A neural-enhanced learning-based sparse recovery method

Abstract

The application discloses a learning type sparse recovery method based on neural enhancement, and comprises the following steps: introducing a long short-term memory unit (LSTM) in each layer of an ALePOM network to construct a NA-ALePOM network; training and optimizing network parameters of the NA-ALePOM network based on a training data set; in the training process, using network parameters adaptively calculated by the LSTM, combining a proximal operator update signal estimation, and outputting a sparse signal; using the trained NA-ALePOM network to process an actual observation signal through single forward propagation, outputting an estimation result of the sparse signal, and realizing sparse recovery. In each layer of the ALePOM, a long short-term memory (LSTM) unit is introduced to adaptively calculate network parameters for each training data. Through this method, the recovery error boundary is further tightened, so that better sparse recovery effect is obtained.

Description

Technical Field

[0001] The present invention relates to the technical field of signal processing and data capture, and particularly to a learning-based sparse recovery method based on neural enhancement. Background Art

[0002] Compressed sensing is an emerging signal processing and data capture technology that can significantly reduce the resources required for storing and transmitting data, and has significant advantages and application values in the field of data processing. As a key link in compressed sensing technology, the sparse recovery algorithm aims to accurately and efficiently reconstruct high-dimensional sparse signals from low-dimensional measurement data, and is a current research hotspot in the field of compressed sensing.

[0003] Generally speaking, the sparse recovery problem is mainly solved by utilizing the sparsity of the target signal. For this purpose, researchers have developed various sparsity-inducing penalty (SIP) functions, including the L0 pseudo-norm, L1 norm, and non-convex SIP (such as the L p pseudo-norm with 0 < p < 1). The L0 pseudo-norm can directly measure the sparsity degree of the signal, but its solution is a typical NP-hard problem, and usually greedy algorithms are used for solution. Greedy algorithms have a relatively fast convergence rate, but they require prior information such as the sparsity of the original signal, and this information is difficult to obtain in advance in practical applications. The L1 norm is a convex relaxation of the L0 pseudo-norm. Convex relaxation algorithms based on the L1 norm do not require prior knowledge and can obtain better global optimal solutions. Therefore, they have been widely used in the field of sparse recovery. With the in-depth research, various more effective solution algorithms have been proposed for the L1 minimization problem, such as the Iterative Shrinkage-Thresholding Algorithm (ISTA) and the Alternating Direction Method of Multipliers. However, these algorithms usually only have a sublinear convergence rate and are difficult to meet the timeliness requirements in some cases. Compared with the L1 norm, non-convex SIP can better approximate the L0 pseudo-norm in sparsity representation and obtain a sparser solution. Therefore, researchers have proposed methods such as the Proximal Operator Method (POM) and the Generalized Gradient Descent Method based on non-convex SIP. However, these methods also face the problem of slow convergence speed during the solution process and are prone to falling into local minima.

[0004] Recently, inspired by deep learning, researchers have innovatively proposed deep unfolding methods for sparse recovery. These methods cleverly transform traditional iterative sparse recovery algorithms (such as ISTA and POM) into deep neural network structures and learn the parameters of these iterative algorithms using a training dataset. Some variants of deep unfolding methods, such as Analytical LISTA (ALISTA) and Analytical LePOM (ALePOM), have received theoretical support in the literature regarding the convergence of the recovery process. Specifically, ALISTA can solve the L1 norm minimization problem with a linear convergence rate, while ALePOM can solve the non-convex SIP minimization problem with a linear convergence rate, and the latter exhibits better sparse recovery performance. However, it is worth noting that the ALePOM network parameters are learned to adapt to the entire training dataset, rather than a single training sample. Therefore, its recovery error boundary may be affected by the diversity of the training data, and its performance still has room for improvement. Summary of the Invention

[0005] The purpose of this invention is to provide a learning-based sparse recovery method based on neural reinforcement. Through this method, the recovery error boundary will be further tightened, thereby obtaining a better sparse recovery effect.

