CN110120026B - Data recovery method based on Schatten Capped p norm - Google Patents

Data recovery method based on Schatten Capped p norm Download PDF

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CN110120026B
CN110120026B CN201910432024.7A CN201910432024A CN110120026B CN 110120026 B CN110120026 B CN 110120026B CN 201910432024 A CN201910432024 A CN 201910432024A CN 110120026 B CN110120026 B CN 110120026B
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李国瑞
郭光�
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Northeastern University Qinhuangdao Branch
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Abstract

The invention discloses a matrix completion method based on Schatten Capped p norm, which comprises the following steps: s1, for the incomplete data matrix of the input
Figure DDA0002069309420000011
Determining its corresponding orthogonal mapping operator
Figure DDA0002069309420000016
The orthogonal mapping operator represents a set of positions of the data matrix D, corresponding items of which are not empty;
Figure DDA0002069309420000012
representing the recovered matrix; s2, Schatten clipped p norm defining matrix
Figure DDA0002069309420000013
Wherein
Figure DDA0002069309420000014
Denotes a truncation parameter, θiDenotes the ith singular value of the matrix, p denotes the power exponent, p ∈ (0, 1)](ii) a S3, solving the optimization problem of the following formula until convergence, and outputting a complete data matrix
Figure DDA0002069309420000015
s.t.EΩ＝XΩ‑DΩAnd X is W, where W is an equivalent variable and γ is a penalty parameter. The method of the invention is used for matrix completion, so that the data matrix is low-rank, the loss of main information can be avoided, and the data recovery precision is high, namely the method has the advantages of low-rank incomplete matrixHas good recovery effect.

