Internet of Things Data Reconstruction Method Based on Structured Low-Rank Tensor Completion

Abstract

The present invention is an Internet of Things data reconstruction method based on structured low-rank tensor completion. First, the monitoring area is discretized into multiple grid points, and a sensor node is deployed inside each grid point. Assuming that the sensor node senses data every other time slot, the data received by the base station within time T forms a third-order tensor. Secondly, the data reconstruction is converted into a basic low-rank tensor completion problem, and a low-rank tensor completion model is constructed. Finally, the block Hankel matrix transformation is performed on the unfolding matrix of each mode i of the third-order tensor, and the basic low-rank tensor completion model is improved into a structured low-rank tensor completion model. The augmented Lagrangian function of the structured low-rank tensor completion model is solved to obtain the third-order tensor, and the Internet of Things data reconstruction is completed. The data collected at continuous moments are arranged in a third-order tensor to make full use of the spatial correlation of the data. The block Hankel matrix transformation is performed on the unfolding matrix of each mode i of the third-order tensor, and data reconstruction is carried out by combining structured and low-rank tensor completion, further exploring and utilizing the spatio-temporal correlation of the data, alleviating the influence of basis mismatch on the reconstruction performance in the sparse constraint-based method, and improving the data reconstruction accuracy.

Description

Technical Field

[0001] The present invention belongs to the technical field of Internet of Things (IoT) data processing, and specifically relates to an IoT data reconstruction method based on structured low-rank tensor completion. Background Art

[0002] As the third wave of the development of the information industry since the emergence of computers and the Internet, the Internet of Things (IoT) has been applied in numerous fields. The IoT device layer consists of a large number of sensor nodes with sensing and communication capabilities. The sensor nodes are randomly deployed in the monitoring area, and each sensor node has certain computing, storage, and communication capabilities to continuously monitor and sense environmental information. Due to factors such as hardware condition limitations, unstable network communication, and harsh environments, data loss inevitably occurs in IoT data. Data loss will cause IoT data to be unable to be normally used for subsequent analysis and applications. Therefore, high-precision reconstruction of complete IoT data has become a research hotspot in this field.

[0003] IoT data reconstruction methods based on sparse constraints can be divided into three categories: reconstruction methods based on Compressed Sensing (CS), reconstruction methods based on Matrix Completion (MC), and reconstruction methods based on tensor completion. Specifically, CS-based methods use a measurement matrix to compress and sample IoT data signals that are sparse in certain transform domains, and use optimization methods to reconstruct the sampled signals. IoT data can be arranged as a two-dimensional matrix according to sensor nodes and acquisition times. The spatio-temporal correlation of the data makes the IoT data matrix have low rank, meeting the prerequisite for matrix completion applications. Therefore, missing IoT data can be reconstructed based on matrix completion. CS- and MC-based methods generally arrange the data collected by spatially distributed sensor nodes in vector form, thus ignoring the spatial correlation between sensor nodes. As a higher-order extension of matrices, tensors can represent high-dimensional data in a more natural and compact way. Therefore, IoT data completion methods naturally extend from low-rank matrix completion to low-rank tensor completion. According to different tensor rank definitions, IoT data reconstruction methods based on various low-rank tensor completions have been successively proposed. Compared with CS- and MC-based reconstruction methods, tensor completion-based reconstruction methods can further explore and utilize the spatio-temporal correlation of IoT data.

[0004] The above first-order, second-order, and high-order sparse constraint methods have achieved very prominent results in the research of Internet of Things (IoT) data reconstruction. However, these methods all a priori assume that the signal is sparse. However, since the true frequency of IoT data is actually specified in the continuous domain, when there is inevitably a basis mismatch between the actual frequency and the assumed basis, it will lead to a decline in the reconstruction performance of the sparse feature-based constraint method. Therefore, this application proposes an IoT data reconstruction method based on structured low-rank tensor completion, further using the spatio-temporal correlation between IoT data to alleviate the impact of basis mismatch on the reconstruction accuracy, thereby improving the data reconstruction accuracy. Summary of the Invention

[0005] Aiming at the deficiencies of the prior art, the technical problem to be solved by the present invention is to propose an IoT data reconstruction method based on structured low-rank tensor completion.

