CN115309814A - Internet of things data reconstruction method based on structured low-rank tensor completion - Google Patents

Internet of things data reconstruction method based on structured low-rank tensor completion Download PDF

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CN115309814A
CN115309814A CN202210943271.5A CN202210943271A CN115309814A CN 115309814 A CN115309814 A CN 115309814A CN 202210943271 A CN202210943271 A CN 202210943271A CN 115309814 A CN115309814 A CN 115309814A
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何静飞
张潇月
刘晓彤
池越
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Hebei University of Technology
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Abstract

The invention relates to an Internet of things data reconstruction method based on structured low-rank tensor completion, which comprises the steps of firstly dispersing a monitoring area into a plurality of grid points, and deploying a sensor node in each grid point; assuming that a sensor node senses data once every other time slot, the data received by a base station in time T form a third-order tensor; secondly, data reconstruction is converted into a basic low-rank tensor completion problem, and a low-rank tensor completion model is constructed; and finally, carrying out block Hankel matrix change on the expansion matrix of each mode i of the third-order tensor, improving the basic low-order tensor completion model into a structured low-order tensor completion model, solving the augmented Lagrangian function of the structured low-order tensor completion model to obtain the third-order tensor, and completing data reconstruction of the Internet of things. Arranging data acquired at continuous moments by three-order tensors, and fully utilizing the spatial correlation of the data; the block Hankel matrix change is carried out on each mode i expansion matrix of the third-order tensor, data reconstruction is carried out through the combination of structuralization and low-rank tensor completion, the space-time correlation of the data is further mined and utilized, the influence of basic mismatch on reconstruction performance in a sparse constraint method is relieved, and the data reconstruction precision is improved.

Description

Internet of things data reconstruction method based on structured low-rank tensor completion
Technical Field
The invention belongs to the technical field of data processing of the Internet of things, and particularly relates to a data reconstruction method of the Internet of things based on structured low-rank tensor completion.
Background
Since the advent of the Internet of Things (IoT), as a computer and the Internet, the third wave of information industry development has spread its application across many areas. The IoT equipment layer is composed of a large number of sensor nodes with sensing and communication capabilities, the sensor nodes are deployed in a monitoring area in a random mode, and each sensor node has certain computing, storage and communication capabilities and can continuously monitor sensing environment information. Due to the influence of factors such as hardware condition limitation, unstable network communication, severe environment and the like, the data loss problem of the internet of things is inevitable. Data missing will cause the data of the internet of things to be unable to be normally used for subsequent analysis and application, so reconstructing the complete IoT data with high precision has become a research hotspot in the field.
IoT data reconstruction methods based on sparse constraints can be divided into three categories: a reconstruction method based on Compressed Sensing (CS for short), a reconstruction method based on Matrix Completion (MC for short), and a reconstruction method based on tensor Completion. In particular, CS-based methods use measurement matrices, compress sampled signals from IoT data that has sparsity in certain transform domains, and reconstruct the sampled signals using optimization methods. The data of the Internet of things can be arranged into a two-dimensional matrix according to the sensor nodes and the acquisition time, and the data space-time correlation enables the data matrix of the Internet of things to have low rank and accords with the premise of matrix completion application, so that missing data of the Internet of things can be reconstructed based on the matrix completion. The CS and MC based methods generally arrange data collected by spatially distributed sensor nodes in a vector form, thereby ignoring spatial correlation between sensor nodes. As the high-order expansion of the matrix, the tensor can express high-dimensional data in a natural and compact mode, so that the internet of things data completion method naturally expands from low-rank matrix completion to low-rank tensor completion. According to different tensor rank definitions, the internet of things data reconstruction method based on various low-rank tensor completions is proposed successively. Compared with reconstruction methods based on CS and MC, the reconstruction method based on tensor completion can further mine the spatiotemporal correlation of IoT data.
The first-order, second-order and high-order sparse constraint methods have very outstanding results in the data reconstruction research of the internet of things, but the methods all assume that signals are sparse a priori. However, since the real frequency of the data of the internet of things is actually specified in the continuous domain, when the base mismatch inevitably exists between the real frequency and the assumed base, the reconstruction performance of the sparse feature constraint-based method is reduced. Therefore, the method for reconstructing the data of the internet of things based on the structured low-rank tensor completion is provided, and the influence of the base mismatch on the reconstruction accuracy is further relieved by utilizing the time-space correlation among the data of the internet of things, so that the data reconstruction accuracy is improved.