Description

Data recovery method based on Schatten Capped p norm
Technical Field
The invention relates to a data recovery method based on a Schatten Capped p norm, and belongs to the technical field of data recovery.
Background
In the fields of machine learning and data mining work, such as computer vision, collaborative filtering, signal processing, recommendation systems, and the like, engineers often recover high-dimensional information (original data) according to low-dimensional features (partial information) at a high rate, and such work can be performed because data abstracted from the original information has the characteristic of sparseness or low rank, and sparseness of vectors corresponds to low rank of matrices. Matrix filling is just one of the most classical applications of low rank nature.
The problem with matrix filling is that assuming in advance that the data matrix is low rank, there is correlation between the matrix elements, and the missing data can be recovered from the observed data based on minimizing the matrix values. For a given incomplete matrix
Figure GDA0002905777270000011
D is low rank, the filling problem of this matrix can be described as follows:
Figure GDA0002905777270000012
matrix array
Figure GDA0002905777270000013
And Ω is a set of positions related to the observation item, i.e., D is observation data (observed data), incomplete data (incomplete data), and X is finally complemented data. Since the rank function is non-convex and discontinuous, the minimization problem of the above equation (1) is an NP-hard problem. A typical solution for equation (1) is to replace the rank function with a kernel norm, since the theory proves that the kernel norm is the tightest convex lower bound of the rank function, and the kernel norm is a convex continuous function. The relation between the rank function and the kernel norm is similar to l0Norm sum l1Norm relationships, although the problem of matrix completion to minimize the nuclear norm is convexGlobally, an optimal solution can be found, but the convex relaxation may have a large deviation from the original data, so a better approximation needs to be found.
For the approximation of the nuclear norm, the property of the convex function is often sacrificed to obtain better effect. Of these, Schatten p norm and Capped norm are of interest to many scholars, and both of these large norms are non-convex approximations of the rank function. However, Schatten p norm considers each singular value, which is not in accordance with the characteristic of low rank (for smaller singular values, noise is often generated and should be removed, if the restoration effect is deteriorated if the noise is reserved), Capped norm only considers the rank size, and may lose some information (Capped norm essentially sets the smaller singular value to 0, and subtracts a small part from the larger singular value, so that although the rank is reduced, some main information is also lost), thereby resulting in poor data restoration effect. In addition, the existing TNNR-APGL algorithm, Logarithm-ADMM algorithm and Logarithm-IRNN algorithm are also commonly used for matrix completion to recover data, but overall, the data recovery quality of the existing TNNR-APGL algorithm, Logarithm-ADMM algorithm and Logarithm-IRNN algorithm is still not ideal. Further improvements are therefore desirable.
Disclosure of Invention
The invention aims to provide a data recovery method based on a Schatten Capped p-norm, which can effectively solve the problems in the prior art and realize high-precision and high-quality recovery of data.
In order to solve the technical problems, the invention adopts the following technical scheme: a data recovery method based on Schatten Capped p norm comprises the following steps:
s1, incomplete data matrix of input original data to be restored
Figure GDA0002905777270000021
Determining its corresponding orthogonal mapping operator
Figure GDA0002905777270000022
Said orthogonal mapping operator represents a set of positions of the data matrix D (of which the corresponding entries are not empty)Theory of things
Figure GDA0002905777270000023
Represents a set of positions where the corresponding entries of the data matrix D are empty, so Ω + Ωc＝ones(m,n))；
Figure GDA0002905777270000024
Representing the recovered matrix;
s2, Schatten clipped p norm defining matrix
Figure GDA0002905777270000025
Wherein
Figure GDA0002905777270000026
Denotes a truncation parameter, θiDenotes the ith singular value of the matrix, p denotes the power exponent, p ∈ (0, 1)]；
S3, solving the optimization problem of the following formula until convergence, and outputting a complete data matrix X, thereby realizing data recovery:
Figure GDA0002905777270000027
s.t.EΩ＝XΩ-DΩ,X＝W
wherein, W is an equivalent variable, and gamma is a penalty parameter.
XΩThe expression Ω and X corresponds to the multiplication of elements, whereas Ω can be understood as the matrix consists of only 0 and 1, 0 denoting the location where an element is missing and 1 denoting the location where an element is left.
Preferably, in step S3, the method of solving the optimization problem by using an alternating direction multiplier method based on the Schatten clamped p norm, Schatten clamped p norm-ADMM, specifically includes the following steps:
first, parameters are set and initialized: . Let DΩ＝W＝Y＝Z＝XΩMu is more than 0, beta is more than 0, rho is more than 1 and less than 2, tau is more than 0, lambda is more than 0, and p is more than 0 and less than or equal to 1; wherein Y, Z is a multiplier term, μ and β are penalty term parameters, and ρ is an update coefficient of the penalty parameter; lambda is a penalty parameter which is set as,
Figure GDA0002905777270000028