[0006] The technical solution adopted by the present invention to solve the above technical problem is as follows:

[0007] An IoT data reconstruction method based on structured low-rank tensor completion, comprising the following steps:

[0008] Step 1: Divide the IoT monitoring area into M×N grid points, and deploy a sensor node inside each grid point; assume that the sensor node senses data every time slot τ and transmits the data to the base station. Therefore, the data received by the base station within time T = L×τ can form a third-order tensor represents the real number field, and M and N are positive integers;

[0009] Due to data loss, only D environmental information measurement values are transmitted to the base station within time T, D << M×N×L. The data received by the base station is passed through is represented by, represents the random sampling operator, represents the data randomly sampled from the third-order tensor contains D environmental information measurement values, and the positions of the unsampled points are filled with zeros; Ω represents the observation set;

[0010] Step 2: Since there is spatial correlation between the horizontal slice data and the side slice data of the third-order tensor composed of IoT data, and there is time correlation between the front slice data, so the tensor has low rank. Therefore, the data reconstruction can be converted into a low-rank tensor completion problem. The expression of the basic low-rank tensor completion model is:

[0011]

[0012] where α i represents the third-order tensor The weights of the nuclear norms of the mode-i unfolding matrices of, where i = 1, 2, 3, satisfy α i > 0 and X (i) represents a third-order tensor of the mode-i unfolding matrix, ||·|| * and ||·|| F represent the matrix nuclear norm and the F-norm respectively, and λ is the regularization parameter;

[0013] Step 3: To more effectively utilize the spatio-temporal correlation of the sensed information in the monitoring environment in IoT data, perform a block Hankel matrix transformation on each mode-i unfolding matrix of the third-order tensor and penalize the nuclear norm of the block Hankel matrix formed by the mode-i unfolding matrix X of the third-order tensor (i) Then, the basic low-rank tensor completion model in Equation (2) is improved to:

[0014]

[0015] where, is the operation operator for converting the matrix into a block Hankel matrix;

[0016] Convert the low-rank tensor completion model in Equation (3) into an equivalent constrained optimization problem in Equation (6) by introducing variable splitting;

[0017]

[0018] Solve the constrained equation (6) using the alternating direction method of multipliers. First, obtain the augmented Lagrangian function of the original objective function. Therefore, the augmented Lagrangian function of Equation (6) is expressed as:

[0019]

[0020] In the formula, β represents the penalty coefficient, β > 0, Ε (i) and both represent Lagrange multipliers;

[0021] Step 4: Solve Equation (7) to obtain the third-order tensor Complete the reconstruction of IoT data.

[0022] Furthermore, the alternating solution process of Equation (7) is:

[0023]

[0024]

[0025]

[0026]

[0027]

[0028]

[0029] Among them, k represents the number of iterations, represents the Lagrange multiplier unfolding matrix of mode i, and fold means restoring the unfolding matrix of mode i to a tensor;

[0030] The sub-problems in equations (8) and (9) can be expanded as:

[0031]

[0032]

[0033] Solve equations (14) and (15) through the conjugate gradient algorithm:

[0034]

[0035]

[0036] In the formula, H represents the Hermitian transpose, and I represents the identity matrix;

[0037] The sub-problem in equation (10) can be expanded as:

[0038]

[0039] Furthermore, solve equation (18) through singular value truncation operation to obtain:

[0040]

[0041] Let Then equation (19) is transformed into:

[0042]

[0043] In equation (20), shrink(A,τ) represents a non-linear function, and the specific operation is to apply the soft threshold operator to the singular values of matrix A; perform singular value decomposition on matrix A, perform soft threshold operator operation on the diagonal of the obtained non-negative real diagonal matrix, then put the soft thresholded vector back into the non-negative real diagonal matrix, and multiply the decomposed components to obtain the matrix after singular value truncation operation, that is, the output result of the non-linear function shrink(A,τ);

[0044] Soft threshold operator is defined as:

[0045]

[0046] where q j represents the j-th singular value of matrix A.