[0006] To achieve the above objectives, the present invention provides the following solution: A learning-based sparse recovery method based on neural reinforcement includes the following steps: S1. Introduce Long Short-Term Memory (LSTM) units into each layer of the ALePOM network to construct an NA-ALePOM network; where LSTM is used to adaptively calculate network parameters based on the estimation results of the previous layer and the input data of the current layer. S2. The network parameters of the NA-ALePOM network are trained and optimized based on the training dataset. During the training process, the network parameters are adaptively calculated and combined with the near-end operator to update the signal estimation and output a sparse signal. S3. Using the trained NA-ALePOM network, the actual observed signal is processed through a single forward propagation to output the estimation result of the sparse signal, thereby achieving sparse recovery.

[0007] Furthermore, in S1, the adaptively calculated network parameters include threshold parameters, step size parameters, and sparse penalty term parameters.

[0008] Furthermore, in S1, the ALePOM network will Represented as learnable parameters The product of the constant matrix B, where matrix B is independent of the number of network layers k and is related to the constant matrix B. The generalized correlation is relatively small, and the operation of the (k+1)th network layer of the ALePOM network is represented as follows: (8) In the formula, Denotes the proximal operator, x (k) For the first k The sparse signal estimate of the layer; B is a constant matrix independent of the number of network layers; y is the given observation data; Indicates the first k The step size for each iteration; , M < N , is the measurement matrix, , Indicates conjugate transpose; The constant matrix B is obtained by solving the following optimization problem: (9) In the formula, This represents the i-th row of B. express The i-th column, , express Zhongyou Definite A matrix composed of columns in the matrix.

[0009] Furthermore, the proximal operator uses the minimum-maximum concavity penalty function as the SIP, denoted as: ; (2) The proximal operator based on the POM algorithm is represented as: (4) Equation (4) can be written as: (5) in, express z The i One element, Let the sign function be denoted by . , , , The representation of the proximal operator is obtained. .

[0010] Furthermore, in S1, the constructed NA-ALePOM network uses LSTM to approximate the results of the first few layers of the network. Integrating into each sparse signal sparse recovery error In the estimation, the specific calculation is expressed as follows: (14) in, This represents the state of the k-th LSTM unit. This represents the hidden state of the k-th LSTM unit. P Indicates the size of the LSTM cell. , , Indicates by Switch to Transformation matrix and transformation vector, and Represents the deviation vector and the deviation. Represents sparse signals The corresponding threshold; The LSTM adaptively calculates the step size parameter of the k-th layer. The update operation expression is: (15) in, and Represents the state of the k-th LSTM unit cell. Convert to step size parameter Transformation matrix and vector, and This represents the deviation vector and deviation amount of the step size parameter. Represents sparse signals The corresponding step size; sparse signal Corresponding sparse penalty term parameters Set as .

[0011] Furthermore, in S1, the sparse signal estimation expression for the (k+1)th layer of the constructed NA-ALePOM is as follows: (16) In the formula, x (k) For the first k Sparse signal estimates for the layer; B is a constant matrix; y For observation vectors; , M < N , is the measurement matrix; This represents the proximal operator.

[0012] Furthermore, in S2, the training and optimization of network parameters specifically includes: Given a training set and network layers K All parameters of NA-ALePOM, i.e., network parameters ,in, The LSTM cell parameters, including weight parameters and bias parameters, are optimized by solving the following problem through backpropagation: (17) in, Indicates that the input is The initial value is NA-AeLPOM k Layer output.

[0013] Furthermore, in S3, given the actual observed signal, the learned parameters are used... The sparse signal is estimated through a single forward propagation of NA-ALPOM.

[0014] The present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements a learning-based sparse recovery method based on neural reinforcement as described above.

