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 present invention relates to a kind of matrix complementing methods based on Schatten Capped p norm, belong to data and restore skill
Art field.
Background technique
In machine learning and data excacation, such as computer vision, collaborative filtering, signal processing, recommender system
Equal fields, engineer often recover high dimensional information (initial data) according to low-dimensional feature (partial information) maximum probability, this
Class work the reason of can carrying out is exactly that the data that take out of raw information have the characteristics that sparse or low-rank, vector it is sparse
Property correspond to matrix low-rank.Matrix fill-in is exactly one of most classic application of low-rank property.
The problem of matrix fill-in processing is that prior hypothesis data matrix is low-rank, has correlation between matrix element, then
The data of missing can be recovered according to the data for the value observation for minimizing matrix.For giving an incomplete matrixD is low-rank, and the filling problem of the matrix can be described as follows:
MatrixAnd Ω is location sets relevant to observation item, i.e. D is observation data (observed
It data), is incomplete data (incomplete data), and X is the data of final completion.Due to rank function be it is non-convex and
It is discontinuous, so the minimization problem of above-mentioned formula (1) is np hard problem.The common solution for formula (1) is by order
Function is replaced with nuclear norm, because theoretical proof nuclear norm is the most tight convex lower bound of rank function, and nuclear norm is convex company
Continuous function.The relationship of rank function and nuclear norm is similar to l0Norm and l1The relationship of norm, although minimizing the matrix of nuclear norm
Completion problem be it is convex, optimal solution can be found in the overall situation, but this convex relaxation may have with original data it is biggish
Deviation, it is therefore desirable to find a kind of better approximate way.
For the approximation of nuclear norm, the property for sacrificing convex function is generally required, in the hope of obtaining better effect.Wherein
Schatten p norm and Capped norm receive the concern of many scholars, this two major classes norm is all that rank function is non-convex
It is approximate.But the characteristics of each singular value is taken into account, do not meet low-rank by Schatten p norm is (for lesser surprise
Different value, often there is noise, it should remove, if retaining meeting so that recovery effects variation), Capped norm if is to only considered
Some information, may be lost that (Capped norm essence is that lesser singular value is set to 0, by biggish surprise by the size of order
Different value subtracts sub-fraction, although reducing order in this way, also has lost some main informations), it is extensive so as to cause data
Multiple effect is poor.In addition, existing TNNR-APGL algorithm, Logarithm-ADMM algorithm and Logarithm-IRNN algorithm
It is also commonly used for matrix completion, carries out the recovery of data, but generally speaking, their data Quality of recovery is still not enough managed
Think.Therefore it requires further improvement.
Summary of the invention
The object of the present invention is to provide a kind of matrix complementing methods based on Schatten Capped p norm, it can
Effectively to solve problems of the prior art, high-precision, the high quality resume of data are realized.
In order to solve the above technical problems, the present invention adopts the following technical scheme that: one kind being based on Schatten Capped p
The matrix complementing method of 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 collection of empty position
It closes (similarlyIndicate that data matrix D respective items are the set of empty position, so Ωc+ Ω=
Eyes (m, n))；Indicate the matrix after restoring；
S2 defines the Schatten Capped p norm of matrix
WhereinIndicate 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 X of completion:
s.t.EΩ=XΩ-DΩ, X=W
Wherein, W is equivalence variable, and γ is punishment parameter.
XΩIndicate that Ω with X corresponding element is multiplied, and Ω can be understood as the matrix and there was only 0 and 1 composition, 0 indicates that element lacks
The position of mistake, 1 indicates the position that element retains.
Preferably, in step S3, the alternating direction multipliers method Schatten based on Schatten Capped p norm is used
Capped p norm-ADMM solves optimization problem, specifically includes the following steps:
Firstly, setting and initiation parameter:
Enable DΩ=W=Y=Z=XