CN115829424A - Traffic data restoration method based on non-parametric non-convex relaxation low-rank tensor completion - Google Patents

Traffic data restoration method based on non-parametric non-convex relaxation low-rank tensor completion Download PDF

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CN115829424A
CN115829424A CN202211497496.9A CN202211497496A CN115829424A CN 115829424 A CN115829424 A CN 115829424A CN 202211497496 A CN202211497496 A CN 202211497496A CN 115829424 A CN115829424 A CN 115829424A
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traffic data
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贺洋
张敏
夏井新
安成川
陆振波
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Southeast University
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Abstract

The invention discloses a traffic data restoration method based on non-parametric non-convex relaxation low-rank tensor completion, which comprises the following steps of: constructing the traffic data containing the missing data into a three-dimensional tensor of position x date x time according to three dimensions of place, date and time
Figure DDA0003964156600000011
Providing a logarithm-based non-parametric non-convex relaxation function, and constructing a low-rank tensor completion-based traffic data restoration model; considering model solving efficiency and removing equality constraint, and constructing an augmented Lagrange function of the model; according to the ADMM framework, the multivariate optimization problem of the model is converted into three univariate quantum optimization problems, and tensor is initialized
Figure DDA0003964156600000012
Updating in sequence
Figure DDA0003964156600000013
Three variables; to be provided with
Figure DDA0003964156600000014
As input, iterative optimization is carried out by utilizing a cross direction multiplier algorithm until a convergence condition is met, and a low-rank tensor is obtained
Figure DDA0003964156600000015
The invention can realize intelligent and accurate restoration of traffic data.

Description

Traffic data restoration method based on non-parametric non-convex relaxation low-rank tensor completion
Technical Field
The invention belongs to the technical field of space-time traffic data restoration, and particularly relates to a traffic data restoration method based on non-parametric non-convex relaxation low-rank tensor completion.
Background
Spatiotemporal traffic data is a key input for many applications in Intelligent Transportation Systems (ITS), such as traffic monitoring and prediction, traffic control, and route guidance, the quality of which directly affects the efficiency of intelligent transportation systems. With the development of traffic awareness technology, the size and dimension of traffic data is increasing. Meanwhile, due to reasons such as sensor faults and communication faults, the problem of traffic data loss is serious, and the practicability and effectiveness of the intelligent traffic system are directly influenced. How to accurately recover accurate and complete traffic data from partial data observation containing deletion by utilizing the space-time correlation of the traffic data has important significance for the field of intelligent transportation.
Traffic data is a high-dimensional time series, usually represented in the form of a matrix of spatio-temporal traffic data. The traffic data matrix is essentially a low rank matrix, reflecting the periodicity in time and the spatial correlation of traffic data. In recent years, many researchers have repaired traffic data matrices by exploiting the inherent low rank nature of traffic data. To better exploit the low rank nature of traffic data, researchers have further decomposed the time dimension into time and date, and represented the traffic data as a three-dimensional tensor in location, time, date. Thus, the traffic data recovery problem translates into a low rank tensor completion problem.
The existing traffic data restoration method based on the low rank tensor mainly comprises the following steps: (1) tensor decomposition based approach. The method for restoring the target low-rank traffic data tensor is characterized in that the rank of the traffic data tensor is determined in advance, the traffic data tensor is decomposed into a plurality of smaller-scale factor tensors by a tensor decomposition method, and the currently mainstream tensor method comprises the following steps: CP tensor decomposition, tucker tensor decomposition, tensor SVD decomposition, etc. (2) rank minimization based approach. The rank minimization-based method does not require exact structural information, and the low-rank traffic data tensor is directly restored by minimizing the rank of the tensor. Since rank minimization is an NP-hard problem, existing methods typically approximate in the form of convex and non-convex relaxation. The model of the convex relaxation approximation mainly uses a nuclear norm minimization. Models that employ a non-convex relaxation approximation include: the truncated nuclear norm is minimized and the Schatten-p norm is minimized.
The prior art has the following disadvantages:
(1) In the tensor decomposition-based method, the rank of the decomposed traffic data tensor needs to be determined in advance, but in the real large-scale road network application, the space-time correlation mode of the space-time traffic data is unknown, so that the rank of the traffic data tensor is difficult to determine in advance.
(2) In the rank minimization-based method, a convex relaxation nuclear norm minimization model has the problem of excessive relaxation, the singular values containing structural information and noise information are assigned with the same weight, and the solution result is often deviated from the optimal value. Models of non-convex relaxation often require additional parameters to control non-convexity, e.g., truncation ratio r in the truncated nuclear norm, p value in the Schatten-p norm. Since the structural information (e.g., rank) of the spatio-temporal traffic data is unknown in practical applications, introducing additional parameters usually requires repeated fine tuning and trial and error to determine the optimal parameter values.
Disclosure of Invention
The technical problem to be solved is as follows: the invention provides a traffic data restoration method based on non-parametric non-convex relaxation low-rank tensor completion by considering the low-dimensional subspace characteristic of high-dimensional space-time traffic data.
The technical scheme is as follows:
a traffic data restoration method based on parameter-free non-convex relaxation low-rank tensor completion comprises the following steps:
s1, constructing traffic data containing missing data into three-dimensional tensor of position multiplied by date multiplied by time according to three dimensions of place, date and time
Figure BDA0003964156580000021
Figure BDA0003964156580000022
Wherein n is 1 Indicating the number of locations of the data-collecting device, n 2 Indicating the number of dates on which the data was collected, n 3 Representing the number of time segments of the data collected on each natural day;
s2, providing a logarithm-based non-parametric non-convex relaxation function, and constructing a low-rank tensor completion-based traffic data restoration model; wherein a tensor is introduced
Figure BDA0003964156580000023
Respectively representing tensors
Figure BDA0003964156580000024
By unfolding the three modes, an auxiliary tensor is introduced
Figure BDA0003964156580000025
Will tensor
Figure BDA0003964156580000026
To the observation information in
Figure BDA0003964156580000027
Performing the following steps;
s3, considering model solving efficiency and removing equality constraint, and constructing an augmented Lagrangian function of the model; according to the ADMM framework, modelThe multivariable optimization problem is converted into three single-variable quantum optimization problems, namely initialization tensor
Figure BDA0003964156580000028
Sequentially updating
Figure BDA0003964156580000029
Three variables;
s4, in the step S3
Figure BDA00039641565800000210
As input, iterative optimization is carried out by using a cross direction multiplier algorithm until a convergence condition is met, and a low-rank tensor is obtained
Figure BDA00039641565800000211
Further, in step S2, a logarithm-based non-parametric non-convex relaxation function is proposed, so as to construct a traffic data restoration model based on low-rank tensor completion, including the following steps:
s21, considering that the penalty of noise information is increased and the penalty of structure information is reduced at the same time, a logarithm-based parameterless non-convex relaxation function is provided, and the function is expressed as:
Figure BDA00039641565800000212
wherein σ i (X) represents the ith singular value of the matrix X, epsilon is more than 0 to ensure positive nature, and the value range of epsilon is 10 -6 ～10 -4 ；
S22, constructing a low-rank tensor completion model for traffic data restoration based on the non-parameter non-convex relaxation function provided in the step S21, wherein the model is expressed as follows:
Figure BDA0003964156580000031
wherein,
Figure BDA0003964156580000032
tensor of representation
Figure BDA0003964156580000033
Along the expansion matrix of the k-th mode, k =1,2,3, α k Is a matrix L k(k) The weight of the regularization term of (a),
Figure BDA0003964156580000034
representing a low rank tensor over an index set Ω of observable data
Figure BDA0003964156580000035
And tensor of observed value
Figure BDA0003964156580000036
Are equal;
s23, considering the requirement of variable independence, introducing tensor
Figure BDA0003964156580000037
Respectively representing tensors
Figure BDA0003964156580000038
By unfolding the three modes, an auxiliary tensor is introduced
Figure BDA0003964156580000039
Tensor is expressed
Figure BDA00039641565800000310
To the observation information in
Figure BDA00039641565800000311
In step S22, the low-rank tensor completion model is further expressed as:
Figure BDA00039641565800000312
further, in step S3, the augmented lagrangian function of the constructed model is represented as:
Figure BDA00039641565800000313
Figure BDA00039641565800000314
wherein,<v. represents the inner product,
Figure BDA00039641565800000315
the number of lagrange multipliers is represented,
Figure BDA00039641565800000316
is the square of the Frobenius norm, ρ k Representing the weight coefficient of the k-th mode.
