Low-rank autoregressive electromagnetic spectrum tensor completion method and device

Abstract

The present invention relates to the field of electromagnetic signal completion, and specifically discloses a low-rank autoregressive electromagnetic spectrum tensor completion method and device. It includes obtaining incomplete electromagnetic spectrum data, constructing a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determining unknown pixel points, performing a translation-invariant wavelet transform, and using the coefficients after the wavelet transform to construct a wavelet coefficient tensor, using the wavelet coefficient tensor to replace data, and introducing a local consistency constraint of time to obtain a low-rank autoregressive tensor completion model, using the ADMM algorithm to solve the low-rank autoregressive tensor completion model, obtaining updated auxiliary variables, updated unknown variables and updated recovery tensors, judging whether the convergence conditions are met or the preset number of iterations is reached, and if so, outputting the completed electromagnetic spectrum data. It is intended to solve the problem of being able to effectively complete electromagnetic spectrum data even when there is a large amount of missing data and a large number of continuous missing time dimensions.

Description

Technical Field

[0001] The present invention relates to the field of electromagnetic signal completion, and in particular to a low-rank autoregressive electromagnetic spectrum tensor completion method and device. Background Art

[0002] The spectrum data of electromagnetic radiation sources has important research value in monitoring the quality of electromagnetic environment and ensuring electromagnetic safety of people's livelihood. The collection of spectrum data is a prerequisite for studying electromagnetic information. However, in actual environments, low-cost spectrum sensors are prone to reading loss due to unexpected hardware failures or communication interruptions. During the collection process, they are also likely to be affected by harsh environments and mixed with noise. At the same time, in order to save collection and storage costs, lower sampling rates or restrictions on the number of sensors can also lead to missing spectrum data. Incomplete electromagnetic data significantly reduces the availability of electromagnetic information.

[0003] Regarding data completion, the first method that appeared was the matrix form that is easy to calculate and process. However, the matrix cannot effectively preserve the multilinear structure of high-order data sets. As an extension of matrices to higher orders, tensor-based completion algorithms have begun to receive widespread attention. According to existing literature, they mainly include methods based on tensor decomposition and low-rank tensor completion based on rank minimization without decomposition structure.

[0004] The method based on tensor decomposition can be transformed into the following optimization problem:

[0005]

[0006] strank(X)≤R

[0007] Where T is the observation tensor, X is the recovery tensor, and Ω represents the index set of observation entries. Represents the rank of a tensor, R is the bound of the rank of a given low-rank tensor X. The differences between different tensor decomposition methods are mainly concentrated in the definition of rank, using the ranks obtained by different decomposition methods to replace the rank of tensor completion, such as CP decomposition, Tucker decomposition, tensor chain decomposition, tensor ring decomposition, etc.

[0008] In practice, tensor rank bounds may not be available in some applications. When there are few observations, the choice of a high rank bound may lead to overfitting. To avoid this, another approach is to borrow the idea of ​​nuclear norm minimization in the matrix completion model and directly minimize the rank of the tensor. For the low-rank tensor completion problem, the general model for directly minimizing the tensor rank can be expressed as:

[0009] rank(X)

[0010] sf X Ω =TΩ

[0011] There are many optimization models for the tensor completion problem, depending on the definition of tensor rank. is a non-convex function, so the optimization problem becomes an NP-hard problem. In different methods, the tensor rank used can be CP rank, Tucker rank, TT rank, TR rank, etc.

[0012] However, so far, most of the research on spectrum data recovery and completion comes from the migration of methods in other directions, and the research on the regularity of spectrum signals themselves is not thorough. At the same time, there are few applications of data recovery under different data missing conditions, especially in some extreme missing conditions (such as too large missing ratio, or more continuous missing). Traditional tensor completion methods lack research on the laws of electromagnetic signals and are not effective under extreme missing conditions. Summary of the invention

[0013] In view of the above problems, an object of the present invention is to provide a low-rank autoregressive electromagnetic spectrum tensor completion method.