[0047] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0048] On the one hand, the method of the present invention arranges the data collected at consecutive moments in the form of a third-order tensor, making full use of the spatial correlation of the data collected by adjacent sensor nodes, which is beneficial to improving the reconstruction accuracy. On the other hand, in order to further improve the utilization of the spatio-temporal correlation of the data, the unfolding matrix of each mode i of the third-order tensor is respectively subjected to a block Hankel matrix transformation, and data reconstruction is carried out through structured processing and low-rank tensor completion, alleviating the negative impact of basis mismatch on the reconstruction performance of the sparse constraint-based method. The experimental comparison results show that, compared with the reconstruction methods based on structured matrix completion and basic low-rank tensor completion, the method of the present invention has smaller reconstruction errors and higher reconstruction accuracy under the same sampling ratio and different sampling ratios. Description of the Drawings

[0049] Figure 1 is a schematic diagram of the unfolding matrix of mode i of the third-order tensor ;

[0050] Figure 2 is a comparison chart of errors for reconstructing NDBC ocean surface temperature data using different methods;

[0051] Figure 3 is a comparison chart of errors for reconstructing Berkeley Lab temperature data using different methods. Detailed Embodiments

[0052] The technical solution of the present invention will be described in detail below in conjunction with the drawings and specific embodiments, but the protection scope of this application is not limited thereto.

[0053] The present invention is an Internet of Things data reconstruction method based on structured low-rank tensor completion (hereinafter referred to as the method, see Figures 1 to 3 ), and the method includes the following steps:

[0054] Step 1: Discretize the Internet of Things monitoring area into M×N grid points, and deploy a sensor node inside each grid point; assume that the sensor node senses data once every time slot τ (the time slot is the difference between adjacent sampling moments) and transmits the data to the base station; the data sensed by all sensor nodes at the same sampling moment form a matrix Therefore, the data received by the base station within time T = L×τ, that is, within consecutive L time slots, form a third-order tensor represents the real number field, and M and N are positive integers;

[0055] Due to data loss, only D environmental information measurement values are transmitted to the base station within time T, where D << M × N × L. Mathematically, the data received by the base station can be regarded as being randomly sampled from the third-order tensor at a certain ratio. The data received by the base station is represented by . denotes the random sampling operator, represents the data randomly sampled from the third-order tensor , which contains D environmental information measurement values, and the positions of the unsampled points are filled with zeros; Ω represents the observation set; the sampling ratio ρ = D / (M × N × L), where 0 < ρ << 1; among them, the data points randomly sampled from the third-order tensor are expressed as:

[0056]

[0057] In the formula, x m,n,l represents the data point in the third-order tensor , where m = 1, 2,..., M, n = 1, 2,..., N, and l = 1, 2,..., L;

[0058] Step 2: Since the data sensed by adjacent sensor nodes is similar within a fixed IoT monitoring area and the data sensed in consecutive time slots has stability, there is spatial correlation between the horizontal slice data and the side slice data of the third-order tensor , and there is temporal correlation between the front slice data. Therefore, the third-order tensor has an approximate low n-rank structure due to the high correlation between the data; thus, the process of reconstructing the third-order tensor using the D environmental information measurement values contained in the data can be transformed into a low-rank tensor completion problem. The expression of the basic low-rank tensor completion model is:

[0059]

[0060] where α i represents the weight of the nuclear norm of the mode-i unfolding matrix of the third-order tensor , i = 1, 2, 3, satisfying α i > 0 and X (i) represents the mode-i unfolding matrix of the third-order tensor , ||·|| * and ||·|| F represent the matrix nuclear norm and the F-norm respectively, and λ is the regularization parameter;

[0061] Step 3. To more effectively utilize the spatio-temporal correlation of the information sensed in the monitoring environment by the Internet of Things data, perform a block Hankel matrix transformation on the matrix obtained by unfolding the third-order tensor in each mode i, and combine the use of structured processing and low-rank tensor completion for data reconstruction to obtain the third-order tensor To promote the low-rank structure, penalize the nuclear norm of the block Hankel matrix formed by the matrix obtained by unfolding the third-order tensor (i) in mode i. Then, the basic low-rank tensor completion model in Equation (2) is improved as follows:

[0062]

[0063] where is the operation operator for converting the matrix into a block Hankel matrix;

[0064] For a certain matrix the block Hankel matrix of matrix Y is defined as:

[0065]

[0066] where k1 and k2 both represent the pencil parameters, and P and Q represent the number of rows and columns of matrix Y, respectively;