Disclosure of Invention
Aiming at the defects in the prior art, the invention aims to solve the technical problem of providing a method for reconstructing data of the Internet of things based on structured low-rank tensor completion.
The technical scheme adopted by the invention for solving the technical problems is as follows:
an Internet of things data reconstruction method based on structured low-rank tensor completion comprises the following steps:
step 1, dividing an internet of things monitoring area into M multiplied by N grid points, wherein a sensor node is deployed in each grid point; assuming that the sensor node senses data once every time slot τ and transmits the data to the base station, the data received by the base station within the time T = L × τ may form a third order tensor
Figure BDA0003786621890000011
Representing a real number domain, wherein M and N are positive integers;
due to data loss, only D environment information measured values are transmitted to the base station in time T, D < M multiplied by N multiplied by L, and the data received by the base station passes
Figure BDA0003786621890000021
It is shown that,
Figure BDA0003786621890000022
a random sampling operator is represented that is,
Figure BDA0003786621890000023
representing tensor from third order
Figure BDA0003786621890000024
The data obtained by the middle random sampling comprises D environment information measurement values, and the positions of the non-sampling points are filled with zeros; Ω represents an observation set;
step 2, three-order tensor formed by data of Internet of things
Figure BDA0003786621890000025
Have spatial correlation between horizontal slice data and between side slice data, have temporal correlation between front slice data, and thus tensor
Figure BDA0003786621890000026
The method has low rank, so that data reconstruction can be converted into a low rank tensor completion problem, and the expression of a basic low rank tensor completion model is as follows:
Figure BDA0003786621890000027
wherein alpha is i Tensor representing the third order
Figure BDA0003786621890000028
I =1,2,3, satisfying α, the weight of the kernel norm of the expanded matrix of (a) i > 0 and
Figure BDA0003786621890000029
X (i) tensor representing the third order
Figure BDA00037866218900000210
The mode i expands the matrix, | · | | non-woven phosphor * And | · | non-conducting phosphor F Respectively representing a matrix nuclear norm and an F-norm, wherein lambda is a regularization parameter;
step 3, in order to more effectively utilize the time-space correlation of the sensing information of the data of the Internet of things in the monitoring environment, the third-order tensor is subjected to
Figure BDA00037866218900000211
The expansion matrix of each mode i carries out block Hankel matrix change, and punishment is given by third-order tensor
Figure BDA00037866218900000212
Mode i of (2) unfolding matrix X (i) And (3) forming a nuclear norm of a block Hankel matrix, and improving a basic low-rank tensor completion model of the formula (2) as follows:
Figure BDA00037866218900000213
wherein the content of the first and second substances,
Figure BDA00037866218900000214
an operator for converting the matrix into a block hankel matrix;
converting the low-rank tensor completion model of the formula (3) into an equivalent constraint optimization problem of the formula (6) by introducing variable splitting;
Figure BDA00037866218900000215
solving the constraint equation (6) by using an alternating direction multiplier method, and firstly obtaining an augmented Lagrange function of an original objective function, so the augmented Lagrange function of the equation (6) is expressed as:
Figure BDA00037866218900000216
in the formula, beta represents a penalty coefficient, beta > 0, E (i) And
Figure BDA00037866218900000217
all represent lagrangian multipliers;
step 4, solving the formula (7) to obtain a third-order tensor
Figure BDA00037866218900000218
And finishing the reconstruction of the data of the Internet of things.
Further, the alternative solving process of equation (7) is:
Figure BDA00037866218900000219
Figure BDA00037866218900000220
Figure BDA00037866218900000221
Figure BDA00037866218900000222
Figure BDA00037866218900000223
Figure BDA0003786621890000031
wherein, k represents the number of iterations,
Figure BDA0003786621890000032
representing lagrange multipliers
Figure BDA0003786621890000033
The fold represents to restore the mode i expansion matrix to tensor;
sub-problem equations (8) and (9) can be expanded as:
Figure BDA0003786621890000034
Figure BDA0003786621890000035
solving equations (14) and (15) by a conjugate gradient algorithm:
Figure BDA0003786621890000036
Figure BDA0003786621890000037
in the formula, H represents Hermite transpose, and I represents a unit matrix;
the subproblem equation (10) can be expanded as:
Figure BDA0003786621890000038
further, the equation (18) is solved through singular value interception operation, so as to obtain:
Figure BDA0003786621890000039
order to
Figure BDA00037866218900000310
Then formula (19) is converted to:
Figure BDA00037866218900000311
in the formula (20), shrnk (a, τ) represents a nonlinear function, and the specific operation is to use a soft threshold operator
Figure BDA00037866218900000312
Applied to the singular values of matrix a; performing singular value decomposition on the matrix A, performing soft threshold operator operation on diagonal lines of the diagonal matrix of the nonnegative real numbers obtained by decomposition, putting vectors subjected to soft threshold back into the diagonal matrix of the nonnegative real numbers, and multiplying the decomposed components to obtain a matrix subjected to singular value interception operation, namely an output result of a nonlinear function shrnk (A, tau);
soft threshold operator
Figure BDA00037866218900000313
Is defined as:
Figure BDA00037866218900000314
in the formula, q j Representing the jth singular value of the matrix a.
