A Distributed Batch Reconstruction Method and System for Time-Varying Graph Signals

Abstract

The present invention discloses a method for distributed batch reconstruction of time-varying graph signals, which includes the following steps: 1) Divide the time-varying graph signals into multiple signal segments in chronological order; 2) Construct the graphs at each moment within each signal segment into a Cartesian product graph, and use the Sobolev difference smoothing of the time-varying graph signals on this product graph to reduce the reconstruction of the time-varying graph signals to an optimization problem; 3) Decompose the Cartesian product graph into a series of subgraphs, calculate the inverse matrix of the corresponding local Hessian matrix at the central nodes of each subgraph, and obtain an approximate inverse matrix of the Hessian matrix corresponding to the above optimization problem through fusion and averaging. Based on the approximate inverse matrix, solve the above optimization problem distributively, thereby completing the reconstruction of the time-varying graph signals within this signal segment; 4) Complete the reconstruction of the time-varying graph signals for all signal segments in sequence. This distributed batch reconstruction method using Sobolev difference smoothing on the product graph is based on the approximate inverse matrix of the Hessian matrix, and this method has the characteristics of low reconstruction error and fast convergence.

Description

Technical Field

[0001] The present invention relates to the technical field of graph signal processing, and particularly to a method and system for distributed batch reconstruction of time-varying graph signals. Background Art

[0002] The reconstruction of large-scale network data faces many challenges, such as high data dimension, irregular data structure, huge data volume, etc. Since classical signal processing methods cannot effectively reconstruct irregular network data, reconstruction algorithms based on graph signal processing theory have attracted much attention in recent years. Aiming at the reconstruction problem of large-scale network data, the present invention is based on the graph signal processing theory and proposes a method for distributed batch reconstruction of time-varying graph signals. This method has low reconstruction error and fast convergence characteristics. Summary of the Invention

[0003] In view of this, the purpose of the present invention is to provide a method for distributed batch reconstruction of time-varying graph signals. Aiming at the deficiencies of existing methods for batch reconstruction of time-varying graph signals, a new method for distributed batch reconstruction of time-varying graph signals is adopted, which has the advantages of low reconstruction error and high convergence rate.

[0004] To achieve the above purpose, the present invention provides the following technical solutions:

[0005] The method for distributed batch reconstruction of time-varying graph signals provided by the present invention includes the following steps:

[0006] Divide the time-varying graph signal X to be reconstructed Reco into multiple signal segments in chronological order;

[0007] Construct a Cartesian product graph for the graphs at all moments within each signal segment, use the Sobolev difference smoothing design penalty term of the time-varying graph signal within this signal segment on this Cartesian product graph, and reduce the reconstruction of the time-varying graph signal within this signal segment to an optimization problem;

[0008] Decompose the Cartesian product graph into a series of subgraphs, calculate the inverse matrix of the corresponding local Hessian matrix for the central nodes of each subgraph, and obtain an approximate inverse matrix of the Hessian matrix corresponding to the above optimization problem through fusion and averaging. Based on the approximate inverse matrix, solve the above optimization problem distributively, thereby completing the reconstruction of the time-varying graph signal within this signal segment;

[0009] Complete the reconstruction of the time-varying graph signals within all signal segments in sequence.

[0010] Further, the Cartesian product graph is constructed according to the following steps:

[0011] Model the graph G representing the spatial correlation between nodes and the graph G representing the temporal correlation between each moment T as a Cartesian product graph through Cartesian product The Cartesian product graph is defined as:

[0012]

[0013] Among them, the node set of the product graph is the set of edges between all nodes \((m, n)\) and adjacent nodes \((m, n')\) n′∈B(n,1) , \((m + 1, n)\) m＜M , \((m - 1, n)\) m＞1 ;

[0014] V T represents the set of all nodes in graph G T ;

[0015] V represents the set of all nodes in graph G;

[0016] m represents the sequence number of the current moment;

[0017] n represents the node number in graph G;

[0018] M represents the number of all moments in the signal segment;

[0019] n' represents the first-order neighbor node of n;

[0020] B(n, 1) represents the set of first-order neighbor nodes of n;

[0021] The combinatorial Laplacian matrix of is expressed as:

[0022]

[0023] Among them, represents the Kronecker product;

[0024] L T represents the combinatorial Laplacian matrix of graph G T ;

[0025] I N represents the identity matrix of order N;

[0026] N represents the number of all nodes in graph G;

[0027] I M represents the identity matrix of order M;

[0028] L represents the combinatorial Laplacian matrix of graph G.