[0067] Any submatrix Y of the block Hankel matrix p is a Hankel matrix and satisfies the following equation:

[0068]

[0069] Convert the low-rank tensor completion model in Equation (3) into an equivalent constrained optimization problem in Equation (6) by introducing variable splitting;

[0070]

[0071] Solve the constrained equation (6) using the alternating direction method of multipliers. First, obtain the augmented Lagrangian function of the original objective function. Therefore, the augmented Lagrangian function of Equation (6) is expressed as:

[0072]

[0073] In the formula, β > 0 represents the penalty coefficient, Ε (i) and both represent Lagrange multipliers;

[0074] Step 4. Solve Equation (7) using the alternating direction method of multipliers to obtain the third-order tensor and complete the Internet of Things data reconstruction; the update and solution process of Equation (7) is as follows:

[0075]

[0076]

[0077]

[0078]

[0079]

[0080]

[0081] Among them, k represents the number of iterations, denotes the mode i unfolding matrix of , and fold means restoring the mode unfolding matrix to a tensor;

[0082] The sub-problems in equations (8) and (9) can be expressed as:

[0083]

[0084]

[0085] Equations (14) and (15) are standard linear least squares problems, which are solved by the conjugate gradient algorithm:

[0086]

[0087]

[0088] In the formula, H represents the Hermitian transpose, and I represents the identity matrix;

[0089] The sub-problem in equation (10) is expressed as:

[0090]

[0091] Equation (18) is solved by singular value truncation operation to obtain:

[0092]

[0093] Let Then equation (19) is transformed into:

[0094]

[0095] In equation (20), shrink(A,τ) is a non-linear function, and the specific operation of the function is to apply the soft threshold operator Applied to the singular values of matrix A; perform singular value decomposition on matrix A, perform a soft threshold operator operation on the diagonal of the obtained non - negative real - valued diagonal matrix (the elements on this diagonal are the singular values of matrix A), distinguish by the τ value, then put the vector after soft thresholding back into the non - negative real - valued diagonal matrix, and multiply the three decomposed components to obtain the matrix after singular value truncation operation, which is the output result of the shrink(A,τ) non - linear function;

[0096] Soft threshold operator Is defined as:

[0097]

[0098] In the formula, q j Represents the j - th singular value of matrix A.

[0099] To verify the effectiveness of the method of the present invention, the method of the present invention and two methods in the prior art including structured matrix completion and basic low - rank tensor completion are respectively used to reconstruct the Internet of Things data; select the ocean surface temperature data collected by the National Data Buoy Center (abbreviated as NDBC) and the temperature data collected by the Berkeley Research Laboratory as test data; since data loss in the Internet of Things is inevitable, a small part of the complete data subset is selected as the real test data; specifically, the NDBC data subset Contains the ocean surface temperature data sensed by 40 sensor nodes in 50 time slots, and the Berkeley Research Laboratory data subset Contains the temperature data sensed by 54 sensor nodes in 50 time slots; in the experimental simulation, the data tensor containing missing information Is obtained by And Random sampling, that is, use the random sampling operator Randomly sample D data on the real test data and discard other data; use the Normalized Mean Absolute Error (NMAE) to characterize the reconstruction error. NMAE reflects the relative difference between the real test data and the reconstructed data. A lower NMAE usually indicates better reconstruction accuracy. The definition of NMAE is:

[0100]

[0101] In formula (22), Represent the real test data and the reconstructed data respectively, and Π represents the sampling index subset of the complete entry set, that is, only the reconstruction error of the missing data is considered when calculating NMAE;

[0102] For each method, the random sampling and reconstruction processes were repeated 10 times, and the average NMAE was calculated; the best parameters for each method were selected individually to ensure the convincingness of the experimental results.