Further, in step S3, the process of transforming the multivariate optimization problem of the model into three univariate quantum optimization problems according to the ADMM framework comprises the following steps:
s31, initializing tensor
Figure BDA00039641565800000317
With the tensor in step S1
Figure BDA00039641565800000318
As input, the tensor is initialized:
Figure BDA00039641565800000319
s32, updating the target tensor
Figure BDA00039641565800000320
To find
Figure BDA00039641565800000321
In particular, the amount of the solvent to be used,
from the initialization tensor
Figure BDA00039641565800000322
Calculating tensors
Figure BDA00039641565800000323
Each modal auxiliary tensor of
Figure BDA00039641565800000324
Figure BDA00039641565800000325
Wherein,
Figure BDA00039641565800000326
is that
Figure BDA00039641565800000327
The mode expansion matrix of (a) is,
Figure BDA00039641565800000328
is that
Figure BDA00039641565800000329
The mode expansion matrix of (a) is,
Figure BDA0003964156580000041
for weighted singular value threshold operators, U (sigma) V T Is an arbitrary matrix
Figure BDA0003964156580000042
The singular value of (a) is decomposed,
Figure BDA0003964156580000043
Figure BDA0003964156580000044
is composed of
Figure BDA0003964156580000045
The mode expansion matrix of (a) is,
Figure BDA0003964156580000046
representation matrix
Figure BDA0003964156580000047
Of the 1 st singular value, epsilon k Is a constant number epsilon k Has a value range of 10 -6 ～10 -4 ，τ＝α k /ρ k ；
Tensor assisted by each modality
Figure BDA0003964156580000048
Computing a low rank tensor
Figure BDA0003964156580000049
Figure BDA00039641565800000410
S33, updating the auxiliary tensor
Figure BDA00039641565800000411
To find
Figure BDA00039641565800000412
By updated
Figure BDA00039641565800000413
Computing
Figure BDA00039641565800000414
Figure BDA00039641565800000415
S34, updating the Lagrange multiplier
Figure BDA00039641565800000416
To find
Figure BDA00039641565800000417
After update
Figure BDA00039641565800000418
Computing
Figure BDA00039641565800000419
Figure BDA00039641565800000420
Wherein,
Figure BDA00039641565800000421
representing a four-dimensional tensor of size 3 x M x N x T,
Figure BDA00039641565800000422
respectively composed of three-dimensional tensors
Figure BDA00039641565800000423
k =1,2,3 is stacked in the fourth dimension,
Figure BDA00039641565800000424
from three identical three-dimensional tensors
Figure BDA00039641565800000425
Stacked on the fourth modality.
Further, in step S4, in step S3
Figure BDA00039641565800000426
As input, iterative optimization is carried out by utilizing a cross direction multiplier algorithm until a convergence condition is met, and a low-rank tensor is obtained
Figure BDA00039641565800000427
Comprises the following steps:
s41, inputting the observed tensor of the partially observed traffic data
Figure BDA00039641565800000428
S42Initializing each parameter:
Figure BDA00039641565800000429
ρ k ＝ρ＝ρ 0 ，ε＝1e-6；
s43, iteratively calculating the following formula until
Figure BDA00039641565800000430
Figure BDA00039641565800000431
Figure BDA00039641565800000432
Figure BDA00039641565800000433
l＝l+1；
Wherein,
Figure BDA00039641565800000434
s44, outputting the repaired complete traffic data low-rank tensor
Figure BDA0003964156580000051
Has the beneficial effects that:
firstly, the traffic data restoration method based on the non-parametric non-convex relaxation low-rank tensor completion is characterized in that a non-convex relaxation based low-rank tensor completion model is constructed, punishment on noise is improved and punishment on structure information is reduced in the traffic data restoration process, and compared with a convex relaxation tensor completion model which treats structure and noise information equally, the traffic data restoration precision is greatly improved.
Secondly, the traffic data restoration method based on the non-parametric non-convex relaxation low-rank tensor completion is characterized in that a non-parametric non-convex relaxation function based on a logarithmic function is designed, no additional parameter is needed in the constructed low-rank tensor completion model, no manual field calibration of structural information (such as rank and the like) of traffic data is needed, and a robust traffic data restoration scheme with higher engineering feasibility is provided. The proposed method can be used for rapidly forming a large range of applications and is low in implementation cost.