[0029] Furthermore, the time-varying graph signal in the product graph The Sobolev difference smoothing on Calculated according to the following formula:

[0030]

[0031] wherein,

[0032] represents the Laplacian matrix on the product graph; on the differential Laplacian matrix;

[0033] μ represents the perturbation parameter;

[0034] Q represents the differential operator matrix.

[0035] Furthermore, the batch reconstruction of the time-varying graph signals within each signal segment is reduced to an optimization problem in the following manner:

[0036]

[0037] In the formula, the undamaged node indication matrix

[0038] represents the observed values of the graph signals at each moment;

[0039] α represents the balance coefficient between the fitting term (the first term) and the regularization term (the second term);

[0040] wherein, the row vector b m is defined as:

[0041]

[0042] wherein, F m is the set of undamaged nodes at the m-th moment, and i is the node number in the graph G;

[0043] wherein, the signal values corresponding to the damaged nodes are set to 0.

[0044] Furthermore, the subgraph decomposition method of the Cartesian product graph is as follows:

[0045] Given a node (m, i) (m = 2, 3,..., M - 1, i ∈ V) on the product graph The subgraph centered on this node is defined as

[0046] wherein, is the node set of the subgraph ;

[0047] k represents the time sequence number in the graph G T ;

[0048] l represents the node number in the graph G;

[0049] ρ(i, l) represents the number of hops of the shortest path between node i and node l;

[0050] is all the connections in the set formed by the edges of the nodes in

[0051] Furthermore, the inverse matrix of the local Hessian matrix is calculated according to the following steps:

[0052] the Hessian matrix corresponding to the time-varying graph signal reconstruction optimization problem in the signal segment in each subgraph the local Hessian matrix of is expressed as:

[0053]

[0054] where (m, i) is the central node of the subgraph, is the indicator matrix of all nodes in the subgraph, and its definition is

[0055]

[0056] where p represents the row index of matrix χ (m,i) of;

[0057] q represents the column index of matrix χ (m,i) of;

[0058] subgraph the central node (m, i) of calculates the pseudo-inverse matrix of, and then fuses and averages the pseudo-inverse matrices of all subgraphs. The resulting matrix is used as the approximate inverse matrix of, and the iterative formula for distributed solution of the batch reconstruction optimization problem is:

[0059]

[0060] represents the approximate inverse matrix of;

[0061] t represents the number of iterations.

[0062] Furthermore, the iterative process of the iterative formula for distributed solution of the batch reconstruction optimization problem is as follows:

[0063] Input: matrix observed signal iteration stop criterion ε, maximum number of iterations K, If t = 0, then:

[0064] 1-7) Each sub-graph central node (m, i) calculates:

[0065] and sends the corresponding signal value to all nodes (k, l) in

[0066] h represents the element serial number of

[0067] 2-7) Each node in

[0068] averages the received signal values and then sends the average values to the central nodes of the sub-graphs where they are located respectively;

[0069]

[0070] If holds or t > K, the program ends; otherwise, let t = t + 1 and return to step 1-7);

[0071] Output: as the reconstructed signal of this paragraph.

[0072] Furthermore, the Q is expressed as follows:

[0073]

[0074] where the time-varying graph signal X Reco contains a number of time instants that is an integer multiple of M.

[0075] Furthermore, the approximate inverse matrix of is expressed as follows:

[0076]

[0077] where represents finding the pseudo-inverse of the matrix.

[0078] The time-varying graph signal distributed batch reconstruction system provided by the present invention includes a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the program, the above method is implemented.