[0015] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects: This invention proposes a sparse recovery method for Neural Augmented ALePOM (NA-ALePOM). It employs a neural augmentation algorithm to adaptively calculate network parameters during the sparse recovery process, thereby reducing the sparse recovery error boundary and improving the accuracy of sparse signal recovery. Specifically, in each layer of ALePOM, a Long Short-Term Memory (LSTM) unit is introduced to adaptively calculate network parameters for each training data set. This method further tightens the sparse recovery error boundary of ALePOM, resulting in better sparse recovery performance. Simulation experiments verify that the proposed method has good sparse recovery performance. Attached Figure Description

[0016] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0017] Figure 1The NA-ALePOM network structure diagram provided by this invention; Figure 2 Parameters of the present invention and and The correlation between them; Among them, (a) is r 1 and The correlation of 1, (b) is r 5 and The correlation of 5, (c) is r 10 and 10 The correlation, (d) is r 14 and 14 The correlation, (e) is u 1 and The correlation of 1, (f) is u 5 and The correlation of 5, (g) is u 10 and 10 The correlation, (h) is u 14 and 14 The correlation; Figure 3 Parameters of the present invention and and Correlation plot between them; Among them, (a) is r 1 and The correlation of 2, (b) is r 6 and The correlation of 7, (c) is r 13 and 14 The correlation, (d) is r 11 and 14 The correlation, (e) is u 1 and The correlation of 2, (f) is u 6 and The correlation of 7, (g) is u 13 and 14 The correlation, (h) is u 11 and 14 The correlation; Figure 4 This is a graph verifying the adaptive calculation capability of the NA-ALePOM method of the present invention. Where (a) is the training / test NMSE, (b) is the stride and threshold of different layers, and (c) is... and Relationship diagram; Figure 5 This is a comparison chart of the sparse signal recovery performance of different algorithms in this invention based on random observations; Where (a) represents the change of NMSE with the number of network layers, (b) represents the change of NMSE with SNR, and (c) represents the change of NMSE with sparsity. Figure 6 This is a comparison chart of the sparse recovery performance of different algorithms for harmonic signals in this invention; Among them, (a) is an example of harmonic recovery results, (b) is the change of NMSE with the number of network layers, and (c) is the change of NMSE with SNR. Detailed Implementation

[0018] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0020] This invention conducts an in-depth study of the ALePOM method and proposes an improved scheme based on neural augmentation, namely the Neural Augmented ALePOM (NA-ALePOM) method. This method introduces Long Short-Term Memory (LSTM) units to further tighten the error boundary of ALePOM and improve sparse recovery performance. First, the recovery error boundary of ALePOM is theoretically analyzed, revealing its potential for performance optimization. Then, LSTM units are introduced into the ALePOM framework. Based on the error estimate between the recovered signal and the true signal for each data sample, the LSTM units adaptively calculate iterative parameters, effectively reducing the recovery error and achieving better sparse recovery results. Finally, simulation experiments comprehensively verify and evaluate the sparse recovery performance of the NA-ALePOM method.

[0021] The structure of this invention is as follows: Section 1 introduces relevant basic knowledge; Section 2 proposes the NA-ALePOM method based on the convergence analysis of ALePOM; Section 3 verifies and evaluates the performance of the proposed method through simulation experiments; Section 4 provides a summary.

[0022] 1. Basic Knowledge Typically, given observation data ,in ( M < N ) is the measurement matrix, n For additive noise, sparsity recovery estimates sparsity by solving the following problem: s The original signal : (1) in, Represents the regularization parameter. Indicated by SIP as parameters, For the sparse penalty term parameters, Representing vectors L 2-norm.

[0023] There are many SIPs that can be used to measure signal sparsity, such as L 1-norm and nonconvex L p (0< p <1) Norm. This invention selects the Minimax Concave Penalty (MCP) function as the SIP, expressed as: ,in, For vectors x The i 1 element, and (2) According to equation (2), the POM algorithm solves equation (1) through multiple iterations, and its first iteration... k +1 time ( k = 0, 1, …, K -1) The iterative solution can be expressed as: (3) in, Initialize as a vector of all zeros. I Represents the identity matrix. Indicates the first k The step size for each iteration. This indicates the conjugate transpose. For the proximal operator, it is represented as: (4) The solution to equation (4) can be written as: (5) in, express z The i One element, Represents a symbolic function.