Drawings
Fig. 1 is a flowchart of a traffic data recovery method based on non-parametric non-convex relaxation low-rank tensor completion according to an embodiment of the present invention.
Fig. 2 is a schematic diagram of a PeMS highway traffic data set (P).
FIG. 3 is a schematic representation of the Seattle highway traffic speed data set (S).
Fig. 4 is a schematic diagram of a traffic speed data set (G) of a city in Guangzhou.
Fig. 5 is a schematic diagram of a burminghan parking lot occupancy data set (B).
Detailed Description
The following examples are presented to enable one of ordinary skill in the art to more fully understand the present invention and are not intended to limit the invention in any way.
Referring to fig. 1, the embodiment discloses a traffic data restoration method based on parameterless non-convex relaxation low rank tensor completion, which includes the following steps:
s1, constructing traffic data containing missing data into three-dimensional tensor of position multiplied by date multiplied by time according to three dimensions of place, date and time
Figure BDA0003964156580000052
And S2, providing a logarithm-based non-parametric non-convex relaxation function, and constructing a low-rank tensor completion-based traffic data restoration model.
And S3, considering model solving efficiency and removing equality constraint, and constructing an augmented Lagrangian function of the model. According to the ADMM framework, the multivariate optimization problem of the model is converted into three simple single-variant quantum optimization problems, and tensor is initialized
Figure BDA0003964156580000053
Updating in sequence
Figure BDA0003964156580000054
Three variables.
S4, in step 3
Figure BDA0003964156580000055
As input, iterative optimization is performed by using an Alternating Direction Method of Multipliers (ADMM) until a convergence condition is satisfied, and a low rank tensor is obtained
Figure BDA0003964156580000056
The specific process of the step 1 is as follows:
the traffic space-time data generally includes label information of three types, i.e., a device position, a data acquisition date, and a data acquisition time, and a three-dimensional tensor of the traffic space-time data, i.e., the position x the date x the time, is constructed
Figure BDA0003964156580000061
To integrate these information, where n 1 Indicating the number of locations of the data-collecting device, n 2 Indicating the number of days the data was collected, n 3 Indicating the number of time segments of data collected per natural day.
The specific process of the step 2 is as follows:
s21, considering that the penalty of noise information is increased and the penalty of structure information is reduced at the same time, a logarithm-based parameterless non-convex relaxation function is provided, and the function is expressed as:
Figure BDA0003964156580000062
wherein σ i (X) denotes the ith singular value of the matrix X, ε > 0 to ensure positive certainty, typically 10-6 to 10-4.
S22, constructing a low-rank tensor completion model for traffic data restoration based on the non-parametric non-convex relaxation function provided in the S21, wherein the model is expressed as follows:
Figure BDA0003964156580000063
wherein,
Figure BDA0003964156580000064
tensor of representation
Figure BDA0003964156580000065
Along the expansion matrix of the k-th mode, k =1,2,3, α k Is a matrix L k(k) The weight of the regularization term of (c),
Figure BDA0003964156580000066
representing low rank tensor over an index set Ω of observable data
Figure BDA0003964156580000067
And observed value tensor
Figure BDA0003964156580000068
Are equal in value.
S23, considering the requirement of variable independence, introducing tensor
Figure BDA0003964156580000069
Respectively representing tensors
Figure BDA00039641565800000610
By unfolding the three modes, an auxiliary tensor is introduced
Figure BDA00039641565800000611
Will tensor
Figure BDA00039641565800000612
To the observation information in
Figure BDA00039641565800000613
In S22, the low rank tensor completion model can be further expressed as:
Figure BDA00039641565800000614
the specific process of the step 3 is as follows:
s31, considering the solving efficiency of the model and removing equality constraint, constructing an augmented Lagrange function of the model, and expressing as follows:
Figure BDA00039641565800000615
Figure BDA00039641565800000616
wherein,<·，·>the inner product is represented by the sum of the two,
Figure BDA00039641565800000617
the number of lagrange multipliers is represented,
Figure BDA00039641565800000618
is the square of the Frobenius norm (i.e., F norm), ρ k Representing the weight coefficient of the k-th mode.