[0079] The beneficial effects of the present invention are as follows:

[0080] The time-varying graph signal distributed batch reconstruction method provided by the present invention includes the following steps: 1) Divide the time-varying graph signal into multiple signal segments in chronological order; 2) Construct the graphs at all moments within each signal segment into a Cartesian product graph, and use the Sobolev differential smoothing of the time-varying graph signal within this segment on this product graph to design a penalty term, reducing the reconstruction of the time-varying graph signal within the segment to an optimization problem; 3) Decompose the Cartesian product graph into a series of subgraphs, calculate the inverse matrix of the corresponding local Hessian matrix at the central node of each subgraph, and obtain an approximate inverse matrix of the Hessian matrix corresponding to the above optimization problem through fusion and averaging. Based on the approximate inverse matrix, solve the above optimization problem distributively, thereby completing the reconstruction of the time-varying graph signal within this signal segment; 4) Complete the reconstruction of the time-varying graph signal within all signal segments in sequence. This distributed batch reconstruction method using Sobolev differential smoothing on the product graph is based on the approximate inverse matrix of the Hessian matrix, and this method features low reconstruction error and fast convergence.

[0081] The method provided by the present invention uses the Cartesian product graph model to characterize the data structure of time-varying graph signals, thereby making full use of the spatio-temporal correlation between nodes. At the same time, by introducing the Sobolev differential smoothing of the time-varying graph signal on the product graph, the condition number of the Hessian matrix of the cost function is improved. Based on the decomposition of the Cartesian product graph, an approximate inverse matrix of the Hessian matrix is calculated. Based on this approximate inverse matrix, a distributed algorithm for solving the batch reconstruction optimization problem is designed. This method has the advantages of low reconstruction error and high convergence rate.

[0082] Other advantages, objectives, and features of the present invention will be described to some extent in the subsequent specification, and to some extent, will be obvious to those skilled in the art based on the study of the following text, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be achieved and obtained through the following specification. Brief Description of the Drawings

[0083] To make the objectives, technical solutions, and beneficial effects of the present invention clearer, the present invention provides the following drawings for illustration:

[0084] Figure 1 It is a flowchart of the time-varying graph signal distributed batch reconstruction method.

[0085] Figure 2 It is a schematic diagram of the subgraph of the Cartesian product graph in the embodiment.

[0086] Figure 3 It is a schematic diagram of the data structure of the daily average PM2.5 concentration in a certain area in the embodiment. Detailed Description of the Embodiment

[0087] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and be able to implement it, but the embodiments cited do not limit the present invention.

[0088] Embodiment 1

[0089] Figure 1 It is a flowchart of a distributed batch reconstruction method for time-varying graph signals. The distributed batch reconstruction method for time-varying graph signals provided in this embodiment includes the following steps:

[0090] 1) Divide the time-varying graph signal into multiple signal segments in chronological order: Divide the time-varying graph signal X Reco in the time dimension in sequence. Each signal segment contains M moments. The time-varying graph signals within each signal segment are represented by X = [x1, x2,..., x M , and it is denoted as where, x M represents the graph signal at the Mth moment;

[0091] 2) Construct a Cartesian product graph for the graphs at all moments within each signal segment, and use the Sobolev difference smoothing design penalty term of the time-varying graph signal on this product graph to reduce the reconstruction of the time-varying graph signal within the segment to an optimization problem:

[0092] Model the graph G representing the spatial correlation between nodes and the graph G T representing the temporal correlation between moments as a Cartesian product graph through Cartesian product The Cartesian product graph is defined as:

[0093]

[0094] where, the product graph node set is the set of all edges between nodes (m, n) and adjacent nodes (m, n′) n′∈B(n,1) , (m + 1, n) m＜M , (m - 1, n) m＞1 ,

[0095] V T represents the set of all nodes in graph G T ;

[0096] V represents the set of all nodes in graph G;

[0097] m represents the current moment sequence number;

[0098] n represents the node sequence number in graph G;

[0099] M represents the number of all moments within the signal segment;

[0100] $n'$ represents the first-order neighbor nodes of $n$;

[0101] $B(n, 1)$ represents the set of the first-order neighbor nodes of $n$;

[0102] The combinatorial Laplacian matrix is expressed as:

[0103]

[0104] where represents the Kronecker product,

[0105] $L$ T represents the combinatorial Laplacian matrix of graph $G$ T ;

[0106] $I$ N represents the identity matrix of order $N$;

[0107] $N$ represents the number of all nodes in graph $G$;