Compared with the prior art, the invention has the beneficial effects that:
on one hand, the method arranges the data acquired at continuous time in a third-order tensor form, makes full use of the spatial correlation of the data acquired by adjacent sensor nodes, and is beneficial to improving the reconstruction precision. On the other hand, in order to further improve the utilization of the space-time correlation of the data, block Hankel matrix change is carried out on each mode i expansion matrix of the third-order tensor, data reconstruction is carried out through structuralization processing and low-rank tensor completion, and the negative influence of base mismatch on reconstruction performance based on the sparse constraint method is relieved. The experimental comparison result shows that compared with a reconstruction method based on structured matrix completion and basic low-rank tensor completion, the method provided by the invention has the advantages that the reconstruction error is smaller and the reconstruction precision is higher under the same adoption proportion and different sampling proportions.
Drawings
FIG. 1 is a diagram of third order tensors
Figure BDA0003786621890000041
The schematic diagram of the mode i expansion matrix of (1);
FIG. 2 is an error contrast plot of NDBC ocean surface temperature data reconstructed using different methods;
fig. 3 is a graph of error versus temperature data for a berkeley laboratory reconstructed using different methods.
Detailed Description
The technical solutions of the present invention are described in detail below with reference to the accompanying drawings and specific embodiments, but the scope of the present invention is not limited thereto.
The invention relates to an Internet of things data reconstruction method (a method for short, see figures 1-3) based on structured low-rank tensor completion, which comprises the following steps:
step 1, dispersing an internet of things monitoring area into M multiplied by N grid points, wherein a sensor node is deployed in each grid point; the sensor node is supposed to sense data once every time slot tau (the time slot is the difference between adjacent sampling moments) and transmit the data to the base station; data sensed by all sensor nodes at the same sampling moment form a matrix
Figure BDA0003786621890000042
Therefore, the data received by the base station in time T = L x tau, namely continuous L time slots, form a third-order tensor
Figure BDA0003786621890000043
Representing a real number domain, wherein M and N are positive integers;
due to data loss, only D environment information measurements are transmitted to the base station in time T, D < M × N × L, then mathematically, the data received by the base station can be viewed as a tensor of third order
Figure BDA0003786621890000044
Is obtained by random sampling in a certain proportion, and the data received by the base station is passed through
Figure BDA0003786621890000045
It is shown that,
Figure BDA0003786621890000046
a random sampling operator is represented that is,
Figure BDA0003786621890000047
representing tensor from third order
Figure BDA0003786621890000048
The data obtained by the middle random sampling comprises D environment information measurement values, and the positions of the non-sampling points are filled with zeros; Ω represents an observation set; the sampling ratio rho = D/(M × N × L), 0 < rho < 1; wherein from the third order tensor
Figure BDA0003786621890000049
Data points obtained by medium random sampling
Figure BDA00037866218900000410
Expressed as:
Figure BDA00037866218900000411
in the formula, x m,n,l Tensor representing the third order
Figure BDA00037866218900000412
The data points in (a), M =1,2,. Cndot, M, N =1,2,. Cndot, N, L =1,2,. Cndot, L;
step 2, because the data sensed by the adjacent sensor nodes in the fixed monitoring area of the Internet of things are similar, the data sensed in the continuous time slots have stability, and therefore the third-order tensor
Figure BDA00037866218900000413
The horizontal slice data and the side slice data have spatial correlation, and the front slice data have time correlation, so that the third order tensor
Figure BDA00037866218900000414
An approximate low n-rank structure due to the high correlation between data; thus, utilizing the data
Figure BDA00037866218900000415
D environmental information measured values contained in the data acquisition unit are subjected to data reconstruction to obtain third-order tensors
Figure BDA00037866218900000416
The process of (2) can be converted into a low-rank tensor completion problem, and the expression of the basic low-rank tensor completion model is as follows:
Figure BDA00037866218900000417