[0014] The second object of the present invention is to provide a low-rank autoregressive electromagnetic spectrum tensor completion device.

[0015] The first technical solution adopted by the present invention is: S1, obtaining incomplete electromagnetic spectrum data;

[0016] S2. construct a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determine unknown pixels, and obtain an observation set;

[0017] S3, performing a translation-invariant wavelet transform according to the observation set, and constructing a wavelet coefficient tensor using the coefficients after the wavelet transform;

[0018] S4. Using the low-rank tensor completion model based on rank minimization as a framework, using the wavelet coefficient tensor for data replacement, and introducing a local consistency constraint of time, a low-rank autoregressive tensor completion model is obtained;

[0019] S5. Solve the low-rank autoregressive tensor completion model using the ADMM algorithm to obtain updated auxiliary variables, updated unknown variables, and updated recovery tensors;

[0020] S6. Determine whether the updated auxiliary variables, updated unknown variables and updated recovery tensors meet the convergence conditions or reach the preset number of iterations. If so, output the completed electromagnetic spectrum data.

[0021] Optionally, S2 includes:

[0022] S21. Construct a data model of high-dimensional tensors;

[0023] S22, using the data model of the high-dimensional tensor to determine the observation period and perform observations to obtain an observation matrix;

[0024] S23, aligning and stacking the observation matrices of different periods to obtain a three-dimensional tensor of time slot-frequency-time period;

[0025] S24. Use the time slot-frequency-time period three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determine the unknown pixel points, and obtain the observation set.

[0026] Optionally, the S3 includes:

[0027] S31, performing a translation invariant wavelet transform according to the observation set to obtain an approximation coefficient matrix and a detail coefficient matrix;

[0028] S32. Stack the approximation coefficient matrix and the detail coefficient matrix to obtain the wavelet coefficient tensor.

[0029] Optionally, the local consistency constraint of the time is,

[0030]

[0031] In the formula, ||Z|| A,H is the total square error of the quantized time series, Z is the time series matrix, z m,t is the (m, t)th data in Z, For z m,t Delay i data, a m,t For The corresponding coefficients, H = {h1,h2,...h d} is a delay set.

[0032] Optionally, the low-rank autoregressive tensor completion model is,

[0033]

[0034] In the formula, X is the original real data, A is the coefficient matrix, and D is the wavelet coefficient tensor. is the constraint for minimizing the tensor rank, Ω is the auxiliary variable, λ is the weight parameter, S(X) is the tensor quantization coefficient, Q(Z) is the tensor quantization matrix, P Ω (Z) is the result of observing the original data matrix, P Ω (Y) is the result of observing the observation matrix, and r is the truncation coefficient.

[0035] The second technical solution adopted by the present invention is: a low-rank autoregressive electromagnetic spectrum tensor completion device, comprising: a data acquisition module, used to acquire incomplete electromagnetic spectrum data;

[0036] A characterization module, used for constructing a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determine unknown pixels, and obtain an observation set;

[0037] A transformation module, used for performing a translation-invariant wavelet transform according to the observation set, and constructing a wavelet coefficient tensor using the coefficients after the wavelet transform;

[0038] A model building module, which is used to use the low-rank tensor completion model based on rank minimization as a framework, use the wavelet coefficient tensor for data replacement, and introduce a local consistency constraint of time to obtain a low-rank autoregressive tensor completion model;

[0039] A solution module, used for solving the low-rank autoregressive tensor completion model by using an ADMM algorithm to obtain updated auxiliary variables, updated unknown variables and updated recovery tensors;

[0040] The judgment module is used to judge whether the updated auxiliary variables, updated unknown variables and updated recovery tensors meet the convergence conditions or reach the preset number of iterations. If so, the completed electromagnetic spectrum data is output.

