration step size, α k is a constant, is the covariance matrix of the non-white noise x2, is the estimated value of the covariance matrix of the non-stationary signal y2;

[0041] S32. Use the gradient to obtain a convergent result, get the estimated value of the graph filter H, and perform eigenvalue decomposition to obtain the estimated value of the eigenvector matrix of the non-stationary graph signal as the eigenbasis.

[0042] Furthermore, in step S4, use the spectral template to relax the optimization problem obtained in step S2 and convert it into a convex optimization problem. Solve to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph. Specifically,

[0043] S41. The l1 norm is the optimal convex approximation of the l0 norm. The l1 norm also constrains sparsity and is easier to optimize and solve than the l0 norm. Therefore, relax the l0 norm to the l1 norm. At this time, graph topology inference is equivalent to learning its eigenvalues under the constraint that the shift operator is sparse. The optimization problem obtained in step S2 is converted to:

[0044]

[0045]

[0046]

[0047] d(S1 - S2) ≤ ε3

[0048] where S1 is the graph shift operator of the stationary signal, S2 is the graph shift operator of the non-stationary signal, Λ2 is the eigenvalue matrix of the non-stationary signal, is the estimated value of the covariance matrix of the stationary signal y1, ε1, ε2, ε3 are constant parameters, is the estimated value of the eigenvector matrix of the non-stationary signal, T is the transpose of the matrix, and d is the distance function;

[0049] S42. Since the distance function d(·,·) can choose the l0 norm or the l1 norm to make the two graphs have the same sparse pattern and weights, or consider the F norm, which will promote similar weights regardless of the sparse pattern; considering that only the two graphs need to be made as similar as possible, the distance function selects the l1 norm, and at this time it is converted into a convex optimization problem:

[0050]

[0051]

[0052]

[0053] ||S1 - S2||1 ≤ ε3

[0054] where S1 is the graph shift operator of the stationary signal, S2 is the graph shift operator of the non-stationary signal, Λ2 is the eigenvalue matrix of the non-stationary signal, is the estimated value of the covariance matrix of the stationary signal y1, ε1, ε2, ε3 are constant parameters, is the estimated value of the eigenvector matrix of the non-stationary signal, T is the transpose of the matrix, and d is the distance function;

[0055] S43. Since the convex optimization problem and the constraint conditions obtained in step S42 are both convex, use the convex optimization toolbox cvx in matlab to directly solve, and obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph.

[0056] Furthermore, in step S5, adjust the parameters to control the sparsity, smoothness, and similarity of the learned graph structure, and obtain the final graph structure. Specifically,

[0057] S51. By adjusting the constant parameter ε1, control the sparsity and smoothness of the graph shift operator S1 of the stationary signal;

[0058] S52. By adjusting the constant parameter ε2, control the sparsity and smoothness of the graph shift operator S2 of the non-stationary signal;

[0059] S53. By adjusting the constant parameter ε3, control the similarity between the graph shift operator S1 of the stationary signal and the graph shift operator S2 of the non-stationary signal;

[0060] S54. Obtain a graph topology that is similar to the true graph structures of both the stationary signal and the non-stationary signal as the final graph structure.

[0061] A system using the network data graph topology joint inference method for stationary signals and non-stationary signals described in any one of the above, including a data mapping module, an optimization problem establishment module, an eigenvalue decomposition module, an optimization problem solving module, and a graph structure generation module,

[0062] Data mapping module: Maps the observed signal into a stationary signal and a non-stationary signal, where the stationary signal is the output of white noise input to a linear graph filter, and the non-stationary signal is the output of non-white noise input to a linear graph filter;

[0063] Optimization problem establishment module: Estimates the stationary signal and the non-stationary signal, respectively obtains their covariance matrices, and obtains an optimization problem;

[0064] Eigenvalue decomposition module: Identifies the graph filter of the non-stationary signal through an iterative algorithm and performs eigenvalue decomposition to obtain eigenbases;

[0065] Optimization problem solution module: Uses a spectral template to relax the optimization problem obtained in step S2 into a convex optimization problem, and solves to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph;

[0066] Graph structure generation module: Adjusts parameters to control the sparsity, smoothness, and similarity of the learned graph structure, and obtains the final graph structure.