[0108] $I$ M represents the identity matrix of order $M$;

[0109] $L$ represents the combinatorial Laplacian matrix of graph $G$;

[0110] The time-varying graph signal on the product graph The Sobolev difference smoothing is defined as:

[0111]

[0112] where

[0113] represents the difference Laplacian matrix on the product graph ;

[0114] $\mu$ represents the perturbation parameter;

[0115] $Q$ represents the difference operator matrix;

[0116] The batch reconstruction of the time-varying graph signal within each signal segment can be reduced to the following optimization problem:

[0117]

[0118] In the formula, the undamaged node indication matrix

[0119] represents the observed values of the graph signal at each moment;

[0120] α represents the balance coefficient between the fitting term (the first term) and the regularization term (the second term);

[0121] Among them, the row vector b m is defined as:

[0122]

[0123] Among them, F m is the set of undamaged nodes at the m-th moment, is the observed value of the graph signal at each moment, and i is the node number in the graph G;

[0124] Among them, the signal value corresponding to the damaged node is set to 0.

[0125] 3) Decompose the Cartesian product graph into a series of subgraphs. The central nodes of each subgraph calculate the inverse matrix of the corresponding local Hessian matrix, and through fusion and averaging, an approximate inverse matrix of the Hessian matrix corresponding to the above optimization problem is obtained. Based on the approximate inverse matrix, the above optimization problem is solved distributively, thereby completing the reconstruction of the time-varying graph signal within this signal segment;

[0126] Cartesian product graph The subgraph decomposition method is as follows:

[0127] Given a node (m, i) (m = 2, 3,..., M - 1, i ∈ V) on the product graph The subgraph centered on this node is defined as

[0128] Among them, is the node set of the subgraph ;

[0129] k represents the time sequence number in the graph G T ;

[0130] l represents the node number in the graph G;

[0131] ρ(i, l) represents the number of hops of the shortest path between node i and node l;

[0132] is All the edges connecting the nodes in form a set. The schematic diagram of the subgraph structure of the Cartesian product graph is as Figure 2 shown.

[0133] The Hessian matrix corresponding to the reconstruction optimization problem of the time-varying graph signal within the signal segment in each subgraph is expressed as:

[0134]

[0135] Among them, (m, i) is the central node of the subgraph, is the indication matrix of all nodes in the subgraph, and its definition is

[0136]

[0137] where p represents the row index of matrix X (m,i) ;

[0138] q represents the column index of matrix X (m,i) ;

[0139] Subgraph calculates the pseudo-inverse matrix of the central node (m, i), and then fuses and averages the pseudo-inverse matrices of all subgraphs. The obtained matrix is used as the approximate inverse matrix of . Then the iterative formula for distributedly solving the batch reconstruction optimization problem is:

[0140]

[0141] represents the approximate inverse matrix of

[0142] t represents the number of iterations;

[0143] The iterative process of formula (7) is as follows:

[0144] Input: Matrix Observation signal Iterative stop criterion ε, maximum number of iterations K, If t = 0, then:

[0145] 1) Each subgraph central node (m, i) calculates:

[0146] and sends the corresponding signal value to all nodes (k, l) in

[0147] h represents the element serial number of

[0148] 2) The nodes in

[0149] average the received signal values and then send the average values to the central nodes of their respective subgraphs;

[0150] If is established or t > K, the program ends; otherwise, let t = t + 1 and return to step 1);

[0151] Output: As the reconstruction signal of this paragraph.

[0152] 4) Sequentially complete the time-varying graph signal reconstruction of all signal segments.

[0153] The time-varying graph signal X in this embodiment Reco contains a positive integer multiple of M time instants.

[0154] Q in this embodiment is expressed as follows:

[0155]

[0156] The approximate inverse matrix of The expression of is as shown in formula (8):

[0157]

[0158] Where represents finding the pseudo-inverse of the matrix.

[0159] In this embodiment, the Cartesian product graph model is adopted to characterize the data structure of the time-varying graph signal, so as to make full use of the spatio-temporal correlation between nodes. At the same time, by introducing the Sobolev difference smoothing of the time-varying graph signal on the product graph, the condition number of the Hessian matrix of the cost function is improved. Based on the Cartesian product graph decomposition, the approximate inverse matrix of the Hessian matrix is calculated. Based on this approximate inverse matrix, a distributed algorithm for solving the batch reconstruction optimization problem is designed, which has the advantages of low reconstruction error and high convergence rate.