[0041] Optionally, the characterization module includes:

[0042] Model building submodule, building high-dimensional tensor data model;

[0043] An observation submodule, using the data model of the high-dimensional tensor to determine the observation period and perform observations to obtain an observation matrix;

[0044] A stacking submodule aligns and stacks the observation matrices of different periods to obtain a three-dimensional tensor of time slot-frequency-time period;

[0045] The characterization submodule uses the time slot-frequency-time period three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determine the unknown pixel points, and obtain the observation set.

[0046] Optionally, the transformation module includes:

[0047] A transformation submodule performs a translation-invariant wavelet transform according to the observation set to obtain an approximation coefficient matrix and a detail coefficient matrix;

[0048] The matrix stacking submodule stacks the approximation coefficient matrix and the detail coefficient matrix to obtain the wavelet coefficient tensor.

[0049] Optionally, the local consistency constraint of the time is,

[0050]

[0051] In the formula, ||Z|| A,H is the total square error of the quantized time series, Z is the time series matrix, z m,t is the (m, t)th data in Z, For z m,t Delay i data, a m,t For The corresponding coefficients, H = {h1,h2,...h d} is a delay set.

[0052] Optionally, the low-rank autoregressive tensor completion model is,

[0053]

[0054] In the formula, X is the original real data, A is the coefficient matrix, and D is the wavelet coefficient tensor. is the constraint for minimizing the tensor rank, Ω is the auxiliary variable, λ is the weight parameter, S(X) is the tensor quantization coefficient, Q(Z) is the tensor quantization matrix, P Ω (Z) is the result of observing the original data matrix, P Ω (Y) is the result of observing the observation matrix, and r is the truncation coefficient.

[0055] Beneficial effects of the above technical solution:

[0056] The present invention acquires incomplete electromagnetic spectrum data, constructs a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determines unknown pixels, performs translation-invariant wavelet transform, and uses the coefficients after wavelet transform to construct a wavelet coefficient tensor, uses the wavelet coefficient tensor to replace data and introduces local consistency constraints of time, obtains a low-rank autoregressive tensor completion model, uses the ADMM algorithm to solve the low-rank autoregressive tensor completion model, obtains updated auxiliary variables, updated unknown variables and updated recovery tensors, and determines whether the convergence conditions are met or the preset number of iterations is reached. If so, the completed electromagnetic spectrum data is output. The purpose is to solve the problem of large amount of missing data and continuous missing of time dimension, and still be able to effectively complete the electromagnetic spectrum data. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 A flow chart of the low-rank autoregressive electromagnetic spectrum tensor completion method provided by the present invention;

[0058] Figure 2 A schematic diagram of the structure of a low-rank autoregressive electromagnetic spectrum tensor completion device provided by the present invention;

[0059] Figure 3 Schematic diagram of the electromagnetic data tensorization process;

[0060] Figure 4 Schematic diagram of the wavelet coefficient tensorization process. DETAILED DESCRIPTION

[0061] The following detailed description of the embodiments of the present invention is further described in detail in conjunction with the accompanying drawings and examples. The detailed description of the following embodiments and the accompanying drawings are used to exemplarily illustrate the principles of the present invention, but cannot be used to limit the scope of the present invention, that is, the present invention is not limited to the preferred embodiments described, and the scope of the present invention is defined by the claims.

[0062] In the description of the present invention, it should be noted that, unless otherwise specified, “plurality” means two or more than two; the terms “first”, “second”, etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance; for ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to the specific circumstances.

[0063] Embodiment 1

[0064] An embodiment of the present invention provides a low-rank autoregressive electromagnetic spectrum tensor completion method, such as Figure 1 As shown, S1, obtaining incomplete electromagnetic spectrum data; S2, constructing a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determining unknown pixels, and obtaining an observation set; S3, performing a translation-invariant wavelet transform according to the observation set, and constructing a wavelet coefficient tensor using the coefficients after the wavelet transform; S4, using a low-rank tensor completion model based on rank minimization as a framework, using the wavelet coefficient tensor to perform data replacement and introducing a local consistency constraint in time to obtain a low-rank autoregressive tensor completion model; S5, using the ADMM algorithm to solve the low-rank autoregressive tensor completion model, to obtain updated auxiliary variables, updated unknown variables, and updated recovery tensors; S6, judging whether the updated auxiliary variables, the updated unknown variables, and the updated recovery tensor meet the convergence conditions or reach the preset number of iterations, and if so, outputting the completed electromagnetic spectrum data.