[0067] The beneficial effects of the present invention are as follows: This network data graph topology joint inference method using stationary signals and non-stationary signals can perform graph topology inference on multiple graph signals, and is no longer limited to inferring a single network. Compared with existing joint inferences, the present invention is a joint inference of stationary signals and non-stationary signals, and is no longer limited to inferring only a certain type of graph signal. This network data graph topology joint inference method using stationary signals and non-stationary signals can perform joint inference on these graph topologies when the underlying graph structure has both stationary signals and non-stationary signals, and these graphs are closely related, can improve the quality of the learned graph, and obtain more accurate results. Brief Description of the Drawings

[0068] Figure 1 is a schematic flowchart of the network data graph topology joint inference method using stationary signals and non-stationary signals according to an embodiment of the present invention;

[0069] Figure 2 is a schematic illustration of the network data graph topology joint inference system using stationary signals and non-stationary signals according to an embodiment. Detailed Description of the Specific Embodiment

[0070] The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0071] Embodiment

[0072] A network data graph topology joint inference method using stationary signals and non-stationary signals, as Figure 1 , includes the following steps,

[0073] S1. Map the observed signal into a stationary signal and a non-stationary signal, where the stationary signal is the output of white noise input to a linear graph filter, and the non-stationary signal is the output of non-white noise input to a linear graph filter.

[0074] The observed signal is two graphs defined on a common set of N nodes, each graph having P1 and P2 observations respectively. The observed signal satisfies a linear diffusion process, that is

[0075] y1 = Hx1, y2 = Hx2

[0076] where y1 is the output of white noise x1 input to a linear graph filter, i.e., the stationary signal, y2 is the output of non-white noise x2 input to a linear graph filter, i.e., the non-stationary signal, and the observed signal is random noise following a Gaussian distribution; H is the graph filter: where h l is the coefficient of the graph filter, S is the graph shift operator, and L is the filter order.

[0077] In the context of graph learning, take the adjacency matrix A as the graph shift operator S. Assume S is symmetric and define S = VΛV T , where the orthogonal matrix V := [v1,…,v N is the eigenvector matrix of S, and the diagonal matrix Λ := diag(λ1,…,λ N ) is the eigenvalue matrix of S.

[0078] Select the adjacency matrix of the undirected graph to constrain the GSO. The feasible set S A specifies additional properties that S must satisfy:

[0079]

[0080] The first condition constrains the non-negativity of the GSO weights, the second condition indicates that the graph is undirected, the third condition is that since the graph has no self-loops, each diagonal element of S must be zero, and the last condition determines the scale of the admissible graph by setting the weight of the first node to 1 and excludes the solution of S = 0.

[0081] S2. Estimate the stationary signal and the non-stationary signal, obtain their respective covariance matrices, and obtain an optimization problem.

[0082] S21. According to the expression of the graph filter, the signal diffusion process is Since the graph filter H is a polynomial of the symmetric graph shift operator S, the graph filter H is also symmetric. Thus, the covariance matrix of the output signal is

[0083] C y := E[yyT = E[Hx(Hx) T = HE[xx T H

[0084] Among them, E is the expectation of the matrix, x is the observed signal, y is the output signal, T is the transpose of the matrix, and h l is the coefficient of the graph filter;

[0085] S22. According to the definition formula of the graph shift operator, the graph filter can also be expressed as

[0086]

[0087] Among them, h l is the coefficient of the graph filter, V := [v1,..., v N is the eigenvector matrix;

[0088] S23. Since the observed signal is white noise, that is Among them, is the covariance matrix of the white noise x1, and I N is the N-order identity matrix. Therefore, the covariance matrix of the stationary signal y1 is

[0089]

[0090] Among them, h l is the coefficient of the graph filter;

[0091] It can be seen from this that the shift operator S of the stationary signal, the graph filter H, and the covariance matrix of the stationary signal y1 have the same eigenvector matrix; for P observed values The estimated value of the covariance matrix of the non-stationary signal y2 is

