Signal recovery method, system, medium and device based on block gauss-seidel algorithm

Abstract

The application discloses a signal recovery method and system based on a block Gauss-Seidel algorithm, a medium and equipment, and the method comprises the following steps: obtaining a noise observation vector and a perception matrix; initializing the number of iterations, an estimated signal, a residual vector and a hyperparameter; setting an iteration relative error, calculating the inner product of the column vector of the perception matrix and the estimated vector obtained in this iteration, calculating the residual vector, and selecting the maximum homogeneous residual; determining the control index set of the target block; randomly extracting the corresponding working column from the coefficient matrix to obtain a submatrix; substituting the target block of the submatrix into the iteration formula of the block Gauss-Seidel algorithm without pseudo-inverse calculation to obtain the estimated value of the next iteration, and repeating the loop iteration until the threshold condition is met to stop the iteration; and outputting the estimated sparse signal obtained in the last iteration. The application improves the convergence rate and recovery effect of signal recovery and improves the recovery performance of the sparse signal.

Description

Technical Field

[0001] This invention relates to the field of communication signal processing technology, and specifically to a signal recovery method, system, medium, and device based on the block Gauss-Seidel algorithm. Background Technology

[0002] In many signal processing applications such as wireless communication, image processing, and pattern recognition, it is necessary to solve a linear system. Traditional iterative methods include the Kaczmarz algorithm, the Gauss-Seidel algorithm, and the Landweber algorithm, among others. However, with the rapid development of science and technology, the efficiency of existing recovery algorithms for the original signal is limited. For example, the traditional single-column Gauss-Seidel method often struggles to meet the application requirements of signal recovery for large-scale, high-dimensional data.

[0003] In 2017, Wu Wei [Wu W. Paving the Randomized Gauss-Seidel[J].2017] proposed the block Gauss-Seidel method, which can update multiple coordinate directions simultaneously in each iteration and also has the advantage of parallel computation, making it efficient for solving large-scale linear systems. However, the recovery effect of existing block Gauss-Seidel algorithms is not ideal, and there is still room for improvement in algorithm efficiency. Considering that signal recovery is an important research direction in the fields of communication and signal applications, it is still urgent to study a signal recovery method with faster speed and better performance. Summary of the Invention

[0004] To overcome the defects and shortcomings of existing technologies, this invention provides a signal recovery method based on the block Gauss-Seidel algorithm. This invention is based on a more efficient block Gauss-Seidel method, namely the pseudo-inverse maximum homogeneous residual block Gauss-Seidel method. This method is significantly superior to the current block Gauss-Seidel method in terms of the number of iterations and computation time, thereby improving the convergence rate and recovery effect of signal recovery and achieving the goal of improving the recovery performance of sparse signals.

[0005] The second objective of this invention is to provide a signal recovery system based on the block Gauss-Seidel algorithm;

[0006] A third objective of this invention is to provide a computer-readable storage medium;

[0007] A fourth objective of this invention is to provide a computer device.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] This invention provides a signal recovery method based on the block Gauss-Seidel algorithm, comprising the following steps:

[0010] Obtain the noise observation vector b and the perception matrix A;

[0011] Initialize the number of iterations k, estimate the signal The residual vector r0 and the hyperparameter η;

[0012] Set the relative error for the k-th iteration. In each iteration of the PMHRBGS algorithm, select an index j and calculate the column vector A corresponding to index j of the perception matrix A. j The estimated vector x obtained in this iteration k The inner product is used to calculate the residual vector r of the k-th iteration. k =b-Ax k The maximum homogeneous residual in the k-th iteration is selected using the max function. The parameter ε was calculated. k =ηH k ;

[0013] According to parameter ε k Determine the control index set of the target block Where, j k Indicates an index. Represents any column of the perception matrix;

[0014] By controlling the index set Select the corresponding working column from the coefficient matrix A, and randomly extract the corresponding working column to form a new matrix, thus obtaining the submatrix.

[0015] Submatrix Substituting the target block into the iterative formula of the Gauss-Seidel algorithm without pseudo-inverse computation, we obtain the estimated value x for the next iteration. k+1 Repeat the loop iteratively until the threshold condition is met and then stop iterating;

[0016] Output the estimated sparse signal obtained in the last iteration.