[0160] Embodiment 2

[0161] In this embodiment, the simulation experiment is carried out according to the following method:

[0162] Adopt a distributed batch reconstruction method for time-varying graph signals to reconstruct the daily average PM2.5 concentration data of a certain area from 93 sensors in the first 220 days before 2015;

[0163] According to the position coordinates of each data acquisition point, use the k-Nearest Neighbors (k-NN) algorithm to construct graph G. The structural schematic diagram of graph G is as Figure 3 shown, where k = 5. At the same time, model the time dimension of each data set as an undirected line graph G T , where the weight of each edge is taken as 0.1, then the Laplacian matrix L of the undirected line graph G T ofT are as follows:

[0164]

[0165] In this simulation, the values of the parameters of the algorithm are respectively μ = 10 -3 , α = 10 -3 , ε = 10 -4 .

[0166] In addition, to simulate node damage, at each moment of the time-varying graph signal, a corresponding number of nodes are randomly selected as damaged nodes according to a given percentage, and the signal values on these damaged nodes are set to 0. Three different node damage rates are selected for experiments, which are 10%, 20%, and 30% respectively.

[0167] In this embodiment, the RMSE index is used to measure the reconstruction error of the reconstruction algorithm at the m-th moment, and the specific definition is

[0168]

[0169] where represents the original graph signal at the m-th moment, represents the reconstructed graph signal at the m-th moment. To facilitate the comparison between algorithms, the average RMSE at each moment is used as the comparison index, that is

[0170]

[0171] where T = 220.

[0172] Table 1 shows the comparison of the average reconstruction error (RMSE) between the method provided in this embodiment and the existing method 1 (batch reconstruction method based on differential smoothing), and the existing method 2 (batch reconstruction based on Sobolev differential smoothing).

[0173] Table 1 Comparison of the average reconstruction error (RMSE) of various algorithms

[0174]

[0175] Table 2 shows the comparison of the average number of iterations between this method and the existing method 1 and the existing method 2.

[0176] Table 2 Comparison of the average number of iterations of various algorithms

[0177]

[0178] The simulation results show that: measured by the average reconstruction error, when the node damage rate is 10%, this method is comparable to the two existing methods; when the node damage rate is 20% and 30%, this method is superior to the two existing methods; measured by the average number of iterations, when the node damage rate is 10%, 20% and 30%, this method is significantly superior to the two existing methods. Therefore, this method has the advantages of low reconstruction error and high convergence rate.

[0179] The above-described embodiments are only preferred embodiments given to fully illustrate the present invention, and the protection scope of the present invention is not limited thereto. Equivalent substitutions or transformations made by those skilled in the art on the basis of the present invention are all within the protection scope of the present invention. The protection scope of the present invention shall be subject to the claims.