[0092] S24. Stationarity means that the graph shift operator S1 of the stationary signal and the covariance matrix of the stationary signal can be simultaneously diagonalized. For a diagonalizable matrix, this is equivalent to commutativity, that is, it is necessary to satisfy Since the exact value of the covariance matrix cannot be known in practice and only the estimated value can be obtained, the equality constraint is relaxed to the convex constraint of the difference, that is to adapt to the difference between the sample and the exact value. Based on this, the optimization problem proposed is:

[0093]

[0094]

[0095]

[0096] d(S1 - S2) ≤ ε3

[0097] Wherein, S1 is the graph shift operator of the stationary signal, S2 is the graph shift operator of the non-stationary signal, Λ2 is the eigenvalue matrix of the non-stationary signal, is the estimated value of the output signal covariance matrix of the stationary signal, ε1, ε2, and ε3 are constant parameters, is the estimated value of the eigenvector matrix of the non-stationary signal, T is the transpose of the matrix, and d is the distance function.

[0098] S3. Identify the graph filter of the non-stationary signal through an iterative algorithm, as shown in Table 1, and perform eigen-decomposition to obtain the eigenbasis;

[0099] S31. Stop after multiple iterations or when the error of two consecutive iterations is below δ. The iterative formula is:

[0100]

[0101]

[0102] Wherein, the gradient of the graph filter is is the iterative estimated value of the graph filter, H k 、H k+1 are graph filters, η is the iterative step size, α k is a constant, is the covariance matrix of the non-white noise x2, is the estimated value of the covariance matrix of the non-stationary signal y2;

[0103] S32. Use the gradient to obtain the convergent result, obtain the estimated value of the graph filter H, and perform eigen-decomposition to obtain the estimated value of the eigenvector matrix of the non-stationary graph signal as the eigenbasis.

[0104] Table 1 Identification of the graph filter of the non-stationary signal using the iterative algorithm

[0105]

[0106] S4. Use the spectral template to relax the optimization problem obtained in step S2 and convert it into a convex optimization problem, and solve to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph;

[0107] S41. The l1 norm is the optimal convex approximation of the l0 norm. The l1 norm also constrains sparsity and is easier to optimize and solve than the l0 norm. Therefore, the l0 norm is relaxed to the l1 norm. At this time, the graph topology inference is equivalent to learning its eigenvalues under the constraint that the shift operator obeys sparsity. The optimization problem obtained in step S2 is converted to:

[0108]

[0109]

[0110]

[0111] d(S1 - S2) ≤ ε3

[0112] where S1 is the graph shift operator of the stationary signal, S2 is the graph shift operator of the non - stationary signal, Λ2 is the eigenvalue matrix of the non - stationary signal, is the estimated value of the covariance matrix of the stationary signal y1, ε1, ε2, and ε3 are constant parameters, is the estimated value of the eigenvector matrix of the non - stationary signal, T is the transpose of the matrix, and d is the distance function;

[0113] S42. Since the distance function d(·, ·) can choose the l0 norm or the l1 norm to make the two graphs have the same sparse pattern and weights, or consider the F norm, which will promote similar weights regardless of the sparse pattern; considering that only the two graphs need to be made as similar as possible, the distance function selects the l1 norm, and at this time it is transformed into a convex optimization problem:

[0114]

[0115]

[0116]

[0117] ||S1 - S2||1 ≤ ε3

[0118] where S1 is the graph shift operator of the stationary signal, S2 is the graph shift operator of the non - stationary signal, Λ2 is the eigenvalue matrix of the non - stationary signal, is the estimated value of the covariance matrix of the stationary signal y1, ε1, ε2, and ε3 are constant parameters, is the estimated value of the eigenvector matrix of the non - stationary signal, T is the transpose of the matrix, and d is the distance function;

[0119] S43. Since the convex optimization problem and the constraint conditions obtained in step S42 are both convex, use the convex optimization toolbox cvx in matlab to directly solve, and obtain the graph shift operator of the stationary graph, the eigenvalues of the non - stationary graph, and the graph shift operator of the non - stationary graph.

[0120] S5. Adjust the parameters to control the sparsity, smoothness, and similarity of the learned graph structure, and obtain the final graph structure.

