CN110120026A - Matrix complementing method based on Schatten Capped p norm - Google Patents
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Abstract
The invention discloses a kind of matrix complementing methods based on Schatten Capped p norm, comprising the following steps: S1, to the incomplete data matrix of inputFind out its corresponding orthogonal mapping operatorThe respective items of the orthogonal mapping operator representation data matrix D are not the set of empty position；Indicate the matrix after restoring；S2 defines the Schatten Capped p norm of matrixWhereinIndicate Truncation Parameters, θiI-th of singular value of representing matrix, p expression power exponent, p ∈ (0,1]；S3 solves the optimization problem of following formula, until convergence, exports the data matrix of completions.t.EΩ=XΩ‑DΩ, X=W, wherein W is equivalence variable, and γ is punishment parameter.Matrix completion is carried out by means of the present invention, so that data matrix is low-rank, and can guarantee that main information does not lose, accuracy of data recovery is high, i.e., the present invention there are good recovery effects for the incomplete matrix with low-rank property.
Description
Technical Field
The invention relates to a matrix completion method based on a Schatten Capped p norm, belonging 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 matrixD is low rank, the filling problem of this matrix can be described as follows:
matrix arrayAnd Ω is a set of positions related to the observation item, i.e., D is observation data (observeaddata), incomplete data (incomplete data), and X is finally complemented data. Due to the fact thatThe rank function is non-convex and discontinuous, so the minimization problem of the above equation (1) is NP-hard. 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 and l1In the norm relationship, although the problem of matrix completion to minimize the kernel norm is convex, and an optimal solution can be found globally, the convex relaxation may have a large deviation from the original data, so that a better approximation mode 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 matrix completion 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 matrix completion method based on Schatten Capped p norm comprises the following steps:
s1, for the incomplete data matrix of the inputDetermining its corresponding orthogonal mapping operatorSaid 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)Represents a set of positions where the corresponding entry of the data matrix D is empty, so Ωc+Ω＝eyes(m，n))；Representing the recovered matrix;
s2, Schatten clipped p norm defining matrix WhereinDenotes 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:
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 the Schatten clamped p norm-based alternating direction multiplier method, specifically includes the following steps:
first, parameters are set and initialized:
let DΩ＝W＝Y＝Z＝XΩMu is more than 0, β is more than 0,1 is more than rho is less than 2, tau is more than 0, lambda is more than 0, 0 is more than p and less than or equal to 1, wherein, Y, Z is multiplier item, mu and β are punishment item parameters, rho is updating coefficient of punishment parameter, lambda is punishment parameter,
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 EnUpdate the matrix X to be restored:
at the same timeAnd isWherein
2) Fix X and W, update error variable EΩ：
Wherein,
3) fixing X and EnUpdating an equivalent variable W:
wherein
4) Updating multiplier terms 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(Schattencapped p norm is a concave function when p is (0, 1)]So that formulaCannot be solved with conventional methods):
first, initialization is performed: order formulaW ═ U ∑ VT，G＝QΔPT，δiFor the ith singular value element of Δ, the following equation can be used to solve for each singular value:
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 forming the new matrix sigma, sigma is sigmaiAnd the other positions are 0;
finally solving the formulaIs equal to Q ∑ 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 stepsAnd (3) solving:
first, initialization is performedv1＝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 obtainedThe optimal solution of (2): if τ is ≦ τ*Then σ*If τ > τ*Then σ*＝x*(ii) a Wherein,
in the present invention, since Schatten clamped p norm is a concave function, when p takes (0, 1)]To formulaThe 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 matrix completion 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 matrix completion 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.