[0024] Because POM only has sublinear convergence (i.e., its computational complexity for achieving better sparse recovery accuracy is high) and its parameters are difficult to set in practical applications, some scholars have proposed LePOM based on the similarity between Equation (3) and neural networks. k The solution of a +1 layer network (iterative) can be expressed as: (6) in, , , For the network k Learnable parameters of the layer.

[0025] It can be seen that if let , , , Then equation (6) is equivalent to equation (3). The difference is that LePOM can achieve a faster convergence speed and higher accuracy of sparse recovery by learning and optimizing these parameters based on training data.

[0026] It can be proven that when LePOM can accurately recover sparse signals, the network parameters... and Gradual satisfaction Therefore, a network with fewer parameters can be further constructed based on LePOM, namely, parameter-coupled LePOM (LePOM with Parameter Coupling, LePOM-PC), to solve equation (1), which is expressed as: (7) Comparing equations (7) and (6), it can be seen that by utilizing the coupling relationship between network parameters, LePOM-PC does not need to learn the parameters in LePOM. This can reduce the difficulty of online training and shorten training time.

[0027] Furthermore, if and The smaller the generalized correlation, the better the performance and the faster the convergence speed of LePOM-PC. Therefore, an Analytical LePOM (ALePOM) can be constructed to further reduce the number of parameters. ALePOM will... Represented as learnable parameters and constant matrix B The product of matrices, where the matrix B With network layers k Irrelevant, and related to The generalized correlation is relatively small, and its first... k The operation of the +1 network layer can be represented as: (8) Among them, matrix B The following optimization problem was solved to obtain the following: (9) in, express B The i OK, express The i List, , express Zhongyou Definite A matrix composed of columns in the matrix.

[0028] 2. Neural Enhancement ALePOM 2.1. Network Design Based on network parameters obtained through training and optimization , and ALePOM achieves faster speed and higher accuracy than POM. Specifically, given a sparsity of... s sparse signal and its observation data (in, , , and (representing the upper power limits of the signal and noise respectively), if the following conditions are met: (10) The upper bound of the recovery error of ALePOM is: (11) in, express B and The generalized correlation coefficient, The coefficients of MCP are the differentiable coefficients. Representing a matrix Frobenius Norm; Since the network parameters of ALePOM are optimized based on all the different training data, its learned threshold... Corresponding to all The maximum error, i.e. (12) in, This indicates taking the upper bound.

[0029] It can be seen from equation (11) that The upper limit of the sparse recovery error is partially determined, therefore, ALePOM's performance theoretically has room for improvement. If each data point could be computed independently... sparse recovery error ALePOM can then adaptively adjust the threshold. To satisfy the upper bound in equation (10) instead of equation (12), the upper bound of the recovery error is further tightened.

[0030] However, due to In practice, it is unknown, and its error cannot be directly obtained. To address this issue, the NA-ALePOM network proposed in this invention utilizes a method that requires no prior knowledge of the network's features. of and to approach , represented as: (13) Assumption Its elements follow a Gaussian distribution with a mean of 0 and a standard deviation of 1 / 250, and each column has L The 2-norm is 1, and the sparsity is . s sparse signal If the non-zero elements follow a Bernoulli distribution, then the true error is... and and The correlation between them is as follows Figure 2 As shown in (a)-(h), the sparsity ratio is defined as p = s / 500. It can be seen that when the signal is sparse, and and The correlation indicates that the estimation based on equation (13) is accurate. To adaptively learn each Corresponding threshold It is feasible. In fact, besides those based on the network... k Layer computing and right Approximation can also be performed based on the approximate results of the first few layers of the network, i.e. ,right To approximate, such as Figure 3 As shown in (a)-(h), it can be observed that... and and ( It is correlated across multiple layers.