wherein alpha is i Tensor representing the third order
Figure BDA00037866218900000418
I =1,2,3, satisfying α i > 0 and
Figure BDA00037866218900000419
X (i) tensor representing the third order
Figure BDA00037866218900000420
The mode i expands the matrix, | · | | non-woven phosphor * And | · | non-counting F Respectively representing a matrix nuclear norm and an F-norm, wherein lambda is a regularization parameter;
step 3, in order to more effectively utilize the time-space correlation of the sensing information of the data of the Internet of things in the monitoring environment, the third-order tensor is subjected to
Figure BDA0003786621890000051
The matrix is expanded by each mode i to change the block Hankel matrix, and data reconstruction is carried out by combining structuralization processing and low-rank tensor complementation to obtain third-order tensor
Figure BDA0003786621890000052
To facilitate low rank structures, the penalties are given by third order tensors
Figure BDA0003786621890000053
Mode i of (2) unfolding matrix X (i) Formed block Chinese characterThe kernel norm of the kerr matrix, the basic low rank tensor completion model of equation (2) is improved as follows:
Figure BDA0003786621890000054
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003786621890000055
an operator for converting the matrix into a block hankel matrix;
for a certain matrix
Figure BDA0003786621890000056
Block-hankel matrix for matrix Y
Figure BDA0003786621890000057
Is defined as follows:
Figure BDA0003786621890000058
wherein k is 1 And k 2 Both represent the penil parameter, and P and Q represent the row number and column number of the matrix Y, respectively;
block Hankel matrix
Figure BDA0003786621890000059
Any one of the sub-matrices Y of p Are all hankel matrices satisfying the following equation:
Figure BDA00037866218900000510
converting the low-rank tensor completion model of the formula (3) into an equivalent constraint optimization problem of the formula (6) by introducing variable splitting;
Figure BDA00037866218900000511
solving the constraint equation (6) by using an alternating direction multiplier method, and firstly obtaining an augmented Lagrange function of an original objective function, so the augmented Lagrange function of the equation (6) is expressed as:
Figure BDA00037866218900000512
in the formula, beta > 0 represents penalty coefficient, E (i) And
Figure BDA00037866218900000513
all represent lagrangian multipliers;
step 4, solving the formula (7) by adopting an alternating direction multiplier method to obtain a third-order tensor
Figure BDA00037866218900000514
Completing data reconstruction of the Internet of things; the update solving process of equation (7) is:
Figure BDA00037866218900000515
Figure BDA00037866218900000516
Figure BDA00037866218900000517
Figure BDA00037866218900000518
Figure BDA00037866218900000519
Figure BDA0003786621890000061
wherein k represents the number of iterations,
Figure BDA0003786621890000062
to represent
Figure BDA0003786621890000063
The mode i expansion matrix of (1), fold represents to restore the mode expansion matrix to tensor;
sub-problem equations (8) and (9) can be expressed as:
Figure BDA0003786621890000064
Figure BDA0003786621890000065
equations (14) and (15) are standard linear least squares problems solved by a conjugate gradient algorithm:
Figure BDA0003786621890000066
Figure BDA0003786621890000067
in the formula, H represents Hermite transpose, and I represents a unit matrix;
the subproblem equation (10) is expressed as:
Figure BDA0003786621890000068
solving equation (18) by singular value intercept operations to obtain:
Figure BDA0003786621890000069
order to
Figure BDA00037866218900000610
Then formula (19) is converted to:
Figure BDA00037866218900000611
in equation (20), shrink (A, τ) is a non-linear function that is specifically operated as a soft threshold operator
Figure BDA00037866218900000612
Applied to the singular values of matrix a; performing singular value decomposition on the matrix A, performing soft threshold operator operation on a diagonal line (elements on the diagonal line are singular values of the matrix A) of the diagonal matrix of the nonnegative real number obtained by decomposition, distinguishing by using a value tau, putting a vector after soft threshold back into the diagonal matrix of the nonnegative real number, and multiplying three decomposed components to obtain a matrix after singular value interception operation, namely an output result of a shrnk (A, tau) nonlinear function;
soft threshold operator
Figure BDA00037866218900000613
Is defined as follows:
Figure BDA00037866218900000614
in the formula, q j Representing the jth singular value of the matrix a.