[0017] As a preferred technical solution, the relative error of the k-th iteration is set, specifically expressed as: Where, x k Let x represent the estimated vector generated in the k-th iteration, and let x represent the true vector.

[0018] As a preferred technical solution, the iterative formula of the block Gauss-Seidel algorithm without pseudo-inverse computation is specifically expressed as follows:

[0019] By calculating an approximation of the pseudo-inverse, the iterative formula for the pseudo-inverse-free maximum homogeneous residual block Gauss-Seidel algorithm is obtained, which is expressed as:

[0020]

[0021] Among them, column index It is any column in the target block. It is the jth ku A unit column vector with elements of 1 and all other elements of 0;

[0022] make The final iterative formula is obtained as follows:

[0023]

[0024] Where, x k Let x represent the estimated value in the current iteration. k+1 This represents the estimated value for the next iteration, and ω represents a constant.

[0025] As the preferred technical solution, ω satisfies the following conditions:

[0026]

[0027] Where Δ represents the convergence guarantee factor parameter, specifically expressed as:

[0028]

[0029] To achieve the second objective mentioned above, the present invention adopts the following technical solution:

[0030] A signal recovery system based on the block Gauss-Seidel algorithm includes: a parameter acquisition module, an initialization module, an iterative relative error setting module, a maximum homogeneous residual calculation module, a parameter calculation module, a target block control index set calculation module, a submatrix construction module, a loop iteration module, and a signal output module.

[0031] The parameter acquisition module is used to acquire the noise observation vector b and the perception matrix A;

[0032] The initialization module is used to initialize the iteration count k and the estimated signal. The residual vector r0 and the hyperparameter η;

[0033] The iterative relative error setting module is used to set the relative error of the kth iteration.

[0034] The maximum homogeneous residual calculation module is used to calculate the maximum homogeneous residual. In each iteration of the PMHRBGS algorithm, an index j is selected, and the column vector A corresponding to index j of the perception matrix A is calculated.j The estimated vector x obtained in this iteration k The inner product is used to calculate the residual vector r of the k-th iteration. k =b-Ax k The maximum homogeneous residual in the k-th iteration is selected using the max function.

[0035] The parameter calculation module is used to calculate the parameter ε. k =ηH k ;

[0036] The control index set calculation module of the target block is used to calculate the parameter ε. k Determine the control index set of the target block Where, j k Indicates an index. Represents any column of the perception matrix;

[0037] The submatrix construction module is used to control the index set. Select the corresponding working column from the coefficient matrix A, and randomly extract the corresponding working column to form a new matrix, thus obtaining the submatrix.

[0038] The loop iteration module is used to submatrix Substituting the target block into the iterative formula of the Gauss-Seidel algorithm without pseudo-inverse computation, we obtain the estimated value x for the next iteration. k+1 Repeat the loop iteratively until the threshold condition is met and then stop iterating;

[0039] The signal output module is used to output the estimated sparse signal obtained in the last iteration.

[0040] As a preferred technical solution, the iterative relative error setting module is used to set the relative error of the k-th iteration, specifically as follows: Where, x k Let x represent the estimated vector generated in the k-th iteration, and let x represent the true vector.

[0041] As a preferred technical solution, the iterative formula of the block Gauss-Seidel algorithm without pseudo-inverse computation is specifically expressed as follows:

[0042] By calculating an approximation of the pseudo-inverse, the iterative formula for the pseudo-inverse-free maximum homogeneous residual block Gauss-Seidel algorithm is obtained, which is expressed as:

[0043]

[0044] Among them, column index It is any column in the target block. It is the jthku A unit column vector with elements of 1 and all other elements of 0;

[0045] make The final iterative formula is obtained as follows:

[0046]

[0047] Where, x k Let x represent the estimated value in the current iteration. k+1 This represents the estimated value for the next iteration, and ω represents a constant.

[0048] As the preferred technical solution, ω satisfies the following conditions:

[0049]

[0050] Where Δ represents the convergence guarantee factor parameter, specifically expressed as:

[0051]

[0052] To achieve the aforementioned third objective, the present invention adopts the following technical solution:

[0053] A computer-readable storage medium storing a program that, when executed by a processor, implements the above-described signal recovery method based on the block Gauss-Seidel algorithm.