[0065] S1. Obtain incomplete electromagnetic spectrum data;

[0066] S2, constructing a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determining unknown pixels, and obtaining an observation set Ω;

[0067] X represents the real electromagnetic spectrum data on N frequency bands collected at M time nodes, with rows corresponding to time points and columns corresponding to frequency bands. The expression is:

[0068]

[0069] Where IJ=M, indicating that M time nodes are divided into J time intervals containing I time nodes, and R is a real number set.

[0070] The observation matrix can be expressed as P Ω (X) means:

[0071]

[0072] Where Ω represents the observation set, m = 1, 2, ..., IJ, n = 1, 2, ..., N.

[0073] The signal spectra observed in continuous time are often correlated and show low-rank characteristics in long-term observations. Spectral data naturally has a high-dimensional tensor structure, and the two-dimensional matrix can be refolded into a three-dimensional tensor form. Based on the two-dimensional observation matrix in the time-frequency domain, a high-dimensional tensor data model can be constructed to represent larger-scale historical spectrum data. The observation matrices of different periods are aligned and stacked to construct a time slot-frequency-time period three-dimensional tensor X∈R I×J×N ,like Figure 3 shown.

[0074] S21. Construct a data model of high-dimensional tensors;

[0075] S22, using the data model of the high-dimensional tensor to determine the observation period and perform observations to obtain an observation matrix;

[0076] S23, aligning and stacking the observation matrices of different periods to obtain a three-dimensional tensor of time slot-frequency-time period;

[0077] S24. Use the time slot-frequency-time period three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determine the unknown pixel points, and obtain the observation set.

[0078] S3. Perform a translation-invariant wavelet transform according to the observation set, and construct a wavelet coefficient tensor using the coefficients after the wavelet transform.

[0079] It should be noted that when using SIDWT (translation-invariant discrete wavelet) to decompose and reconstruct the signal, sampling and interpolation are not required. Therefore, it can not only ensure translation invariance but also reduce the complexity of calculation, which is an improvement on the discrete wavelet framework. According to the definition of SIDWT, the original observation matrix will obtain 1 approximation coefficient and 3S detail coefficients after S-level wavelet transform, and the size of all coefficient matrices is the same as the input matrix.

[0080] The (3S+1) coefficient matrices after the translation-invariant wavelet transform are stacked to form a wavelet coefficient tensor. The specific process is as follows: Figure 4 shown.

[0081] S31, performing a translation invariant wavelet transform according to the observation set to obtain an approximation coefficient matrix and a detail coefficient matrix;

[0082] S32. Stack the approximation coefficient matrix and the detail coefficient matrix to obtain the wavelet coefficient tensor.

[0083] S4. Using the low-rank tensor completion model based on rank minimization as a framework, the wavelet coefficient tensor is used for data replacement, and a local consistency constraint of time is introduced to obtain a low-rank autoregressive tensor completion model.

[0084] Taking the low-rank tensor completion model based on rank minimization as the framework, the wavelet coefficient tensor is used for replacement, the autoregressive model AR(p) is introduced as the local consistency constraint in time, and a low-rank autoregressive tensor completion model based on translation invariant wavelet transform is constructed.

[0085] Define a temporal local consistency constraint to capture short-term trends in the EM data:

[0086]

[0087] In the formula, ||Z|| A,H is the total square error of the quantized time series, Z is the time series matrix, z m,t is the (m, t)th data in Z, For z m,t Delay i data, a m,t For The corresponding coefficients, whose set can be written as the coefficient matrix A∈R M×d ,H={h1,h2,...h d} is a delay set.