[0103] In the experiment, the third-order tensor of the method of the present invention The weights [α1, α2, α3] of the three modes were set to [0.33, 0.33, 0.34]; Figure 2 、 3 are the reconstruction errors for the NDBC data and the Berkeley Lab data at different sampling ratios respectively; it can be seen from the figure that at the same sampling ratio and different sampling ratios, the reconstruction error of the data using the method of the present invention is less than that of the other two methods; for the Internet of Things data reconstruction method based on structured matrix completion, the data collected by the sensor nodes are arranged in a two-dimensional matrix, which can only reflect the spatial correlation, while the method of the present invention arranges the data collected by the sensor nodes in the monitoring area in a third-order tensor, avoiding the destruction of the spatio-temporal correlation of the data and being conducive to improving the reconstruction accuracy; compared with the basic low n-rank tensor completion, the method of the present invention performs a block Hankel matrix transformation on the unfolded matrix of each mode of the third-order tensor to enhance the data structure and further strengthen the utilization of the spatio-temporal correlation of the data. Even in the case where only very few data can be obtained at an extremely low sampling ratio, the method of the present invention can reconstruct the complete data with high accuracy.

[0104] Those not described in the present invention are applicable to the prior art.

Claims

1. An Internet of Things data reconstruction method based on structured low-rank tensor completion, characterized in that The method includes the following steps: Step 1: Discretize the Internet of Things monitoring area into M×N grid points, and deploy a sensor node inside each grid point; assume that the sensor node senses data every time slot τ and transmits the data to the base station. Therefore, the data received by the base station within time T = L×τ forms a third-order tensor denotes the real number field, and M and N are positive integers; Due to data loss, only D environmental information measurement values are transmitted to the base station within time T, where D << M × N × L. Then the data received by the base station is represented by denotes denotes the random sampling operator denotes the data randomly sampled from the third-order tensor containing D environmental information measurement values, with the positions of unsampled points filled with zeros; Ω represents the observation set Step 2, third-order tensor There is spatial correlation between the horizontal slice data and between the side slice data, and there is temporal correlation between the front slice data. Therefore, the tensor has low rank, so the data reconstruction can be converted into a low-rank tensor completion problem. The expression of the basic low-rank tensor completion model is as follows: Among them, α i represents the weight of the nuclear norm of the mode-i unfolding matrix of the third-order tensor , where i = 1, 2, 3, and α satisfies i > 0 and X (i) represents the mode-i unfolding matrix of the third-order tensor . ||·|| * and ||·|| F represent the matrix nuclear norm and the F-norm respectively, and λ is the regularization parameter; Step 3. Perform a block Hankel matrix transformation on the matrix obtained by unfolding the third-order tensor for each mode i, and penalize the nuclear norm of the block Hankel matrix formed by the matrix X obtained by unfolding the third-order tensor (i) in mode i. Then, the basic low-rank tensor completion model in Equation (2) is improved as follows: Among them, is the operation operator for converting a matrix into a block Hankel matrix; By introducing variable splitting, the low-rank tensor completion model of Equation (3) is transformed into an equivalent constrained optimization problem of Equation (6): Using the alternating direction method of multipliers to solve the constraint equation (6), first, the augmented Lagrangian function of the original objective function needs to be obtained. Therefore, the augmented Lagrangian function of Equation (6) is expressed as: where β represents a penalty coefficient, β > 0, Ε (i) and both represent Lagrange multipliers; Step 4: Solve equation (7) to obtain a third-order tensor Complete the reconstruction of IoT data.

2. The method for reconstructing Internet of Things data based on structured low-rank tensor completion according to claim 1, wherein The alternating solution process of Equation (7) is: where k represents the number of iterations, represents the Lagrange multiplier of the mode i unfolding matrix, and fold represents restoring the mode i unfolding matrix to a tensor; The sub-problems of Equations (8) and (9) can be expanded as: Solving Equations (14) and (15) by the conjugate gradient algorithm: In the formula, H represents the Hermitian transpose, and I represents the identity matrix; The sub-problem of Equation (10) can be expanded as:

3. The method for reconstructing Internet of Things data based on structured low-rank tensor completion according to claim 1, wherein Solving Equation (18) through singular value truncation operation to obtain: Let Then equation (19) is transformed into: In Equation (20), shrink(A,τ) represents a non-linear function. The specific operation is to apply the soft threshold operator to the singular values of matrix A; perform singular value decomposition on matrix A, perform soft threshold operator operation on the diagonal of the resulting non-negative real diagonal matrix, then put the soft thresholded vector back into the non-negative real diagonal matrix, and multiply the decomposed components to obtain the matrix after singular value truncation operation, which is the output result of the non-linear function shrink(A,τ); Soft threshold operator It is defined as: where q j represents the j-th singular value of matrix A.

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