[0031] Therefore, considering and Based on the structural similarity between sequence correlation and recurrent neural networks, the constructed NA-ALePOM network uses LSTM to process previous estimates. Integrate into In the estimation, the specific calculation can be expressed as: (14) in, This represents the state of the k-th LSTM unit. This represents the hidden state of the k-th LSTM unit. P Indicates the size of the LSTM cell. , , Indicates by Switch to Transformation matrix and transformation vector, and Represents the deviation vector and the deviation. Represents sparse signals The corresponding threshold.

[0032] Furthermore, as can be seen from equation (10), the step size of another network parameter of ALePOM is... With signal sparsity s as well as B and correlation Related. To make it equally adaptable to different... The proposed NA-ALePOM method is effective for... The update operation is as follows: (15) in, and Indicates by Convert to Transformation matrix and vector, and The deviation vector and deviation amount represent the step size. Represents sparse signals The corresponding step size.

[0033] Based on equations (14) and (15), the proposed NA-ALePOM method's first... k +1 level operation can be represented as: (16) In order to further reduce the number of network parameters and facilitate network training, sparse signals are used. Corresponding sparse penalty term parameters According to equation (10), it is set as follows: .

[0034] In summary, Figure 1 The network structure diagram of the proposed NA-ALePOM method is given, where the upper part represents the network structure of ALePOM, and the network parameters of its different layers are shown. , and It is calculated from the output of the LSTM unit.

[0035] 2.2 Network Training The proposed NA-ALPePOM network parameters first need to be learned using a training dataset, and then can be applied to practical sparse recovery problems. Specifically, given a training set... and network layers K All parameters of NA-ALePOM, i.e., network parameters (in The LSTM element parameters (including weight parameters and bias parameters) are optimized by solving the following problem through backpropagation: (17) in, Indicates that the input is The initial value is NA-AeLPOM K Layer output.

[0036] Finally, given the actual observed signal, the learned parameters are used... The sparse signal can be estimated through a single forward propagation of NA-ALPOM.

[0037] 3. Simulation Verification This section uses simulation experiments to compare and verify the performance and advantages of the proposed method. All experiments use normalized mean square error (NMSE) in decibels (dB) to evaluate the performance of different methods, defined as: (18) in, This represents an estimated value. Represents the actual value.

[0038] 3.1 Recovery of Sparse Signals from Random Observations First, the sparse signal recovery performance of the method of the present invention under random observation conditions is verified. This is to generate observation data. Observation matrix The element values ​​satisfy a random Gaussian distribution. The L2 norm of each column is set to 1, resulting in a sparse signal. The probability that each element is non-zero is 0.1, and each non-zero element value follows a standard normal distribution. To train the network, the training dataset consists of 10,000 pairs of sparse signals and their corresponding observations, while the test set consists of 1,000 pairs of sparse signals and their corresponding observations. All observations... s All are based on the same observation matrix and random sparse signals produce.

[0039] Given M = 250, N = 500, number of network layers K = 14, LSTM unit size P = 64, the trained NA-ALePOM algorithm was tested using a test set, and the step size and threshold calculated by NA-ALePOM for each sample in each layer were recorded. The results are as follows. Figure 4 As shown. By Figure 4 As shown in (a), NA-ALePOM can achieve good convergence with the increase of training iterations, and the training and testing NMSEs both reach below -100 dB. Figure 4 (b) shows that the step size and threshold parameters of NA-ALePOM vary within a certain range, and the threshold gradually approaches 0 as the number of network layers increases, indicating that the values ​​of the iteration parameters are related to the sparse signal to be recovered, rather than remaining fixed in ALePOM. Figure 4 As shown in (c), in different network layers, and The parameters are all proportional, indicating that the NA-ALePOM network can adaptively learn the parameters that satisfy equation (10) without the upper limit in equation (12), thus having a more compact upper limit for the recovery error than ALePOM.