[0088] ||Z|| A,H By m The autoregressive model is used to use the coefficient a m The total square error is quantified, and the time variation can be minimized by determining the coefficient matrix A, so that the completed data has better time consistency.

[0089] Completion methods based on rank minimization usually use the nuclear norm of the tensor |X|| * or the truncated nuclear norm X|| r,* An approximation is made to minimize the tensor rank, where r is the cutoff parameter.

[0090] By integrating the rank minimization model and the temporal local consistency constraint, the low-rank autoregressive tensor completion model is proposed as

[0091]

[0092] where \(X\) is the original true data, \(A\) is the coefficient matrix, \(D\) is the wavelet coefficient tensor, is the constraint of tensor rank minimization, \(W\) is the auxiliary variable, \(\lambda\) is the weight parameter used to adjust the proportion of the truncated nuclear norm constraint and the temporal consistency constraint, \(S(X)\) is the tensorized coefficient, \(Q(Z)\) is the tensorized matrix, \(P\) Ω (Z) is the result of observing the original data matrix, \(P\) Ω (Y) is the result of observing the observation matrix, and \(r\) is the truncation coefficient.

[0093] \(Y\in\mathbb{R}\) (IJ×N) is the observed partial matrix, \(r\in\mathbb{N}\) + is the truncation coefficient, satisfying \(r < \min\{I, J, N\}\). In the constraint conditions, \(D = W\) ensures the correlation between \(D\) and the wavelet coefficients of \(X\), and \(X = Q(Z)\) ensures the correlation between \(X\) and the partial observation matrix \(Y\).

[0094] S5. Use the ADMM algorithm to solve the low-rank autoregressive tensor completion model to obtain the updated auxiliary variable, the updated unknown variable, and the updated recovered tensor.

[0095] The ADMM algorithm is often used to solve the tensor completion problem based on nuclear norm minimization. However, due to the introduction of \(A\) in the model, the traditional ADMM algorithm cannot be directly applied. Therefore, in this method, the minimization problem is decomposed into the solution of two sub-problems:

[0096] 1) Fix \(A\) k , and update \(D\) k+1 , \(Z\) k+1 :

[0097]

[0098] where \(k\) represents the iteration number of the algorithm running.

[0099] 2) Fix \(D\) k+1 , \(Z\) k+1 , and update \(A\) k+1 :

[0100]

[0101] Starting from the initialized \((D\) 0 , \(Z\) 0 , \(A\) 0 ), through the iterative solution of the two sub-problems, achieve Updates.

[0102] When A k When fixed, D k+1 ,Z k+1 The solution is transformed into a general low-rank tensor problem, and the augmented Lagrangian function is as follows:

[0103]

[0104] Among them, β, ρ are the weights of each penalty term, ε∈R (3S+1)×(LJ)×N and T∈R I×J×N represents the Lagrange multiplier. It should be noted that P Ω (Z) = P Ω (Y) and W = S (X) are used as fixed constraints to maintain the accuracy of the observation part and the accuracy of the wavelet coefficients, which will be updated by assignment during iteration. According to the augmented Larangrangian function, the ADMM solution can be decomposed into the following sub-problems:

[0105]

[0106] W k+1 :=S(X k+1 )

[0107] ε k+1 :=ε k +β(D k+1 -W k+1 )

[0108] T k+1 :=T k +ρ(X k+1 -Q(Z k+1 ))

[0109] Step 1: Update D

[0110] The optimization problem of D is a truncated nuclear norm minimization problem, and the truncated nuclear norm of any known tensor is the weighted sum of the truncated nuclear norm of the tensor expansion matrix:

[0111]

[0112] in, Here, D1, D2, and D3 are defined as the three modal expansion matrices of D. Then, each D p The solution can be written as:

[0113]

[0114] Among them, fold (k)(·) indicates that the matrix is ​​folded into a tensor according to the modulus k, and D(·) represents the wide-area singular value threshold operator related to the minimization of the truncated nuclear norm. Finally, D can be updated by the following formula:

[0115]

[0116] Step 2: Update Z

[0117] The solution for Z can be written as:

[0118]

[0119] Among them, Q -1 (·) represents the inverse operator of Q(·).