In order to verify the effectiveness of the method, the method and two methods in the prior art including the structured matrix completion and the basic low-rank tensor completion are respectively utilized to reconstruct the data of the Internet of things; selecting ocean surface temperature collected by a National Data Buoy Center (NDBC for short) and temperature Data collected by Berkeley research laboratory as test Data; because data loss in the Internet of things is inevitable, a small part of complete data subset is selected as real test data; specifically, NDBC data subsets
Figure BDA0003786621890000071
Comprises ocean surface temperature data sensed by 40 sensor nodes in 50 time slots, a Berkeley research laboratory data subset
Figure BDA0003786621890000072
Temperature data sensed by 54 sensor nodes in 50 time slots are contained; in the experimental simulation, the data tensor containing the missing information
Figure BDA0003786621890000073
By pairs
Figure BDA0003786621890000074
And
Figure BDA0003786621890000075
obtaining by random sampling, i.e. using random sampling operators
Figure BDA0003786621890000076
Randomly collecting D data on the real test data, and discarding other data; the reconstruction Error is characterized by a Normalized Mean Absolute Error (NMAE), which reflects the relative difference between the real test data and the reconstructed data, with a lower NAME generally indicating a better reconstruction accuracy, defined as:
Figure BDA0003786621890000077
in the formula (22), the reaction mixture is,
Figure BDA0003786621890000078
respectively representing real test data and reconstructed data, wherein pi represents a sampling index subset of a complete entry set, namely the reconstruction error of only missing data is considered when NMAE is calculated;
for each method, the random sampling and reconstruction process was repeated 10 times and the average NMAE was calculated; the optimal parameters for each method were chosen individually to ensure convincing experimental results.
In the experiment, the third order tensor of the method of the invention
Figure BDA0003786621890000079
Weight of three modes [ alpha ] 1 ,α 2 ,α 3 ]Set to [0.33, 0.34](ii) a FIGS. 2 and 3 are respectively the reconstruction errors for NDBC data and Berkeley laboratory data at different sampling ratios; as can be seen from the figure, under the same sampling proportion and different sampling proportions, the reconstruction error of the data by utilizing the method of the invention is smaller than that of the other two methods; the data collected by the sensor nodes are arranged in the two-dimensional matrix, only the spatial correlation can be reflected, and the data collected by the sensor nodes distributed in the monitoring area are arranged in the third-order tensor, so that the damage to the time-space correlation of the data is avoided, and the reconstruction precision is improved; compared with the basic low n-rank tensor completion method, the method of the invention completes the third-order tensor
Figure BDA00037866218900000710
The block Hankel matrix change is carried out on each mode expansion matrix to enhance the data structure, the utilization of the data space-time correlation is further enhanced, and even under the condition that only a few data can be obtained at an extremely low sampling ratio, the method can reconstruct the complete data at high precision.
Nothing in this specification is said to apply to the prior art.

Claims (3)

1. An Internet of things data reconstruction method based on structured low-rank tensor completion is characterized by comprising the following steps:
step 1, dispersing an internet of things monitoring area into M multiplied by N grid points, wherein a sensor node is deployed in each grid point; suppose that the sensor node senses data once every time slot tau and transmits the data to the base station, so that the data received by the base station in the time T = L x tau forms a third-order tensor
Figure FDA0003786621880000011
Figure FDA0003786621880000012
Representing a real number field, wherein M and N are positive integers;
only D environmental information measured values are transmitted to the base station in time T due to data loss, D < M multiplied by N multiplied by L, and then the data received by the base station passes
Figure FDA0003786621880000013
It is shown that the process of the present invention,
Figure FDA0003786621880000014
a random sampling operator is represented that is,
Figure FDA0003786621880000015
representing tensor from third order
Figure FDA0003786621880000016
The data obtained by the middle random sampling comprises D environment information measurement values, and the positions of the non-sampling points are filled with zeros; Ω represents an observation set;
step 2, third order tensor
Figure FDA0003786621880000017
Have spatial correlation between horizontal slice data and between side slice data, have temporal correlation between front slice data, and thus tensor
Figure FDA0003786621880000018
The method has low rank, so that data reconstruction can be converted into a low rank tensor completion problem, and the expression of a basic low rank tensor completion model is as follows:
Figure FDA0003786621880000019