and then, repeatedly updating and iterating the following steps until the iteration number Iter or the difference of the variables of the two iterations is less than a certain amount:
1) fixed variables W and EΩUpdate the matrix X to be restored:
Figure GDA0002905777270000029
at the same time
Figure GDA0002905777270000031
And is
Figure GDA0002905777270000032
Wherein
Figure GDA0002905777270000033
2) Fix X and W, update error variable EΩ：
Figure GDA0002905777270000034
Wherein the content of the first and second substances,
Figure GDA0002905777270000035
3) fixing X and EΩUpdating an equivalent variable W:
Figure GDA0002905777270000036
wherein
Figure GDA0002905777270000037
4) Updating multiplier entries Z, YΩAnd penalty parameters μ and β:
YΩ＝YΩ-μ(XΩ-EΩ-DΩ)
Z＝Z-β(W-X)
μ＝ρμ
β＝ρβ
specifically, the formula is solved by the following steps
Figure GDA0002905777270000038
(since Schatten clamped p norm is a concave function, when p takes on (0, 1)]Equation of time
Figure GDA0002905777270000039
Cannot be solved with conventional methods):
first, initialization is performed: order formula
Figure GDA00029057772700000310
In W ═ U Σ VT，G＝QΔPT，δiFor the ith singular value element of Δ, the following equation can be used to solve for each singular value:
Figure GDA00029057772700000311
setting and initializing a parameter lambda to be more than 0;
secondly, solving sigma corresponding to singular value elements of each deltaiThe value on the main diagonal of sigma, sigma constituting the new matrix is sigmaiAnd the other positions are 0;
finally solving the formula
Figure GDA00029057772700000312
Is equal to Q sigma PT。
In the method, the iteration times can be set according to experience, and are converged often for dozens of times in an experiment, or the iteration can be stopped by using the difference of the variables of two iterations before and after being smaller than a certain amount.
The invention solves the optimization problem by using an alternating direction multiplier method based on the Schatten clamped p norm, has small calculation complexity and can conveniently carry out parallel operation on large-scale data.
More preferably, the formula is represented by the following steps
Figure GDA0002905777270000041
And (3) solving:
first, initialization is performed
Figure GDA0002905777270000042
v1＝v+λpvp-1；
Secondly, solve the optimum x*: when deltaiLess than v1When x*Equal to 0; when deltaiIs equal to v1When x*Is equal to v; when deltaiGreater than v1When x*Is determined by the following method:
(1) by x: (0) Initialization deltai；
(2) And (3) iterative calculation:
x(i+1)＝δi-λp(x(i))p-1
after convergence, an optimal solution x is obtained*；
Finally, a formula is obtained
Figure GDA0002905777270000043
The optimal solution of (2): if τ is ≦ τ*Then σ*If τ > τ*Then σ*＝x*(ii) a Wherein the content of the first and second substances,
Figure GDA0002905777270000044
in the present invention, since Schatten clamped p norm is a concave function, when p takes (0, 1)]To formula
Figure GDA0002905777270000045
The optimization problem can not be solved by the conventional method, so the inventor proposes the above solving method, which is also the innovation difficulty of the invention. Tong (Chinese character of 'tong')The method not only realizes the solution of the formula, but also has very high precision.
In the aforementioned data recovery method based on the Schatten Capped p norm, the value of p is greater than or equal to 0.6 and less than or equal to 0.9. Therefore, the optimal solution can be obtained more quickly, and the efficiency and the precision of matrix completion are improved.
In the aforementioned data recovery method based on Schatten Capped p-norm, the value of τ is greater than or equal to 30. Thereby improving the accuracy of matrix recovery.
In the aforementioned data recovery method based on the Schatten Capped p norm, λ < τ2-pAnd/p (1-p), thereby effectively reducing information loss and enabling the accuracy of data recovery to be higher.
Compared with the prior art, the invention provides a new norm: schatten p- τ norm, which essentially selects the singular values σ of the finally recovered matrix X by setting the truncation parameteriTherefore, the method of the invention can be used for matrix completion, the data matrix is low-rank, the loss of main information can be avoided, and the data recovery quality is improved, namely, the invention has better recovery effect on incomplete matrixes with low-rank property.
The technical difficulty of the invention lies in that tau is added into a new norm Schatten p-tau norm which is a concave function when p is (0, 1)]To formula
Figure GDA0002905777270000051
The optimization problem can not be solved by a conventional method, and the calculation is more complex, so that the inventor proposes the solving method, and the data recovery effect of the invention is better.
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FIG. 1 is a work flow diagram of one embodiment of the present invention;
FIG. 2 is a schematic diagram of the effect of using the method of the present invention to recover pictures (the first row of pictures represents the pixel missing at random positions with different ratios, and the second row represents the recovery result corresponding to using the method of the present invention);
fig. 3 is a schematic diagram showing the effect of completing four different pictures by using different algorithms (each row is used for completing four different pictures with different data loss, and the same column is used for the same algorithm);
FIG. 4 is a graph of RE and PSNR as a function of p, preserving the 50% randomly positioned pixel loss for different singular values;
FIG. 5 is a diagram of PSNR corresponding to different algorithms in Table 1 for picture completion;
fig. 6 is a schematic diagram of corresponding REs when performing image completion by using different algorithms in table 1.