[0120] In order to solve Z∈R M×T , any z m ∈R T Write:

[0121] in,

[0122]

[0123] but:

[0124]

[0125] Therefore, z m The least squares solution is:

[0126]

[0127] in

[0128] Step 3: Update X

[0129] The solution for X can be written as:

[0130]

[0131] Based on this, the least squares solution of X can be written as:

[0132]

[0133] Step 4: Update A

[0134] A∈R M×d is the coefficient matrix in the autoregressive process, and we can derive:

[0135]

[0136] in so:

[0137]

[0138] in, * represents the pseudo-inverse.

[0139] S6. Determine whether the updated auxiliary variables, the updated unknown variables and the updated recovery tensor meet the convergence conditions or reach the preset number of iterations. If so, output the completed electromagnetic spectrum data.

[0140] If not satisfied, return to S5 to continue iteration.

[0141] The present embodiment is a low-rank autoregressive electromagnetic spectrum tensor completion method, which includes obtaining incomplete electromagnetic spectrum data, constructing a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determining unknown pixels, and obtaining an observation set. According to the observation set, a translation-invariant wavelet transform is performed, and a wavelet coefficient tensor is constructed using the coefficients after the wavelet transform. A low-rank tensor completion model based on rank minimization is used as a framework, and the wavelet coefficient tensor is used to perform data replacement and introduce a local consistency constraint in time to obtain a low-rank autoregressive tensor completion model. The low-rank autoregressive tensor completion model is solved using an ADMM algorithm to obtain updated auxiliary variables, updated unknown variables, and updated recovery tensors; it is determined whether the updated auxiliary variables, the updated unknown variables, and the updated recovery tensor meet convergence conditions or reach a preset number of iterations. If so, the completed electromagnetic spectrum data is output. The present invention takes into account the low rank of the wavelet coefficients of the electromagnetic signal and adopts the wavelet coefficient tensor as a low rank constraint to improve the effect of tensor completion; it adopts the local consistency of time as another constraint condition, and can still achieve effective completion of electromagnetic spectrum data when the amount of missing data is large and there are many continuous missing data in the time dimension.

[0142] Embodiment 2

[0143] An embodiment of the present invention provides a low-rank autoregressive electromagnetic spectrum tensor completion device 200, such as Figure 2 As shown, including:

[0144] A data acquisition module 201 is used to acquire incomplete electromagnetic spectrum data;

[0145] A characterization module 202, for constructing a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determine unknown pixels, and obtain an observation set;

[0146] A transformation module 203, configured to perform a translation-invariant wavelet transform according to the observation set, and construct a wavelet coefficient tensor using the coefficients after the wavelet transform;

[0147] A model building module 204 is used to use the low-rank tensor completion model based on rank minimization as a framework, use the wavelet coefficient tensor to perform data replacement and introduce a local consistency constraint of time to obtain a low-rank autoregressive tensor completion model;

[0148] A solution module 205 is used to solve the low-rank autoregressive tensor completion model using an ADMM algorithm to obtain updated auxiliary variables, updated unknown variables and updated recovery tensors;

[0149] The judgment module 206 is used to judge whether the updated auxiliary variables, the updated unknown variables and the updated recovery tensor meet the convergence conditions or reach the preset number of iterations. If so, the completed electromagnetic spectrum data is output.

[0150] Furthermore, the characterization module 202 includes:

[0151] A model building submodule 2021 is used to build a data model of a high-dimensional tensor;

[0152] An observation submodule 2022 is used to determine an observation period and perform observation using the data model of the high-dimensional tensor to obtain an observation matrix;

[0153] A stacking submodule 2023 is used to align and stack the observation matrices of different periods to obtain a time slot-frequency-time period three-dimensional tensor;

[0154] The characterization submodule 2024 is used to characterize by using the time slot-frequency-time period three-dimensional tensor to determine the unknown pixel point and obtain the observation set.