And S32, converting the multivariable optimization problem of the model in the S3 into three simple univariate quantum optimization problems according to the ADMM framework. The method mainly comprises the following four steps:
(1) Initializing tensors
Figure BDA0003964156580000071
With the tensor in step S1
Figure BDA0003964156580000072
As input, the tensor is initialized:
Figure BDA0003964156580000073
(2) Updating a target tensor
Figure BDA0003964156580000074
To find
Figure BDA0003964156580000075
From the initialization tensor
Figure BDA0003964156580000076
Calculating tensors
Figure BDA0003964156580000077
Each modal auxiliary tensor of
Figure BDA0003964156580000078
Figure BDA0003964156580000079
Wherein,
Figure BDA00039641565800000710
is that
Figure BDA00039641565800000711
The mode expansion matrix of (a) is,
Figure BDA00039641565800000712
is that
Figure BDA00039641565800000713
The mode expansion matrix of (a) is,
Figure BDA00039641565800000714
for weighted singular value threshold operators, U (sigma) V T As an arbitrary matrix
Figure BDA00039641565800000715
The singular value of (a) is decomposed,
Figure BDA00039641565800000716
Figure BDA00039641565800000717
is composed of
Figure BDA00039641565800000718
The mode expansion matrix of (a) is,
Figure BDA00039641565800000719
representation matrix
Figure BDA00039641565800000720
Of the 1 st singular value, epsilon k Is a constant, usually 10-6 to 10-4, τ = α k /ρ k 。
Tensor assisted by modalities
Figure BDA00039641565800000721
Computing a low rank tensor
Figure BDA00039641565800000722
Figure BDA00039641565800000723
(3) Updating an auxiliary tensor
Figure BDA00039641565800000724
To find
Figure BDA00039641565800000725
By updated
Figure BDA00039641565800000726
Computing
Figure BDA00039641565800000727
Figure BDA00039641565800000728
(4) Updating lagrange multipliers
Figure BDA00039641565800000729
To find
Figure BDA00039641565800000730
After update
Figure BDA00039641565800000731
Computing
Figure BDA00039641565800000732
Figure BDA00039641565800000733
Wherein,
Figure BDA00039641565800000734
representing a four-dimensional tensor of size 3 x M x N x T,
Figure BDA00039641565800000735
respectively composed of three-dimensional tensors
Figure BDA00039641565800000736
k =1,2,3 are stacked in the fourth dimension,
Figure BDA00039641565800000737
from three identical three-dimensional tensors
Figure BDA00039641565800000738
Stacked on the fourth modality.
The specific process of the step 4 is as follows:
s41 updated in step S3
Figure BDA00039641565800000739
As inputs, variables are paired based on the ADMM method
Figure BDA00039641565800000740
Sequentially and iteratively updating until the convergence condition is met, and obtaining the repaired complete traffic data low-rank tensor
Figure BDA0003964156580000081
The pseudo-code of the algorithm is as follows:
Figure BDA0003964156580000082
case analysis
As shown in fig. 2 to 5, traffic data of an embodiment of the present invention is derived from the following four data sets:
(P) PeMS Highway traffic data set. This data set contained traffic collected from 228 annular detectors at 5 minute resolution (i.e., 288 time intervals per day) by the Performance Measurement System (PeMS) in region 7 of california during the 5 and 6 months of the 2012 operating days. The tensor size is 228 × 288 × 44.
(S) Seattle highway traffic speed data set. This data set contains the highway traffic speeds of 323 5-minute resolution (i.e., 288 time intervals per day) coil detectors four weeks before 1 month of seattle, usa. The tensor size is 323 × 288 × 28.
(G) A city traffic speed data set in Guangzhou city. This data set contains traffic speeds collected from 214 road segments in Guangzhou, china at a resolution of over two months (8/1/9/30/2016) and 10 minutes (i.e., 144 time intervals per day). The tensor size is 214 × 144 × 61.
(B) Birmingham parking lot occupancy data set. This data set recorded that 30 parking lots in birmingham city were separated from 8:00 to 17: occupancy (i.e., number of stops) every half hour between 00. The tensor size is 30 × 18 × 77.
(2) And (4) generating different deletion scenes and deletion rates by experimental deletion data.