[0121] S51. By adjusting the constant parameter ε1, control the sparsity and smoothness of the graph shift operator S1 of the stationary signal;

[0122] S52. By adjusting the constant parameter ε2, control the sparsity and smoothness of the graph shift operator S2 of the non-stationary signal;

[0123] S53. By adjusting the constant parameter ε3, control the similarity between the graph shift operator S1 of the stationary signal and the graph shift operator S2 of the non-stationary signal;

[0124] S54. Obtain a graph topology that is similar to the true graph structures of both the stationary signal and the non-stationary signal as the final graph structure.

[0125] Such as Figure 2 , the embodiment also provides a system using the network data graph topology joint inference method for stationary signals and non-stationary signals described in any one of the above, including a data mapping module, an optimization problem establishment module, an eigen-decomposition module, an optimization problem solving module, and a graph structure generation module.

[0126] Data mapping module: Map the data in the observed nodes into stationary signals and non-stationary signals. Among them, the stationary signal is the output of white noise input to a linear graph filter, and the non-stationary signal is the output of non-white noise input to a linear graph filter;

[0127] Optimization problem establishment module: Estimate the stationary signal and the non-stationary signal, respectively obtain their covariance matrices, and obtain the optimization problem;

[0128] Eigen-decomposition module: Identify the graph filter of the non-stationary signal through an iterative algorithm and perform eigen-decomposition to obtain the eigen-basis;

[0129] Optimization problem solving module: Use the spectral template to relax the optimization problem obtained in step S2 into a convex optimization problem, and solve to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph;

[0130] Graph structure generation module: Adjust the parameters to control the sparsity, smoothness, and similarity of the learned graph structure, and obtain the final graph structure.

[0131] The method and system for jointly inferring the topology of network data graphs using stationary and non-stationary signals can perform graph topology inference on multiple graph signals compared to inferring graph topology alone, and is no longer limited to inferring a single network. Compared with existing joint inferences, the present invention is a joint inference of stationary and non-stationary signals and is no longer limited to inferring only a certain type of graph signal. The method for jointly inferring the topology of network data graphs using stationary and non-stationary signals can solve the problem of jointly inferring the topologies of these graphs when the underlying graph structure has both stationary and non-stationary signals and these graphs are closely related, which can improve the quality of the learned graphs and obtain more accurate results.

[0132] A specific example of the method for jointly inferring the topology of network data graphs using stationary and non-stationary signals in the embodiment is as follows:

[0133] The method for jointly inferring the topology of network data graphs using stationary and non-stationary signals is applied to the joint inference of the data graph topology of the brain functional network. In neuroscience, the brain functional network of a patient can be observed. When the patient performs a certain activity, different functional parts of the brain cooperate with each other, and some of the corresponding neural networks exhibit stationary signals while some exhibit non-stationary signals. The specific steps are as follows.

[0134] S1. Simultaneously monitor different parts of the brain to obtain electroencephalogram signals as observation signals. The observation signals pass through a linear graph filter, and the outputs are stationary and non-stationary signals.

[0135] S2. Estimate the stationary and non-stationary signals, respectively obtain their covariance matrices, and obtain an optimization problem.

[0136] S3. Identify the graph filter of the non-stationary signal through an iterative algorithm and perform eigen-decomposition to obtain eigen-bases.

[0137] S4. Use a spectral template to relax the previous optimization problem into a convex optimization problem, and solve to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph.

[0138] S5. Adjust the parameters to control the sparsity, smoothness, and similarity of the learned graph structure, and obtain the final graph structure. That is to say, through this model, higher-quality neural network structure diagrams of different functional parts of the brain can be obtained simultaneously.

[0139] The above embodiments are only used to illustrate the technical idea of the present invention and cannot be used to limit the protection scope of the present invention. Any modification made to the technical solution according to the technical idea proposed by the present invention falls within the protection scope of the present invention.