[0040] Furthermore, Figure 5 The performance comparison of NA-ALePOM with existing typical deep unfolded networks ALISTA, ALePOM, and NA-ALISTA under different conditions is presented. Figure 5(a) shows the performance comparison of different methods in the absence of noise (first, four networks with different numbers of network layers were trained based on the aforementioned training dataset, and then their NMSE on the aforementioned test dataset was calculated). Figure 5 (b) shows the performance comparison of different methods on the test dataset with a signal-to-noise ratio (SNR) of 0–65 dB (network layers are fixed). K = 8. All four networks are trained on the aforementioned noiseless training dataset. 1000 sparse signals are randomly generated for each SNR, and corresponding noisy measurement data are generated. Figure 5 (c) represents the differences in signal sparsity between different methods. s Performance comparison at values ​​from 5 to 100 (assuming the measurement data does not contain noise). Figure 5 It can be seen that: (1) Without noise, the NMSE of NA-ALePOM decreases proportionally with the increase of the number of network layers, and is always less than that of other comparative networks; (2) For noisy data, when the SNR is less than 20 dB, the performance of NA-ALePOM is similar to that of ALISTA, ALePOM and NA-ALISTA. When the SNR is greater than 30 dB, the NMSE of NA-ALePOM decreases by about 30 dB, 10 dB and 20 dB compared with the comparative methods, and its performance is close to that of the noise-free case; (3) As the number of non-zero elements of the signal increases, the NMSE of all deep unfolded networks gradually increases, but the NMSE of NA-ALePOM is always less than that of other comparative methods, indicating that it has a strong adaptability to signal sparsity.

[0041] 3.2 Harmonic Signal Sparse Recovery Here, the performance of the proposed NA-ALePOM on the one-dimensional sparse harmonic recovery problem is tested, which has wide applications in practice.

[0042] Usually by s A time-domain sampled signal composed of superimposed sparse harmonic signals can be represented as: (19) in, and They represent the first i The complex amplitude and frequency of each harmonic. From a set Selected includes M A subset of elements, where N is the number of grids used to divide the frequency range to be estimated.

[0043] Equation (19) can be expressed in vector form. The observation matrix Given a partial Fourier matrix, its mn-th element... for: (20) Signal x * In position Nf i There are non-zero terms at position +1. The values ​​of these non-zero terms are as follows: (in i = 1, 2, ..., s Therefore, if the number of sine waves... s much smaller N Therefore, sparse recovery techniques can be used to analyze sparse signals. x * It is accurately estimated in the measurement. In this way, the frequency and amplitude of the sine wave can be obtained, a process commonly known as harmonic recovery.

[0044] In order to apply the proposed NA-ALePOM method and other typical learning methods to perform one-dimensional harmonic recovery in this experiment, a specific observation matrix was set according to formula (19). (in M = 128, N = 256), and generated various harmonic signals with different frequencies and complex amplitudes, thus constructing a training dataset. Specifically, it was set according to a Bernoulli distribution with a probability of 0.1. x * The non-zero terms in the equation are obtained by independent and identically distributed sampling from a complex Gaussian distribution with a mean of 0 and a standard deviation of 1.

[0045] Given LSTM cell size P The one-dimensional harmonic recovery performance of different methods is 64. Figure 6 As shown. Among them, Figure 6 (a) gives a K = 8 and SNR = 20dB recovery example; Figure 6 Figures (b) and (c) show the relationship between the average NMSE of 1000 test sparse signals and the number of network layers and the signal-to-noise ratio, respectively. The results show that: (1) the proposed method can accurately recover the frequency and amplitude of a sine wave using sparse measurements; and (2) the proposed method consistently outperforms the comparative methods.

[0046] The present invention also provides a non-transitory computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the learning-based sparse recovery method based on neural reinforcement as described above.

[0047] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.

[0048] Specific examples have been used to illustrate the principles and implementation methods of this invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of this invention. Furthermore, those skilled in the art will recognize that, based on the ideas of this invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of this invention.