[0054] To achieve the fourth objective mentioned above, the present invention adopts the following technical solution:

[0055] A computer device includes a processor and a memory for storing a processor-executable program, wherein when the processor executes the program stored in the memory, it implements the signal recovery method based on the block Gauss-Seidel algorithm described above.

[0056] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0057] (1) This invention solves the technical problem of solving the pseudo-inverse of the submatrix during the iteration of the traditional block Gauss-Seidel algorithm by eliminating the need to calculate the pseudo-inverse, thereby effectively reducing the running time of the algorithm.

[0058] (2) The present invention can flexibly control the size of the target block, solves the technical problem that the greedy block Gauss-Seidel algorithm cannot autonomously select a comprehensive target column, and achieves the goal of reducing the relative error between the iteratively estimated signal and the original signal.

[0059] (3) Based on the multi-core parallel mechanism, this invention solves the technical problem that the single-column Gauss-Seidel algorithm can only optimize one coordinate direction in one iteration, thus achieving a faster signal recovery rate. Attached Figure Description

[0060] Figure 1 This is a flowchart illustrating the signal recovery method based on the block Gauss-Seidel algorithm of the present invention.

[0061] Figure 2 This diagram illustrates the comparison between the consistency recovery error and the number of algorithm iterations of the PMHRBGS algorithm of this invention and the existing MRBGS, PGBGS, and GBGS algorithms.

[0062] Figure 3 This diagram illustrates the comparison between the relative recovery error and the number of algorithm iterations of the PMHRBGS algorithm of this invention and the existing MRBGS, PGBGS, and GBGS algorithms.

[0063] Figure 4 This diagram illustrates the comparison of the relative recovery error and algorithm running time between the present invention and existing MRBGS, PGBGS, and GBGS algorithms. Detailed Implementation

[0064] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0065] Example 1

[0066] The term "PMHRBGS" as used below refers to the improved Gauss-Seidel algorithm for maximum homogeneous residual blocks without pseudo-inverses, which is adopted in this invention.

[0067] The following explanations of some terms used in the embodiments of this application are provided to facilitate understanding by those skilled in the art.

[0068] Gauss-Seidel Algorithm: The Gauss-Seidel algorithm is a mainstream algorithm for computing large-scale overdetermined linear systems. Because it is easy to implement and parallelize on computers, it is widely used in engineering fields such as distributed computing, regression analysis, signal processing, and image restoration.

[0069] Block Gauss-Seidel Algorithm: The block Gauss-Seidel algorithm can update multiple coordinate directions simultaneously in each iteration, which greatly speeds up the traditional single-column Gauss-Seidel algorithm.

[0070] Greedy Block Gauss-Seidel Algorithm (GBGS): It uses a greedy strategy to select the unit coordinate direction that most urgently needs to be updated and adaptively selects target blocks of different sizes.

[0071] The Greedy Block Gauss-Seidel Algorithm Without Pseudo-Inverses (PGBGS) avoids the computation of pseudo-inverses of submatrices, thus accelerating the greedy block Gauss-Seidel algorithm.

[0072] Maximum Residual Block Gauss-Seidel Algorithm (MRBGS): An improvement on the greedy block Gauss-Seidel algorithm, it eliminates the adaptive selection of column indexes and instead flexibly controls the size of the target block.

[0073] like Figure 1 As shown, this embodiment provides a signal recovery method based on the block Gauss-Seidel algorithm, including the following steps:

[0074] (1) Obtain the noise observation vector and perception matrix

[0075] (2) Data initialization, including initializing the number of iterations k and the estimated signal. The residual vector r0 and the hyperparameter η∈(0,1];

[0076] (3) Set the relative error of the kth iteration (The estimated vector x generated in the kth iteration) k (The relative distance between the real vector x and the actual vector x) In each iteration of the PMHRBGS algorithm, an index j is selected, and the column vector A corresponding to index j of the perception matrix A is calculated. j The estimated vector x obtained in this iteration k The inner product is used to calculate the residual vector r of the k-th iteration. k =b-Ax k Then, the maximum homogeneous residual in the k-th iteration is selected using the max function. This allows us to obtain the parameter ε. k =ηH k , where η∈(0,1];

[0077] (4) Based on the parameter ε from the previous step k The control index set of the target block can be determined. (Any column of the perception matrix is ​​denoted as) ), by controlling the index set Select the corresponding working column from coefficient matrix A, and randomly extract the corresponding working column from coefficient matrix A to form a new matrix, thus obtaining a submatrix. By selecting the block closest to the maximum homogeneous residual;