The foregoing description of the inventionIn the matrix completion method based on the Schatten Capped p-norm,therefore, the information loss can be effectively reduced, and the accuracy of data recovery is 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, and when p is (0, 1)]To formulaThe 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 matrix completion method based on Schatten Capped p-norm, as shown in fig. 1, comprising the following steps:
s1, for the incomplete data matrix of the inputDetermining its corresponding orthogonal mapping operatorSaid 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)Represents a set of positions where the corresponding entry of the data matrix D is empty, so Ωc+Ω＝eyes(m，n))；Representing the recovered matrix;
s2, Schatten clipped p norm defining matrix WhereinDenotes 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:
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 optimization problem may be solved by using an alternating direction multiplier method based on the Schatten clamped p norm, Schatten clamped p norm-ADMM (or may be optimized by using other existing methods), which specifically includes the following steps:
first, parameters are set and initialized:
let DΩ＝W＝Y＝Z＝XΩ，μ＞0，β＞0，1＜ρ＜2，τ＞0，λ＞0，0＜p is less than or equal to 1, wherein Y, Z is a multiplier item, mu and β are penalty item parameters, rho is an updating coefficient of the penalty parameter, lambda is the penalty parameter,
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:
at the same timeAnd isWherein
2) Fix X and W, update error variable EΩ：
Wherein,
3) fixing X and EΩUpdating an equivalent variable W:
wherein
4) Updating multiplier terms 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(Schattencapped p norm is a concave function when p is (0, 1)]So that formulaCannot be solved with conventional methods):
first, initialization is performed: order formulaIn W ═ U Σ VT，G＝QΔPτ，δiFor the ith singular value element of Δ, the following equation can be used to solve for each singular value:
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 forming the new matrix sigma, sigma is sigmaiAnd the other positions are 0;
finally solving the formulaIs equal to Q ∑ 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 stepsAnd (3) solving:
first, initialization is performedv1＝v+λpvp-1；
Secondly, solve the optimum x*: when deltaiLess than v1When x*Equal to 0; when deltaiIs equal to v1Xw 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 obtainedThe optimal solution of (2): if τ is ≦ τ*Then σ*If τ > τ*Then σ*＝x*(ii) a Wherein,
in the present invention, since Schatten clamped p norm is a concave function, when p takes (0, 1)]To formulaThe 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 matrix completion 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 matrix completion 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 matrix completion method based on Schatten Capped p-norm,
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 i5-6500CPU 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 commonly used discrimination indexes of image recovery quality, represent the Ratio of signal power and destructive Noise power influencing the representation precision of the signal power, the larger the value is, the better the recovery quality is, and the comparison of the algorithm of the invention is calculated by using a PSNR function embedded in matlab. RE (relative error) represents the ratio of the absolute error to the original dataA smaller value indicates a better recovery quality.
TABLE 1 recovery of different matrix completion algorithms at different data loss rates
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
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. A matrix completion method based on Schatten Capped p norm is characterized by comprising the following steps:
s1, for the incomplete data matrix of the inputDetermining its corresponding orthogonal mapping operatorSaid orthogonal mapping operator representing a data matrixA set of locations for which the corresponding item of D is not empty;representing the recovered matrix;
s2, Schatten clipped p norm defining matrix WhereinDenotes 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:
s.t.EΩ＝XΩ-DΩ，X＝W
wherein, W is an equivalent variable, and gamma is a penalty parameter.
2. The matrix completion method based on the Schatten clamped p-norm as claimed in claim 1, wherein in step S3, the optimization problem is solved by using an alternating direction multiplier method based on the Schatten clamped p-norm, specifically comprising the following steps:
first, parameters are set and initialized:
let DΩ＝W＝Y＝Z＝XΩMu is more than 0, β is more than 0,1 is more than rho is less than 2, tau is more than 0, lambda is more than 0, 0 is more than p and less than or equal to 1, wherein, Y, Z is multiplier item, mu and β are punishment item parameters, rho is updating coefficient of punishment parameter, lambda is punishment parameter,
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:
at the same timeAnd isWherein
2) Fix X and W, update error variable EΩ：
Wherein,
3) fixing X and EΩUpdating an equivalent variable W:
wherein
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
First, initialization is performed: order formulaW ═ U ∑ VT，G＝QΔPT，δiFor the ith singular value element of Δ, the following equation can be used to solve for each singular value:
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 forming the new matrix sigma, sigma is sigmaiAnd the other positions are 0;
finally solving the formulaIs equal to Q ∑ PT。
3. The method of claim 2, wherein the formula is complemented by the following stepsAnd (3) solving:
first, initialization is performedv1＝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 obtainedThe optimal solution of (2): if τ is ≦ τ*Then σ*If τ > τ*Then σ*＝x*(ii) a Wherein,
4. the matrix completion 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 matrix completion 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 method of any of claims 1 to 3, wherein the Schatten Capped p-norm-based matrix completion method,
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| CN111159638A (en) * | 2019-12-26 | 2020-05-15 | 华南理工大学 | Power distribution network load missing data recovery method based on approximate low-rank matrix completion |
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