Claims

1. A method for jointly inferring the topology of network data graphs using stationary signals and non-stationary signals, characterized in that: It includes the following steps: S1. Map the observed signal into a stationary signal and a non-stationary signal, where the stationary signal is the output of white noise input into a linear graph filter, and the non-stationary signal is the output of non-white noise input into a linear graph filter; S2. Estimate the stationary signal and the non-stationary signal to obtain their respective covariance matrices, and obtain an optimization problem: Stationarity means that the graph shift operator S1 of a stationary signal and the covariance matrix of the stationary signal y1 can be simultaneously diagonalized. For diagonalizable matrices, this is equivalent to commutativity, that is, it is necessary to satisfy Since the exact value of the covariance matrix cannot be known in practice and only an estimated value can be obtained, the equality constraint is relaxed to a convex constraint on the difference, that is to accommodate the difference between the sample and the exact value. Based on this, the optimization problem proposed is: Among them, S1 is the graph shift operator of the stationary signal, S2 is the graph shift operator of the non-stationary signal, and Λ2 is the eigenvalue matrix of the non-stationary signal. is the estimated value of the output signal covariance matrix of the stationary signal, and ε1, ε2, and ε3 are constant parameters. is the estimated value of the eigenvector matrix of the non-stationary signal, T is the transpose of the matrix, and d is the distance function. S3. Identify the graph filter of the non-stationary signal through an iterative algorithm and perform eigen-decomposition to obtain eigen-bases; S4. Use a spectral template to relax the optimization problem obtained in step S2 into a convex optimization problem, and solve to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph; S5. Adjust parameters to control the sparsity, smoothness, and similarity of the learned graph structure, and obtain the final graph structure.

2. The network data graph topology joint inference method using stationary signals and non-stationary signals according to claim 1, wherein: In step S1, mapping the observed signal into a stationary signal and a non-stationary signal, specifically, The observed signal is two graphs defined on a common set of N nodes, and each graph has P1 and P2 observed values respectively. The observed signal satisfies a linear diffusion process, that is y1 = Hx1, y2 = Hx2 Among them, y1 is the output of the white noise x1 as the observation signal input to the linear graph filter, that is, the stationary signal, y2 is the output of the non-white noise x2 as the observation signal input to the linear graph filter, that is, the non-stationary signal, and the observation signal is a random noise obeying the Gaussian distribution; H is the graph filter: Among them, h l is the coefficient of the graph filter, S is the graph shift operator, and L is the filter order.

3. The network data graph topology joint inference method using stationary signals and non-stationary signals according to claim 1, characterized in that: In step S2, estimating the stationary signal and the non-stationary signal to obtain their respective covariance matrices, specifically, S21. According to the expression of the graph filter, the signal diffusion process is Since the graph filter H is a polynomial of the symmetric graph shift operator S, the graph filter H is also symmetric. Thus, the covariance matrix of the output signal is C y := E[yy T = E[Hx(Hx) T = HE[xx T H where E is the expectation of the matrix, x is the observed signal, y is the output signal, T is the transpose of the matrix, and h l is the coefficient of the graph filter; S22. According to the definition formula of the graph shift operator, the graph filter can also be expressed as where h l is the coefficient of the graph filter, V := [v1,..., v N is the eigenvector matrix, and the diagonal matrix Λ is the eigenvalue matrix of the graph shift operator S; S23. Since the observed signal is white noise, that is where is the covariance matrix of white noise x1, and I N is the N-order identity matrix, so the covariance matrix of the stationary signal y1 is where h l is the coefficient of the graph filter; It can be seen that the covariance matrices of the shift operator S of the stationary signal, the graph filter H, and the stationary signal y1 have the same eigenvector matrix; for P observations The estimated value of the covariance matrix of the non-stationary signal y2 is 4. The method for jointly inferring the topology of network data graphs using stationary signals and non-stationary signals as described in claim 1, wherein: In step S3, identifying the graph filter of the non-stationary signal through an iterative algorithm and performing eigen-decomposition to obtain eigen-bases, specifically, S31. Stop after multiple iterations or when the error between two consecutive iterations is below δ. The iterative formula is: Among them, the gradient of the graph filter is is the iterative estimated value of the graph filter, H k 、H k+1 are graph filters, η is the iteration step size, α k is a constant, is the covariance matrix of the non-white noise x2, is the estimated value of the covariance matrix of the non-stationary signal y2; S32. Obtain a convergent result using the gradient, obtain an estimated value of the graph filter H, and perform eigen decomposition to obtain an estimated value of the non-stationary graph signal eigenvector matrix as the eigenbasis.