wherein alpha is i Tensor representing the third order
Figure FDA00037866218800000110
I =1,2,3, satisfying α, the weight of the kernel norm of the expanded matrix of (a) i > 0 and
Figure FDA00037866218800000111
X (i) tensor representing the third order
Figure FDA00037866218800000112
The mode i of (1) expands the matrix, | | caldol |, L * And | · | non-conducting phosphor F Respectively representing a matrix kernel norm and an F-norm, wherein lambda is a regularization parameter;
step 3, for the third order tensor
Figure FDA00037866218800000113
The expansion matrix of each mode i carries out block Hankel matrix change, and punishment is given by third-order tensor
Figure FDA00037866218800000114
Mode i of (2) unfolding matrix X (i) And (3) forming a kernel norm of a block Hankel matrix, and improving a basic low-rank tensor completion model of the formula (2) into the following steps:
Figure FDA00037866218800000115
wherein the content of the first and second substances,
Figure FDA00037866218800000116
an operator for converting the matrix into a block hankel matrix;
converting the low rank tensor completion model of equation (3) into an equivalence constraint optimization problem of equation (6) by introducing variable splitting:
Figure FDA00037866218800000117
solving the constraint equation (6) by using an alternating direction multiplier method, and firstly obtaining an augmented Lagrange function of an original objective function, so the augmented Lagrange function of the equation (6) is expressed as:
Figure FDA00037866218800000118
in the formula, beta represents a penalty coefficient, beta > 0, E (i) And
Figure FDA00037866218800000119
all represent lagrangian multipliers;
step 4, solving the formula (7) to obtain a third-order tensor
Figure FDA00037866218800000120
And finishing the reconstruction of the data of the Internet of things.
2. The method for reconstructing data of the internet of things based on the structured low-rank tensor completion as claimed in claim 1, wherein the alternative solution process of equation (7) is as follows:
Figure FDA0003786621880000021
Figure FDA0003786621880000022
Figure FDA0003786621880000023
Figure FDA0003786621880000024
Figure FDA0003786621880000025
Figure FDA0003786621880000026
wherein k represents the number of iterations,
Figure FDA0003786621880000027
representing lagrange multipliers
Figure FDA0003786621880000028
The fold represents to restore the mode i expansion matrix to tensor;
sub-problem equations (8) and (9) can be expanded as:
Figure FDA0003786621880000029
Figure FDA00037866218800000210
solving equations (14) and (15) by a conjugate gradient algorithm:
Figure FDA00037866218800000211
Figure FDA00037866218800000212
in the formula, H represents Hermite transpose, and I represents a unit matrix;
the subproblem equation (10) can be expanded as:
Figure FDA00037866218800000219
3. the method for reconstructing data of the internet of things based on structured low-rank tensor completion as claimed in claim 1, wherein equation (18) is solved by singular value intercept operation to obtain:
Figure FDA00037866218800000214
order to
Figure FDA00037866218800000215
Then formula (19) is converted to:
Figure FDA00037866218800000216
in the formula (20), shrnk (a, τ) represents a nonlinear function, and the specific operation is to use a soft threshold operator
Figure FDA00037866218800000217
Applied to the singular values of the matrix a; performing singular value decomposition on the matrix A, performing soft threshold operator operation on diagonal lines of a non-negative real number diagonal matrix obtained by decomposition, putting vectors subjected to soft threshold back into the non-negative real number diagonal matrix, and multiplying decomposed components to obtain a matrix subjected to singular value interception operation, namely an output result of a nonlinear function shrink (A, tau);
soft threshold operator
Figure FDA00037866218800000218
Is defined as follows:
Figure FDA0003786621880000031
in the formula, q j Representing the jth singular value of the matrix a.
CN202210943271.5A 2022-08-08 2022-08-08 Internet of things data reconstruction method based on structured low-rank tensor completion Pending CN115309814A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116450636A (en) * 2023-06-20 2023-07-18 石家庄学院 Internet of things data completion method, equipment and medium based on low-rank tensor decomposition
CN116955334A (en) * 2023-06-27 2023-10-27 香港理工大学深圳研究院 Structural health monitoring sensing data loss recovery method based on low-rank Hank matrix

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116450636A (en) * 2023-06-20 2023-07-18 石家庄学院 Internet of things data completion method, equipment and medium based on low-rank tensor decomposition
CN116450636B (en) * 2023-06-20 2023-08-18 石家庄学院 Internet of things data completion method, equipment and medium based on low-rank tensor decomposition
CN116955334A (en) * 2023-06-27 2023-10-27 香港理工大学深圳研究院 Structural health monitoring sensing data loss recovery method based on low-rank Hank matrix
CN116955334B (en) * 2023-06-27 2024-06-04 香港理工大学深圳研究院 Structural health monitoring sensing data loss recovery method based on low-rank Hank matrix

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