In order that the above objects, features and advantages of the present invention can be more clearly understood, a more particular description of the invention will be rendered by reference to the appended drawings. It should be noted that the embodiments and features of the embodiments of the present application may be combined with each other without conflict.
In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention, however, the present invention may be practiced in other ways than those specifically described herein, and therefore the scope of the present invention is not limited by the specific embodiments disclosed below.
Detailed Description
The embodiment of the invention comprises the following steps: a data recovery method based on Schatten Capped p-norm, as shown in fig. 1, includes the following steps:
s1, incomplete data matrix of input original data to be restored
Figure GDA0002905777270000052
Determining its corresponding orthogonal mapping operator
Figure GDA0002905777270000053
Said orthogonal mapping operator represents a set of positions of the data matrix D for which the corresponding entries are not empty (in the same way)
Figure GDA0002905777270000054
Represents a set of positions where the corresponding entries of the data matrix D are empty, so Ω + Ωc＝ones(m,n))；
Figure GDA0002905777270000055
Representing the recovered matrix;
s2, Schatten clipped p norm defining matrix
Figure GDA0002905777270000056
Wherein
Figure GDA0002905777270000057
Denotes a truncation parameter, θiDenotes the ith singular value of the matrix, p denotes the power exponent, p ∈ (0, 1)]；
S3, solving the optimization problem of the following formula until convergence, and outputting a complete data matrix X, thereby realizing data recovery:
Figure GDA0002905777270000061
s.t.EΩ＝XΩ-DΩ,X＝W
wherein, W is an equivalent variable, and gamma is a penalty parameter.
XΩThe expression Ω and X corresponds to the multiplication of elements, whereas Ω can be understood as the matrix consists of only 0 and 1, 0 denoting the location where an element is missing and 1 denoting the location where an element is left.
Preferably, in step S3, the method of solving the optimization problem using an alternating direction multiplier method based on the Schatten clamped p norm, Schatten clamped p norm-ADMM (which may also be optimized by using other existing methods), specifically includes the following steps:
first, parameters are set and initialized: . Let DΩ＝W＝Y＝Z＝XΩMu is more than 0, beta is more than 0, rho is more than 1 and less than 2, tau is more than 0, lambda is more than 0, and p is more than 0 and less than or equal to 1; wherein Y, Z is a multiplier term, μ and β are penalty term parameters, and ρ is an update coefficient of the penalty parameter; lambda is a penalty parameter which is set as,
Figure GDA0002905777270000062
and then, repeatedly updating and iterating the following steps until the iteration number Iter or the difference of the variables of the two iterations is less than a certain amount:
1) fixed variables W and EΩUpdate the matrix X to be restored:
Figure GDA0002905777270000063
at the same time
Figure GDA0002905777270000064
And is
Figure GDA0002905777270000065
Wherein
Figure GDA0002905777270000066
2) Fix X and W, update error variable EΩ：
Figure GDA0002905777270000067
Wherein the content of the first and second substances,
Figure GDA0002905777270000068
3) fixing X and EΩUpdating an equivalent variable W:
Figure GDA0002905777270000069
wherein
Figure GDA00029057772700000610
4) Updating multiplier entries Z, YΩAnd penalty parameters μ and β:
YΩ＝YΩ-μ(XΩ-EΩ-DΩ)
Z＝Z-β(W-X)
μ＝ρμ
β＝ρβ
specifically, the formula is solved by the following steps
Figure GDA0002905777270000071
(since Schatten clamped p norm is a concave function, when p takes on (0, 1)]Equation of time
Figure GDA0002905777270000072
Cannot be solved with conventional methods):
first, initialization is performed: order formula
Figure GDA0002905777270000073
In W ═ U Σ VT，G＝QΔPT，δiFor the ith singular value element of Δ, the following equation can be used to solve for each singular value:
Figure GDA0002905777270000074
setting and initializing a parameter lambda to be more than 0;
secondly, solving sigma corresponding to singular value elements of each deltaiThe value on the main diagonal of sigma, sigma constituting the new matrix is sigmaiAnd the other positions are 0;
finally solving the formula
Figure GDA0002905777270000075
Is equal to Q sigma PT。
In the method, the iteration times can be set according to experience, and are converged often for dozens of times in an experiment, or the iteration can be stopped by using the difference of the variables of two iterations before and after being smaller than a certain amount.
Preferably, the formula can be represented by the following steps
Figure GDA0002905777270000076
And (3) solving:
firstly, the methodInitialization of
Figure GDA0002905777270000077
v1＝v+λpvp-1；
Secondly, solve the optimum x*: when deltaiLess than v1When x*Equal to 0; when deltaiIs equal to v1When x*Is equal to v; when deltaiGreater than v1When x*Is determined by the following method:
(1) by x(0)Initialization deltai；
(2) And (3) iterative calculation:
x(i+1)＝δi-λp(x(i))p-1
after convergence, an optimal solution x is obtained*；
Finally, a formula is obtained
Figure GDA0002905777270000081
The optimal solution of (2): if τ is ≦ τ*Then σ*If τ > τ*Then σ*＝x*(ii) a Wherein the content of the first and second substances,
Figure GDA0002905777270000082
in the present invention, since Schatten clamped p norm is a concave function, when p takes (0, 1)]To formula
Figure GDA0002905777270000083
The optimization problem can not be solved by the conventional method, so the inventor proposes the above solving method, which is also the innovation difficulty of the invention. By the method, the formula is solved, and the precision is very high.