[0155] Furthermore, the transformation module 203 includes:

[0156] The transformation submodule 2031 is used to perform a translation invariant wavelet transform according to the observation set, to obtain an approximation coefficient matrix and a detail coefficient matrix;

[0157] The matrix stacking submodule 2032 is used to stack the approximation coefficient matrix and the detail coefficient matrix to obtain the wavelet coefficient tensor.

[0158] Furthermore, the local consistency constraint of the time is:

[0159]

[0160] In the formula, ||Z|| A,H is the total square error of the quantized time series, Z is the time series matrix, z m,t is the (m, t)th data in Z, For z m,t Delay i data, am,t For The corresponding coefficients, H = {h1,h2,...h d} is a delay set.

[0161] Furthermore, the low-rank autoregressive tensor completion model is:

[0162]

[0163] In the formula, X is the original real data, A is the coefficient matrix, and D is the wavelet coefficient tensor. is the constraint for minimizing the tensor rank, Ω is the auxiliary variable, λ is the weight parameter, S(X) is the tensor quantization coefficient, Q(Z) is the tensor quantization matrix, P Ω (Z) is the result of observing the original data matrix, P Ω (Y) is the result of observing the observation matrix, and r is the truncation coefficient.

[0164] The present embodiment is a low-rank autoregressive electromagnetic spectrum tensor completion device, which is used to obtain incomplete electromagnetic spectrum data through a data acquisition module; a characterization module is used to construct a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determine unknown pixels, and obtain an observation set; a transformation module is used to perform a translation-invariant wavelet transform according to the observation set, and construct a wavelet coefficient tensor using the coefficients after the wavelet transform; a model construction module is used to use a low-rank tensor completion model based on rank minimization as a framework, use the wavelet coefficient tensor to perform data replacement and introduce a local consistency constraint in time to obtain a low-rank autoregressive tensor completion model; a solution module is used to solve the low-rank autoregressive tensor completion model using an ADMM algorithm to obtain updated auxiliary variables, updated unknown variables, and updated recovery tensors; a judgment module is used to judge whether the updated auxiliary variables, the updated unknown variables, and the updated recovery tensor meet the convergence conditions or reach a preset number of iterations, and if so, output the completed electromagnetic spectrum data. The present invention takes into account the low rank of the wavelet coefficients of the electromagnetic signal and adopts the wavelet coefficient tensor as a low rank constraint to improve the effect of tensor completion; it adopts the local consistency of time as another constraint condition, and can still achieve effective completion of electromagnetic spectrum data when the amount of missing data is large and there are many continuous missing data in the time dimension.

[0165] The above is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art can easily think of changes or substitutions within the technical scope disclosed by the present invention, which should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A low-rank autoregressive electromagnetic spectrum tensor completion method, characterized in that: The following steps are involved: S1. Obtain incomplete electromagnetic spectrum data; S2. construct a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determine unknown pixels, and obtain an observation set; S3, performing a translation-invariant wavelet transform according to the observation set, and constructing a wavelet coefficient tensor using the coefficients after the wavelet transform; S4. Using the low-rank tensor completion model based on rank minimization as a framework, using the wavelet coefficient tensor for data replacement, and introducing a local consistency constraint of time, a low-rank autoregressive tensor completion model is obtained; S5. Solve the low-rank autoregressive tensor completion model using the ADMM algorithm to obtain updated auxiliary variables, updated unknown variables, and updated recovery tensors; S6, judging whether the updated auxiliary variables, the updated unknown variables and the updated recovery tensor meet the convergence condition or reach the preset number of iterations, and if so, outputting the completed electromagnetic spectrum data; The local consistency constraint of the time is, In the formula, is the total square error of the quantized time series, Z is the time series matrix, z m,t is the (m, t)th data in Z, For z m,t Delay i data, a m,t For The corresponding coefficients are, is a set of time delays; The low-rank autoregressive tensor completion model is: In the formula, X is the original real data, A is the coefficient matrix, and D is the wavelet coefficient tensor. is the constraint for minimizing the tensor rank, is an auxiliary variable, λ is a weight parameter, S(X) is a tensor quantization coefficient, Q(Z) is a tensor quantization matrix, is the result of observing the original data matrix, is the result of observing the observation matrix, and r is the truncation coefficient.