In order to test the missing data repair capability of the present invention, two data loss modes were configured: random deletions and non-random deletions. According to the mechanism of random missing and non-random missing data, a certain number of observed values are used as missing values, 20%, 40%, 60% and 80% of data of each missing scene are respectively masked, and the rest observed values are used as input data.
The TC-PFNC visualization method proposed in this embodiment is used to visualize missing data time series and corresponding repair time series with extreme missing rates of 60% for the four data sets.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.

Claims (5)

1. A traffic data restoration method based on non-parametric non-convex relaxation low-rank tensor completion is characterized by comprising the following steps of:
s1, constructing traffic data containing missing data into three-dimensional tensor of position multiplied by date multiplied by time according to three dimensions of place, date and time
Figure FDA0003964156570000011
Figure FDA0003964156570000012
Wherein n is 1 Indicating the number of locations of the data-collecting device, n 2 Indicating the number of dates on which the data was collected, n 3 Representing the number of time segments of the data collected in each natural day;
s2, a logarithm-based non-parameter non-convex relaxation function is provided, and a low-rank tensor completion-based traffic data restoration model is constructed according to the logarithm-based non-parameter non-convex relaxation function; wherein a tensor is introduced
Figure FDA0003964156570000013
Respectively representing tensors
Figure FDA0003964156570000014
By unfolding the three modes, an auxiliary tensor is introduced
Figure FDA0003964156570000015
Tensor is expressed
Figure FDA0003964156570000016
To the observation information in
Figure FDA0003964156570000017
Performing the following steps;
s3, considering model solving efficiency and removing equality constraint, and constructing an augmented Lagrangian function of the model; according to the ADMM framework, converting the multivariate optimization problem of the model into three univariate quantum optimization problems, and initializing tensor
Figure FDA0003964156570000018
Updating in sequence
Figure FDA0003964156570000019
Three variables;
s4, in step S3
Figure FDA00039641565700000110
As input, iterative optimization is carried out by utilizing a cross direction multiplier algorithm until a convergence condition is met, and a low-rank tensor is obtained
Figure FDA00039641565700000111
2. The method as claimed in claim 1, wherein the step S2 of providing a logarithm-based non-parametric non-convex relaxation function to construct the traffic data restoration model based on the low-rank tensor completion comprises the steps of:
s21, considering that the penalty of noise information is increased and the penalty of structure information is reduced at the same time, a logarithm-based parameterless non-convex relaxation function is provided, and the function is expressed as:
Figure FDA00039641565700000112
wherein σ i (X) represents the ith singular value of the matrix X, epsilon is more than 0 to ensure positive nature, and the value range of epsilon is 10 -6 ～10 -4 ；
S22, constructing a low-rank tensor completion model for traffic data restoration based on the non-parametric non-convex relaxation function provided in the step S21, wherein the model is expressed as:
Figure FDA00039641565700000113
wherein,
Figure FDA00039641565700000114
tensor of representation
Figure FDA00039641565700000115
Along the expansion matrix of the k-th mode, k =1,2,3, α k Is a matrix L k(k) The weight of the regularization term of (a),
Figure FDA00039641565700000116
representing low rank tensor over an index set Ω of observable data
Figure FDA00039641565700000117
And observed value tensor
Figure FDA00039641565700000118
Are equal in value;
s23, considering the requirement of variable independence, introducing tensor
Figure FDA0003964156570000021
Respectively representing tensors
Figure FDA0003964156570000022
Along the expansion of three modes, auxiliary tensor is introduced
Figure FDA0003964156570000023
Will tensor
Figure FDA0003964156570000024
To the observation information in
Figure FDA0003964156570000025
In step S22, the low-rank tensor completion model is further expressed as:
Figure FDA0003964156570000026
3. the traffic data restoration method based on the parameterless non-convex relaxation low-rank tensor completion as claimed in claim 1, wherein in step S3, the augmented lagrangian function of the constructed model is represented as:
Figure FDA0003964156570000027
Figure FDA0003964156570000028
wherein,<·，·>the inner product is represented by the sum of the two,
Figure FDA0003964156570000029
the lagrange multiplier is represented by a number of lagrange multipliers,
Figure FDA00039641565700000210
is the square of the Frobenius norm, rho k Representing the weight coefficient of the k-th mode.