In the aforementioned data recovery method based on the Schatten Capped p norm, the value of p is greater than or equal to 0.6 and less than or equal to 0.9. Therefore, the optimal solution can be obtained more quickly, and the efficiency and the precision of matrix completion are improved.
In the aforementioned data recovery method based on Schatten Capped p-norm, the value of τ is greater than or equal to 30. Thereby improving the accuracy of matrix recovery.
In the aforementioned data recovery method based on the Schatten Capped p norm, λ < τ2-p/p(1-p)
In addition, the inventor also carries out matrix filling on different data to verify the effectiveness of the method provided by the invention.
Different matrix completion algorithms include:
the algorithm Schatten clamped p norm-ADMM is based on the ADMM method of Schatten clamped p norm provided by the invention;
TNNR-APGL algorithm: APGL matrix completion algorithm based on truncation kernel norm punishment;
the Schatten p-ADMM algorithm, wherein the Schatten p norm is an effective approximation to a rank function;
Logiathm-ADMM algorithm: ADMM algorithm based on Logiathm penalty;
the Log arithm-IRNN algorithm is an IRNN algorithm based on Log arithm punishment;
the clamped-L1-IRNN has non-convex property, and the optimized IRNN algorithm is solved by using sub-differentiation.
The experiment is simulated on matlab by using a desktop computer with i5-6500 CPU and 4G memory.
One, random position missing data
And comparing the recovery conditions of different algorithms on the pictures with different random loss ratios. Fig. 2 shows the recovery result of the algorithm of the present invention. The pixel missing positions are randomly set by a ratio, and the three channels are set to the same missing position. The recovery conditions of different algorithms are shown in table 1, and it can be seen from table 1 that the recovery effect of TNNR-APGL is slightly better than that of the method of the present invention when 20% of random data is retained, and the method of the present invention is optimal in other cases; PSNR and RE recovered by different algorithms are shown in fig. 5 and 6. PSNR (Peak Single-to-Noise Ratio) and SNR are common discrimination indexes of image recovery quality, represent the Ratio of signal power and destructive Noise power influencing the representation precision, the larger the value is, the better the recovery quality is, and the comparison of the algorithm of the invention leads to that the recovery quality is betterCalculated using the Matlab embedded psnr function. RE (relative error) represents the ratio of the absolute error to the original data
Figure GDA0002905777270000091
A smaller value indicates a better recovery quality.
TABLE 1 recovery of different matrix completion algorithms at different data loss rates
Figure GDA0002905777270000092
Two, block position missing data
In practical application, an image is usually a data matrix with a low rank effect, main information of the image is concentrated in a few larger singular values at the front, so that matrix completion for the image is a common experimental method, and the experiment completes the matrix completion by processing each channel separately by using a common RGB three-channel image, without considering correlation between channels.
The algorithm of the present invention and other five algorithms are to approximate a low rank function by adding a non-convex penalty term to an objective function, as shown in fig. 3, (h) is the result recovered by the algorithm of the present invention, (a) is the original image without pixel missing, (b) is pixel missing with different shapes: there are large areas of triangular area missing, block missing of different areas, and text occlusion missing. By comparing different algorithms, the algorithm disclosed by the invention can recover the pictures in the column (b) with a better recovery effect in various scenes. The specific comparison data are detailed in the PSNR comparison given in table 2, and as can be seen from table 2, except that the recovery effect of the volcanic image occluded by the characters is Capped-IRNN is slightly better, the recovery effect of the method of the present invention is the best.
TABLE 2 recovery of different matrix completion algorithms under different block position deletions
Figure GDA0002905777270000101
In addition, for the selection of the p value in the Schatten p-tau norm, the inventors also performed the following experiments to screen:
in the experimental process, the inventor finds that the p value of the norm of the penalty term is not as small as possible, but is a certain value in the middle of (0, 1) to achieve the best recovery quality.
As shown in fig. 4, the present invention selects an image lena (beauty image) and an algorithm of the present invention, (a) is a pixel at a position where random 50% of the original image is removed, PSNR and RE at different values of p are obtained by using the algorithm of the present invention, and (b) (c) (d) is a pixel at a position where random 50% of the original image is removed and the previous 30,20,10 singular values are retained after the original image SVD is decomposed, and similarly PSNR and RE at different values of p are obtained by using the algorithm of the present invention.
As can be seen from fig. 4, the stricter the lower rank pictures are, the better the quality recovered by the method of the present invention is. Meanwhile, the value of p with the best recovery quality is not at two ends of the definition domain, but the optimal value of p is obtained in the middle section, namely the value of p is not close to 0 or close to 1, but the value between [0.6 and 0.9] has better effect.