Claims

1. A learning-based sparse recovery method based on neural reinforcement, characterized in that, Includes the following steps: S1. Introduce Long Short-Term Memory (LSTM) units into each layer of the ALePOM network to construct an NA-ALePOM network; where LSTM is used to adaptively calculate network parameters based on the estimation results of the previous layer and the input data of the current layer. S2. The network parameters of the NA-ALePOM network are trained and optimized based on the training dataset. During the training process, the network parameters are adaptively calculated and combined with the near-end operator to update the signal estimation and output a sparse signal. S3. Using the trained NA-ALePOM network, the actual observed signal is processed through a single forward propagation to output the estimation result of the sparse signal, thereby achieving sparse recovery.

2. The learning-based sparse recovery method based on neural reinforcement according to claim 1, characterized in that, In S1, the adaptively calculated network parameters include threshold parameters, step size parameters, and sparse penalty term parameters.

3. The learning-based sparse recovery method based on neural reinforcement according to claim 1, characterized in that, In S1, the ALePOM network will Represented as learnable parameters The product of the constant matrix B, where matrix B is independent of the number of network layers k and is related to the constant matrix B. The generalized correlation is relatively small, and the operation of the (k+1)th network layer of the ALePOM network is represented as follows: (8) In the formula, Denotes the proximal operator, x (k) For the first k The sparse signal estimate of the layer; B is a constant matrix independent of the number of network layers; y is the given observation data; Indicates the first k The step size for each iteration; , M < N , is the measurement matrix, , Indicates conjugate transpose; The constant matrix B is obtained by solving the following optimization problem: (9) In the formula, This represents the i-th row of B. express The i-th column, , express Zhongyou Definite A matrix composed of columns in the matrix.

4. The learning-based sparse recovery method based on neural reinforcement according to claim 3, characterized in that, The proximal operator uses the minimum maximum concavity penalty function as the SIP, denoted as: ; (2) The proximal operator based on the POM algorithm is represented as: (4) Equation (4) can be written as: (5) in, express z The i One element, Let the sign function be denoted by . , , , The representation of the proximal operator is obtained. .

5. The learning-based sparse recovery method based on neural reinforcement according to claim 4, characterized in that, In step S1, the constructed NA-ALePOM network uses LSTM to approximate the results of the first few layers of the network. Integrating into each sparse signal sparse recovery error In the estimation, the specific calculation is expressed as follows: (14) in, This represents the state of the k-th LSTM unit. This represents the hidden state of the k-th LSTM unit. P Indicates the size of the LSTM cell. , , Indicates by Switch to Transformation matrix and transformation vector, and Represents the deviation vector and the deviation. Represents sparse signals The corresponding threshold; The LSTM adaptively calculates the step size parameter of the k-th layer. The update operation expression is: (15) in, and Represents the state of the k-th LSTM unit cell. Convert to step size parameter Transformation matrix and vector, and This represents the deviation vector and deviation amount of the step size parameter. Represents sparse signals The corresponding step size; sparse signal Corresponding sparse penalty term parameters Set as .

6. The learning-based sparse recovery method based on neural reinforcement according to claim 5, characterized in that, In S1, the sparse signal estimation expression of the (k+1)th layer of the constructed NA-ALePOM is as follows: (16) In the formula, x (k) For the first k Sparse signal estimates for the layer; B is a constant matrix; y For observation vectors; , M < N , is the measurement matrix; This represents the proximal operator.

7. The learning-based sparse recovery method based on neural reinforcement according to claim 6, characterized in that, In S2, the training and optimization of network parameters specifically includes: Given a training set and network layers K All parameters of NA-ALePOM, i.e., network parameters ,in, The LSTM cell parameters, including weight parameters and bias parameters, are optimized by solving the following problem through backpropagation: (17) in, Indicates that the input is The initial value is NA-AeLPOM k Layer output.

8. The learning-based sparse recovery method based on neural reinforcement according to claim 7, characterized in that, In step S3, given the actual observed signal, the learned parameters are used... The sparse signal is estimated through a single forward propagation of NA-ALPOM.

9. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements a learning-based sparse recovery method based on neural reinforcement as described in any one of claims 1 to 8.

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