5. The network data graph topology joint inference method using stationary signals and non-stationary signals according to any one of claims 1-4, characterized in that: In step S4, using a spectral template to relax the optimization problem obtained in step S2 into a convex optimization problem, and solve to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph, specifically, S41. The l1 norm is the optimal convex approximation of the l0 norm. The l1 norm also constrains sparsity and is easier to optimize and solve than the l0 norm. Therefore, relax the l0 norm to the l1 norm. At this time, graph topology inference is equivalent to learning its eigenvalues under the constraint that the shift operator obeys sparsity. The optimization problem obtained in step S2 is transformed into: Among them, S1 is the graph shift operator of the stationary signal, S2 is the graph shift operator of the non-stationary signal, and Λ2 is the eigenvalue matrix of the non-stationary signal. is the estimated value of the covariance matrix of the stationary signal y1, and ε1, ε2, and ε3 are constant parameters. is the estimated value of the eigenvector matrix of the non-stationary signal, T is the transpose of the matrix, and d is the distance function. S42. Since the distance function d(·,·) can choose the l0 norm or the l1 norm to make the two graphs have the same sparse pattern and weights, or consider the F norm. Regardless of the sparse pattern, it will promote similar weights; considering that only the two graphs need to be as similar as possible, the distance function chooses the l1 norm. At this time, it is transformed into a convex optimization problem: Among them, S1 is the graph shift operator of the stationary signal, S2 is the graph shift operator of the non-stationary signal, and Λ2 is the eigenvalue matrix of the non-stationary signal. is the estimated value of the covariance matrix of the stationary signal y1, and ε1, ε2, and ε3 are constant parameters. is the estimated value of the eigenvector matrix of the non-stationary signal, T is the transpose of the matrix, and d is the distance function. S43. Since the convex optimization problem and the constraint conditions obtained in step S42 are both convex, directly solve using the convex optimization toolbox cvx in matlab to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph.

6. The network data graph topology joint inference method using stationary signals and non-stationary signals as described in claim 5, characterized in that: In step S5, adjust parameters to control the sparsity, smoothness, and similarity of the learned graph structure, and obtain the final graph structure, specifically, S51. Control the sparsity and smoothness of the graph shift operator S1 of the stationary signal by adjusting the constant parameter ε1; S52. Control the sparsity and smoothness of the graph shift operator S2 of the non-stationary signal by adjusting the constant parameter ε2; S53. By adjusting the constant parameter ε3, control the similarity between the graph shift operator S1 of the stationary signal and the graph shift operator S2 of the non-stationary signal; S54. Obtain a graph topology that is similar to both the true graph structures of the stationary signal and the non-stationary signal as the final graph structure.

7. A system using the network data graph topology joint inference method of utilizing stationary signals and non-stationary signals according to any one of claims 1-6, characterized in that: It includes a data mapping module, an optimization problem establishment module, a feature decomposition module, an optimization problem solving module, and a graph structure generation module. Data mapping module: Map the observed signal into a stationary signal and a non-stationary signal. Among them, the stationary signal is the output of white noise input to a linear graph filter, and the non-stationary signal is the output of non-white noise input to a linear graph filter; Optimization problem establishment module: Estimate the stationary signal and the non-stationary signal, respectively obtain their covariance matrices, and obtain the optimization problem; Feature decomposition module: Identify the graph filter of the non-stationary signal through an iterative algorithm and perform eigen-decomposition to obtain the eigenbasis; Optimization problem solving module: Use the spectral template to relax the optimization problem obtained in step S2 into a convex optimization problem, and solve to obtain the graph shift operator of the stationary graph, the eigenvalues of the non-stationary graph, and the graph shift operator of the non-stationary graph; Graph structure generation module: Adjust the parameters to control the sparsity, smoothness, and similarity of the learned graph structure, and obtain the final graph structure.

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