Claims (6)

1. An image data recovery method based on Schatten Capped p-norm, characterized by comprising the following steps:
s1, incomplete data matrix D epsilon R of input original image data to be restoredm×nFind its corresponding orthogonal mapping operator
Figure DEST_PATH_IMAGE002
The orthogonal mapping operator represents a set of positions of the data matrix D, corresponding items of which are not empty; x is formed by Rm×nRepresenting the recovered matrix;
s2, Schatten clipped p norm defining matrix
Figure FDA0003324598620000012
Figure FDA0003324598620000013
Where τ ∈ R > 0 denotes the truncation parameter, θiDenotes the ith singular value of the matrix, p denotes the power exponent, p ∈ (0, 1)]；
S3, solving the optimization problem of the following formula until convergence, and outputting a complete data matrix X, thereby realizing image data recovery:
Figure FDA0003324598620000014
s.t.EΩ＝XΩ-DΩ,X＝W
wherein, W is an equivalent variable, and gamma is a penalty parameter.
2. The method for restoring image data according to claim 1, wherein in step S3, an optimization problem is solved by using an alternating direction multiplier method based on the Schatten clamped p norm, Schatten clamped p norm-ADMM, which specifically includes the following steps:
first, parameters are set and initialized:
let DΩ＝W＝Y＝Z＝XΩMu is more than 0, beta is more than 0, rho is more than 1 and less than 2, tau is more than 0, lambda is more than 0, and p is more than 0 and less than or equal to 1; wherein Y, Z is a multiplier term, μ and β are penalty term parameters, and ρ is an update coefficient of the penalty parameter; lambda is a penalty parameter which is set as,
Figure FDA0003324598620000015
and then, repeatedly updating and iterating the following steps until the iteration number Iter or the difference of the variables of the two iterations is less than a certain amount:
1) fixed variables W and EΩUpdate the matrix X to be restored:
Figure FDA0003324598620000016
at the same time
Figure FDA0003324598620000017
And is
Figure FDA0003324598620000018
Wherein
Figure FDA0003324598620000019
2) Fix X and W, update error variable EΩ：
Figure FDA00033245986200000110
Wherein the content of the first and second substances,
Figure FDA00033245986200000111
3) fixing X and EΩUpdating an equivalent variable W:
Figure FDA0003324598620000021
wherein
Figure FDA0003324598620000022
4) Updating multiplier entries Z, YΩAnd penalty parameters μ and β:
YΩ:＝YΩ-μ(XΩ-EΩ-DΩ)
Z:＝Z-β(W-X)
μ:＝ρμ
β:＝ρβ
specifically, the formula is solved by the following steps
Figure FDA0003324598620000023
First, initialization is performed: order formula
Figure FDA0003324598620000024
In W ═ U Σ VT，G＝QΔPT，δiFor the ith singular value element of Δ, the following equation can be used to solve for each singular value:
Figure FDA0003324598620000025
setting and initializing a parameter lambda to be more than 0;
secondly, solving sigma corresponding to singular value elements of each deltaiThe value on the main diagonal of the new matrix Σ, Σ is σiAnd the other positions are 0;
finally solving the formula
Figure FDA0003324598620000026
Is equal to Q sigma PT。
3. The method for restoring image data based on the Schatten Capped p-norm as claimed in claim 2, wherein the formula is represented by the following steps
Figure FDA0003324598620000027
And (3) solving:
first, initialization is performed
Figure FDA0003324598620000028
v1＝v+λpvp-1；
Secondly, solve the optimum x*: when deltaiLess than v1When x*Equal to 0; when deltaiIs equal to v1When x*Is equal to v; when deltaiGreater than v1When x*Determined by the following method:
(1) by x(0)Initialization deltai；
(2) And (3) iterative calculation:
x(i+1)＝δi-λp(x(i))p-1
after convergence, an optimal solution x is obtained*；
Finally, a formula is obtained
Figure FDA0003324598620000031
The optimal solution of (2): if τ is ≦ τ*Then σ*If τ > τ*Then σ*＝x*(ii) a Wherein the content of the first and second substances,
Figure FDA0003324598620000032
4. the image data recovery method based on the Schatten Capped p-norm as claimed in any one of claims 1 to 3, wherein the value of p is greater than or equal to 0.6 and less than or equal to 0.9.
5. The image data recovery method based on the Schatten Capped p-norm as claimed in any one of claims 1 to 3, wherein τ is greater than or equal to 30.
6. The image data recovery method based on the Schatten Capped p-norm as claimed in claim 2 or 3,
Figure FDA0003324598620000033
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Publication number Priority date Publication date Assignee Title
CN111028162B (en) * 2019-11-26 2023-03-31 广东石油化工学院 Image missing recovery method based on truncated Schattenp-norm
CN111159638B (en) * 2019-12-26 2023-12-08 华南理工大学 Distribution network load missing data recovery method based on approximate low-rank matrix completion

Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102881002A (en) * 2012-07-11 2013-01-16 天津大学 Video background recovery method based on movement information and matrix completion
CN103345729A (en) * 2013-06-30 2013-10-09 浙江贝尔技术有限公司 Image restoration method based on truncation nuclear norm regularization
CN103679660A (en) * 2013-12-16 2014-03-26 清华大学 Method and system for restoring image
CN109102477A (en) * 2018-08-31 2018-12-28 哈尔滨工业大学 A kind of high-spectrum remote sensing restoration methods based on the constraint of non-convex low-rank sparse
CN109658362A (en) * 2018-12-30 2019-04-19 东北大学秦皇岛分校 Three-dimensional tensor complementing method based on Capped nuclear norm
CN109671030A (en) * 2018-12-10 2019-04-23 西安交通大学 A kind of image completion method based on the optimization of adaptive rand estination Riemann manifold

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20170061452A1 (en) * 2015-08-31 2017-03-02 Wal-Mart Stores, Inc. System for forecasting using low-rank matrix completion and method therefor
US20180306884A1 (en) * 2017-04-21 2018-10-25 Mayo Foundation For Medical Education And Research Accelerated dynamic magnetic resonance imaging using low rank matrix completion

Patent Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102881002A (en) * 2012-07-11 2013-01-16 天津大学 Video background recovery method based on movement information and matrix completion
CN103345729A (en) * 2013-06-30 2013-10-09 浙江贝尔技术有限公司 Image restoration method based on truncation nuclear norm regularization
CN103679660A (en) * 2013-12-16 2014-03-26 清华大学 Method and system for restoring image
CN109102477A (en) * 2018-08-31 2018-12-28 哈尔滨工业大学 A kind of high-spectrum remote sensing restoration methods based on the constraint of non-convex low-rank sparse
CN109671030A (en) * 2018-12-10 2019-04-23 西安交通大学 A kind of image completion method based on the optimization of adaptive rand estination Riemann manifold
CN109658362A (en) * 2018-12-30 2019-04-19 东北大学秦皇岛分校 Three-dimensional tensor complementing method based on Capped nuclear norm

Non-Patent Citations (3)

* Cited by examiner, † Cited by third party
Title
Matrix completion with capped nuclear norm via majorized proximal minimization;shenfen kuang 等;《Neurocomputing》;20181117;第316卷;190-201 *
低秩矩阵恢复算法综述;史加荣 等;《计算机应用研究》;20130219;第30卷(第6期);1601-1605 *
基于范数正则化矩阵补全的无线传感网定位算法;肖甫 等;《计算机研究与发展》;20160115;第53卷(第1期);216-227 *

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