4. The method for restoring traffic data based on non-parametric non-convex relaxation low-rank tensor completion as claimed in claim 3, wherein in step S3, the process of converting the multivariate optimization problem of the model into three univariate quantum optimization problems according to the ADMM framework comprises the following steps:
s31, initializing tensor
Figure FDA00039641565700000211
With the tensor in step S1
Figure FDA00039641565700000212
As input, the tensor is initialized:
Figure FDA00039641565700000213
s32, updating the target tensor
Figure FDA00039641565700000214
To find
Figure FDA00039641565700000215
In particular, the amount of the solvent to be used,
from the initialization tensor
Figure FDA00039641565700000216
Calculating tensors
Figure FDA00039641565700000217
Each modal auxiliary tensor of
Figure FDA00039641565700000218
Figure FDA00039641565700000219
Wherein,
Figure FDA00039641565700000220
is that
Figure FDA00039641565700000221
The mode expansion matrix of (a) is,
Figure FDA00039641565700000222
is that
Figure FDA00039641565700000223
The mode expansion matrix of (a) is,
Figure FDA00039641565700000224
in order to weight the singular value threshold operator,
Figure FDA00039641565700000225
as an arbitrary matrix
Figure FDA00039641565700000226
The singular value of (a) is decomposed,
Figure FDA00039641565700000227
Figure FDA00039641565700000228
is composed of
Figure FDA00039641565700000229
The mode expansion matrix of (a) is,
Figure FDA00039641565700000230
representation matrix
Figure FDA00039641565700000231
Of the 1 st singular value, ε k Is a constant number epsilon k Has a value range of 10 -6 ～10 -4 ，τ＝α k /ρ k ；
Tensor assisted by modalities
Figure FDA0003964156570000031
Computing a low rank tensor
Figure FDA0003964156570000032
Figure FDA0003964156570000033
S33, updating the auxiliary tensor
Figure FDA0003964156570000034
To find
Figure FDA0003964156570000035
By renewed
Figure FDA0003964156570000036
Computing
Figure FDA0003964156570000037
Figure FDA0003964156570000038
S34, updating Lagrange multiplier
Figure FDA0003964156570000039
To find
Figure FDA00039641565700000310
After update
Figure FDA00039641565700000311
Computing
Figure FDA00039641565700000312
Figure FDA00039641565700000313
Wherein,
Figure FDA00039641565700000314
representing a four-dimensional tensor of size 3 x M x N x T,
Figure FDA00039641565700000315
respectively composed of three-dimensional tensors
Figure FDA00039641565700000316
The result is a stack in the fourth dimension,
Figure FDA00039641565700000317
from three identical three-dimensional tensors
Figure FDA00039641565700000318
Stacked on the fourth modality.
5. The method as claimed in claim 1, wherein the step S4 is performed by the step S3
Figure FDA00039641565700000319
As input, iterative optimization is carried out by utilizing a cross direction multiplier algorithm until a convergence condition is met, and a low-rank tensor is obtained
Figure FDA00039641565700000320
Comprises the following steps:
s41, inputting the partially observed traffic data observation tensor
Figure FDA00039641565700000321
S42, initializing each parameter:
Figure FDA00039641565700000322
p k ＝ρ＝ρ 0 ，ε＝1e-6；
s43, iteratively calculating the following formula until
Figure FDA00039641565700000323
Figure FDA00039641565700000324
Figure FDA00039641565700000325
Figure FDA00039641565700000326
l＝l+1；
Wherein,
Figure FDA00039641565700000327
s44, outputting the repaired complete traffic data low-rank tensor
Figure FDA00039641565700000328
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117271988A (en) * 2023-11-23 2023-12-22 广东工业大学 Tensor wheel-based high-dimensional signal recovery method and device
CN118445284A (en) * 2024-07-08 2024-08-06 中国科学院合肥物质科学研究院 Beidou space-time traffic data recovery method based on transformation-induced low-rank tensor

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN117271988A (en) * 2023-11-23 2023-12-22 广东工业大学 Tensor wheel-based high-dimensional signal recovery method and device
CN117271988B (en) * 2023-11-23 2024-02-09 广东工业大学 Tensor wheel-based high-dimensional signal recovery method and device
CN118445284A (en) * 2024-07-08 2024-08-06 中国科学院合肥物质科学研究院 Beidou space-time traffic data recovery method based on transformation-induced low-rank tensor

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