2. The low-rank autoregressive electromagnetic spectrum tensor completion method according to claim 1, characterized in that: The S2 includes: S21. Construct a data model of high-dimensional tensors; S22, using the data model of the high-dimensional tensor to determine the observation period and perform observations to obtain an observation matrix; S23, aligning and stacking the observation matrices of different periods to obtain a three-dimensional tensor of time slot-frequency-time period; S24. Use the time slot-frequency-time period three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determine the unknown pixel points, and obtain the observation set.

3. The low-rank autoregressive electromagnetic spectrum tensor completion method according to claim 1, characterized in that: The S3 includes: S31, performing a translation invariant wavelet transform according to the observation set to obtain an approximation coefficient matrix and a detail coefficient matrix; S32. Stack the approximation coefficient matrix and the detail coefficient matrix to obtain the wavelet coefficient tensor.

4. A low-rank autoregressive electromagnetic spectrum tensor completion device, characterized in that: include: A data acquisition module, used for acquiring incomplete electromagnetic spectrum data; A characterization module, used for constructing a three-dimensional tensor to characterize the incomplete electromagnetic spectrum data, determine unknown pixels, and obtain an observation set; A transformation module, used for performing a translation-invariant wavelet transform according to the observation set, and constructing a wavelet coefficient tensor using the coefficients after the wavelet transform; A model building module, which is used to use the low-rank tensor completion model based on rank minimization as a framework, use the wavelet coefficient tensor for data replacement, and introduce a local consistency constraint of time to obtain a low-rank autoregressive tensor completion model; A solution module, used for solving the low-rank autoregressive tensor completion model by using an ADMM algorithm to obtain updated auxiliary variables, updated unknown variables and updated recovery tensors; A judgment module, used to judge whether the updated auxiliary variables, the updated unknown variables and the updated recovery tensor meet the convergence conditions or reach the preset number of iterations, and if so, output the completed electromagnetic spectrum data; The local consistency constraint of the time is, In the formula, ||Z|| A,H is the total square error of the quantized time series, Z is the time series matrix, z m,t is the (m, t)th data in Z, For z m,t Delay i data, a m,t For The corresponding coefficients are, is a set of time delays; The low-rank autoregressive tensor completion model is: In the formula, X is the original real data, A is the coefficient matrix, and D is the wavelet coefficient tensor. is the constraint for minimizing the tensor rank, is an auxiliary variable, λ is a weight parameter, S(X) is a tensor quantization coefficient, Q(Z) is a tensor quantization matrix, is the result of observing the original data matrix, is the result of observing the observation matrix, and r is the truncation coefficient.

5. The low-rank autoregressive electromagnetic spectrum tensor completion device according to claim 4, characterized in that: The characterization module comprises: Model building submodule, used to build data models of high-dimensional tensors; An observation submodule, used to determine an observation period and perform observations using the data model of the high-dimensional tensor to obtain an observation matrix; A stacking submodule, used for aligning and stacking the observation matrices of different periods to obtain a time slot-frequency-time period three-dimensional tensor; The characterization submodule is used to characterize the incomplete electromagnetic spectrum data using the time slot-frequency-time period three-dimensional tensor, determine the unknown pixel points, and obtain the observation set.

6. The low-rank autoregressive electromagnetic spectrum tensor completion device according to claim 4, characterized in that: The transformation module comprises: A transformation submodule, used for performing a translation invariant wavelet transform according to the observation set to obtain an approximation coefficient matrix and a detail coefficient matrix; The matrix stacking submodule is used to stack the approximation coefficient matrix and the detail coefficient matrix to obtain the wavelet coefficient tensor.

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