Claims

1. A distributed batch reconstruction method for time-varying graph signals, characterized in that: It includes the following steps: The time-varying graph signal X to be reconstructed Reco is divided into multiple signal segments in chronological order; Construct a Cartesian product graph for all moments within each signal segment. Use the Sobolev difference smoothing design penalty term of the time-varying graph signal within this signal segment on the Cartesian product graph, and reduce the reconstruction of the time-varying graph signal within this signal segment to an optimization problem; Decompose the Cartesian product graph into a series of subgraphs. The central node of each subgraph calculates the inverse matrix of the corresponding local Hessian matrix, and through fusion and averaging, an approximate inverse matrix of the Hessian matrix corresponding to the above optimization problem is obtained. Based on the approximate inverse matrix, the above optimization problem is solved distributively, thereby completing the reconstruction of the time-varying graph signal within this signal segment; Successively complete the reconstruction of the time-varying graph signals within all signal segments; The Cartesian product graph is constructed according to the following steps: The graph G representing the spatial correlation between each node and the graph G representing the temporal correlation between each moment T is modeled as a Cartesian product graph through the Cartesian product The Cartesian product graph is defined as: Among them, the product graph node set is the set of edges between all nodes (m, n) and adjacent nodes (m, n′) n′∈B(n,1) , (m + 1, n) m<M , (m - 1, n) m>1 ; V T represents the set of all nodes in graph G T ; V represents the set of all nodes in graph G; m represents the current moment serial number; n represents the node serial number in graph G; M represents the number of all moments within the signal segment; n′ represents the first-order neighbor node of n; B(n,1) represents the set of the first-order neighbor nodes of n; Combined Laplacian matrix It is expressed as: Among them, represents the Kronecker product; L T represents the combinatorial Laplacian matrix of graph G T ; I N represents the identity matrix of order N; N represents the number of all nodes in graph G; I M represents the identity matrix of order M; L represents the combinatorial Laplacian matrix of graph G; The batch reconstruction of the time-varying graph signal within each signal segment is reduced to an optimization problem in the following manner: In the formula, the undamaged node indication matrix represent the observed values of the figure signals at each moment; α represents the balance coefficient between the fitting term, i.e., the first term, and the regularization term, i.e., the second term; Among them, the row vector b m is defined as: Among them, F m is the set of undamaged nodes at the m-th moment, and i is the node number in the graph G; Among them, the signal value corresponding to the damaged node is set to 0.

2. The time-varying graph signal distributed batch reconstruction method according to claim 1, wherein: The time-varying graph signal within the signal segment On the product graph Sobolev difference smoothing Is calculated according to the following formula: Among them, Denote the product graph on the differential Laplacian matrix; μ represents the perturbation parameter; Q represents the difference operator matrix.

3. The time-varying graph signal distributed batch reconstruction method according to claim 1, characterized in that: The Cartesian product graph is decomposed into subgraphs as follows: Given product graph For a node (m, i) on , where m = 2, 3, …, M - 1 and i ∈ V, the subgraph centered at this node is defined as Among them, is the node set of the sub-graph ; k represents the sequence number of the moment in graph G T ; l represents the node serial number in graph G; ρ(i,l) represents the number of hops of the shortest path between node i and node l; is the set of all edges connecting the nodes in 4. The time-varying graph signal distributed batch reconstruction method according to claim 1, wherein: The inverse matrix of the local Hessian matrix is calculated according to the following steps: Hessian matrix corresponding to the problem of optimizing the reconstruction of time-varying graph signals within a signal segment In each subgraph The local Hessian matrix Is expressed as: where (m, i) is the central node of the subgraph, is the indication matrix of all nodes in the subgraph, and its definition is where p represents the row index of matrix χ (m,i) ; q represents the column label of matrix χ (m,i) ; Sub - graph The central node (m, i) of calculates the pseudo - inverse matrix, and then fuses and averages the pseudo - inverse matrices of all sub - graphs. The resulting matrix is used as the approximate inverse matrix. Then the iterative formula for distributed solution of the batch reconstruction optimization problem is: denote approximate inverse matrix of; t represents the number of iterations.

5. The time-varying graph signal distributed batch reconstruction method according to claim 4, wherein: The iterative process of the iterative formula for distributively solving the batch reconstruction optimization problem is as follows: Input: Matrix Observation signal Iteration stop criterion ε, maximum number of iterations K If t = 0, then: 1 - 7) Each central node (m,i) of the subgraph calculates: and send the corresponding signal values to all the nodes (k, l) in h represents the element number; 2-7) Each node in 3-7) Each sub-graph central node (m, i) forms a new sub-graph signal by combining the received signal value and its own signal value: If is established or t > K, the program ends; otherwise, let t = t + 1 and return to step 1-7); Output: As the reconstruction signal for this paragraph.

6. The time-varying graph signal distributed batch reconstruction method according to claim 2, wherein: The Q is represented as follows: Among them, the time-varying graph signal X Reco contains a number of time instants that is a positive integer multiple of M.

7. The time-varying graph signal distributed batch reconstruction method according to claim 4, wherein: The matrix of the approximate inverse matrix is expressed as follows: Among them, represents the calculation of the pseudo-inverse of a matrix.

8. A time-varying graph signal distributed batch reconstruction system, comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, characterized in that When the processor executes the program, it implements the method described in any one of claims 1 to 7.

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