[0078] (5) Submatrix Substituting the target block into the iterative formula of the block Gauss-Seidel algorithm (PMHRBGS) without pseudo-inverse computation, we obtain the estimated value x for the next iteration. k+1 Repeat the loop iteratively until the threshold condition is met and then stop iterating;

[0079] The iterative formula for the block Gauss-Seidel algorithm is expressed as follows:

[0080]

[0081] This embodiment calculates an approximation of the pseudo-inverse, thus obtaining the iterative formula for the pseudo-inverse-free maximum homogeneous residual block Gauss-Seidel algorithm, which is expressed as:

[0082]

[0083] Among them, column index It is any column in the target block. It is the jth ku A unit column vector with elements of 1 and all other elements of 0;

[0084] make The iterative formula obtained in this embodiment is:

[0085]

[0086] ω satisfies the following condition:

[0087]

[0088] Among them, the convergence guarantee factor parameter This condition is sufficient, but not necessary, to guarantee the convergence of this algorithm.

[0089] (6) Output estimated signal x k The estimated sparse signal obtained in the last iteration is output.

[0090] This embodiment improves signal recovery efficiency and enables signal recovery without pseudo-inverse computation, as well as flexible control over the number of working columns in the target block, while also enabling parallel computation.

[0091] While existing BGS algorithms can update multiple coordinate directions simultaneously, they require very time-consuming calculations of the pseudo-inverse of the submatrix of the perception matrix. This embodiment tests the performance of the PMBRS algorithm. In multiple experiments, simulations are conducted to compare the algorithmic efficiency and recovery effect of the three previously proposed BGS algorithms, namely GBGS, PGBGS, and MRBGS.

[0092] In this embodiment, each element in the coefficient matrix A is set to be independent and identically distributed and follow a standard normal distribution, and each element in the original signal x is set to be independent and identically distributed and follow a standard normal distribution. The experimental results in this embodiment are averaged over 100 independent experiments.

[0093] like Figure 2 As shown, the consistency recovery error of the PMHRBGS algorithm proposed in this embodiment compared with existing MRBGS, PGBGS, and GBGS algorithms is illustrated. The comparison is made with the number of algorithm iterations, where M = 5000 and N = 500. As can be seen from the figure, the PMHRBGS algorithm in this embodiment reduces the consistency recovery error more significantly in one iteration compared to the other three currently available BGS algorithms, demonstrating better performance.

[0094] like Figure 3 As shown, the relative recovery error of the PMHRBGS algorithm proposed in this embodiment compared with existing MRBGS, PGBGS, and GBGS algorithms is illustrated. Comparing the number of algorithm iterations, where M=5000 and N=500, it can be seen from the figure that the PMHRBGS algorithm in this embodiment reduces the relative recovery error more in one iteration compared to the other three currently available BGS algorithms, and the PMHRBGS algorithm performs better.

[0095] like Figure 4 As shown, the relative recovery error is displayed. Compared to existing algorithms MRBGS, PGBGS, and GBGS, the PMHRBGS algorithm proposed in this embodiment converges to the given error size more quickly, demonstrating better algorithm efficiency, where M = 5000 and N = 500. In the above embodiment, each element in the perception matrix A is set to be independent and identically distributed, following a standard normal distribution; the original signal x is randomly generated using the MATLAB command randn(500,1); the observation vector b is generated from A*x. The experimental results in this embodiment are averaged across 100 independent experiments.

[0096] Example 2

[0097] This embodiment provides a signal recovery system based on the block Gauss-Seidel algorithm, including: a parameter acquisition module, an initialization module, an iterative relative error setting module, a maximum homogeneous residual calculation module, a parameter calculation module, a target block control index set calculation module, a submatrix construction module, a loop iteration module, and a signal output module;

[0098] In this embodiment, the parameter acquisition module is used to acquire the noise observation vector b and the perception matrix A;

[0099] In this embodiment, the initialization module is used to initialize the iteration count k and the estimated signal. The residual vector r0 and the hyperparameter η;

[0100] In this embodiment, the iterative relative error setting module is used to set the relative error of the k-th iteration;

[0101] In this embodiment, the maximum homogeneous residual calculation module is used to calculate the maximum homogeneous residual. In each iteration of the PMHRBGS algorithm, an index j is selected, and the column vector A corresponding to index j of the perception matrix A is calculated. j The estimated vector x obtained in this iteration k The inner product is used to calculate the residual vector r of the k-th iteration. k =b-Ax k The maximum homogeneous residual in the k-th iteration is selected using the max function.

[0102] In this embodiment, the parameter calculation module is used to calculate the parameter ε. k =ηH k ;

[0103] In this embodiment, the control index set calculation module of the target block is used to calculate the control index set based on the parameter ε. k Determine the control index set of the target block Where, j k Indicates an index. Represents any column of the perception matrix;

[0104] In this embodiment, the submatrix construction module is used to control the index set. Select the corresponding working column from the coefficient matrix A, and randomly extract the corresponding working column to form a new matrix, thus obtaining the submatrix.

[0105] In this embodiment, the loop iteration module is used to process the submatrix Substituting the target block into the iterative formula of the Gauss-Seidel algorithm without pseudo-inverse computation, we obtain the estimated value x for the next iteration. k+1 Repeat the loop iteratively until the threshold condition is met and then stop iterating;

[0106] In this embodiment, the signal output module is used to output the estimated sparse signal obtained in the last iteration.

[0107] In this embodiment, the iterative relative error setting module is used to set the relative error of the k-th iteration, specifically as follows: Where, x k Let x represent the estimated vector generated in the k-th iteration, and let x represent the true vector.

[0108] In this embodiment, the iterative formula of the block Gauss-Seidel algorithm without pseudo-inverse computation is specifically expressed as follows:

[0109] By calculating approximate values ​​of pseudo-inverses, the iterative formula for the pseudo-inverse-free maximum homogeneous residual block Gauss-Seidel algorithm is obtained, which is expressed as:

[0110]

[0111] Among them, column index It is any column in the target block. It is the jth ku A unit column vector with elements of 1 and all other elements of 0;

[0112] make The final iterative formula is obtained as follows:

[0113]

[0114] Where, x k Let x represent the estimated value in the current iteration. k+1 ω represents the estimated value for the next iteration, and ω represents a constant.

[0115] In this embodiment, the condition that ω satisfies is:

[0116]

[0117] Where Δ represents the convergence guarantee factor parameter, specifically expressed as:

[0118]

[0119] Example 3

[0120] This embodiment provides a storage medium, which may be a ROM, RAM, disk, optical disk, or other storage medium. The storage medium stores one or more programs. When the programs are executed by a processor, they implement the signal recovery method based on the block Gauss-Seidel algorithm of Embodiment 1.

[0121] Example 4

[0122] This embodiment provides a computing device, which may be a desktop computer, laptop computer, smartphone, PDA handheld terminal, tablet computer or other terminal device with display function. The computing device includes a processor and a memory. The memory stores one or more programs. When the processor executes the program stored in the memory, it implements the signal recovery method based on the block Gauss-Seidel algorithm of Embodiment 1.

[0123] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A signal recovery method based on the block Gauss-Seidel algorithm, characterized in that, Includes the following steps: Obtain the noise observation vector b and the perception matrix A; Initialize the number of iterations k, estimate the signal The residual vector r0 and the hyperparameter η; Set the relative error for the k-th iteration. In each iteration of the PMHRBGS algorithm, select an index j and calculate the column vector A corresponding to index j of the perception matrix A. j The estimated vector x obtained in this iteration k The inner product is used to calculate the residual vector r of the k-th iteration. k =b-Ax k The maximum homogeneous residual in the k-th iteration is selected using the max function. The parameter ε was calculated. k =ηH k ; According to parameter ε k Determine the control index set of the target block Where, j k Indicates an index. Represents any column of the perception matrix; By controlling the index set Select the corresponding working column from the coefficient matrix A, and randomly extract the corresponding working column to form a new matrix, thus obtaining the submatrix. Submatrix Substituting the target block into the iterative formula of the Gauss-Seidel algorithm without pseudo-inverse computation, we obtain the estimated value x for the next iteration. k+1 Repeat the loop iteratively until the threshold condition is met and then stop iterating; Output the estimated sparse signal obtained in the last iteration.

2. The signal recovery method based on the block Gauss-Seidel algorithm according to claim 1, characterized in that, The relative error of the k-th iteration is set as follows: Where, x k Let x represent the estimated vector generated in the k-th iteration, and let x represent the true vector.

3. The signal recovery method based on the block Gauss-Seidel algorithm according to claim 1, characterized in that, The iterative formula for the block Gauss-Seidel algorithm without pseudo-inverse computation is specifically expressed as follows: By calculating approximate values ​​of pseudo-inverses, the iterative formula for the Gauss-Seidel algorithm for maximum homogeneous residual blocks without pseudo-inverses is obtained, which is expressed as: Among them, column index It is any column in the target block. It is the jth ku A unit column vector with elements 1 and all other elements 0; make The final iterative formula is obtained as follows: Where, x j Let x represent the estimated value in the current iteration. k+1 This represents the estimated value for the next iteration, and ω represents a constant.

4. The signal recovery method based on the block Gauss-Seidel algorithm according to claim 3, characterized in that, The condition that ω satisfies is: Where Δ represents the convergence guarantee factor parameter, specifically expressed as:

5. A signal recovery system based on the block Gauss-Seidel algorithm, characterized in that, include: The module includes: parameter acquisition module, initialization module, iterative relative error setting module, maximum homogeneous residual calculation module, parameter calculation module, target block control index set calculation module, submatrix construction module, loop iteration module, and signal output module. The parameter acquisition module is used to acquire the noise observation vector b and the perception matrix A; The initialization module is used to initialize the iteration count k and the estimated signal. The residual vector r0 and the hyperparameter η; The iterative relative error setting module is used to set the relative error of the kth iteration. The maximum homogeneous residual calculation module is used to calculate the maximum homogeneous residual. In each iteration of the PMHRBGS algorithm, an index j is selected, and the column vector A corresponding to index j of the perception matrix A is calculated. j The estimated vector x obtained in this iteration k The inner product is used to calculate the residual vector r of the k-th iteration. k =b-Ax k The maximum homogeneous residual in the k-th iteration is selected using the max function. The parameter calculation module is used to calculate the parameter ε. k =ηH k ; The control index set calculation module of the target block is used to calculate the parameter ε. k Determine the control index set of the target block Where, j k Indicates an index. Represents any column of the perception matrix; The submatrix construction module is used to control the index set. Select the corresponding working column from the coefficient matrix A, and randomly extract the corresponding working column to form a new matrix, thus obtaining the submatrix. The loop iteration module is used to submatrix Substituting the target block into the iterative formula of the Gauss-Seidel algorithm without pseudo-inverse computation, we obtain the estimated value x for the next iteration. k+1 Repeat the loop iteratively until the threshold condition is met and then stop iterating; The signal output module is used to output the estimated sparse signal obtained in the last iteration.

6. The signal recovery system based on the block Gauss-Seidel algorithm according to claim 5, characterized in that, The iterative relative error setting module is used to set the relative error of the k-th iteration, specifically as follows: Where, x k Let x represent the estimated vector generated in the k-th iteration, and let x represent the true vector.

7. The signal recovery system based on the block Gauss-Seidel algorithm according to claim 5, characterized in that, The iterative formula for the block Gauss-Seidel algorithm without pseudo-inverse computation is specifically expressed as follows: By calculating approximate values ​​of pseudo-inverses, the iterative formula for the Gauss-Seidel algorithm for maximum homogeneous residual blocks without pseudo-inverses is obtained, which is expressed as: Among them, column index It is any column in the target block. It is the jth ku A unit column vector with elements of 1 and all other elements of 0; make The final iterative formula is obtained as follows: Where, x j Let x represent the estimated value in the current iteration. k+1 This represents the estimated value for the next iteration, and ω represents a constant.

8. The signal recovery system based on the block Gauss-Seidel algorithm according to claim 7, characterized in that, The condition that ω satisfies is: Where Δ represents the convergence guarantee factor parameter, specifically expressed as:

9. A computer-readable storage medium storing a program, characterized in that, When the program is executed by the processor, it implements the signal recovery method based on the block Gauss-Seidel algorithm as described in any one of claims 1-4.

10. A computer device comprising a processor and a memory for storing a processor-executable program, characterized in that, When the processor executes the program stored in the memory, it implements the signal recovery method based on the block Gauss-Seidel algorithm as described in any one of claims 1-4.

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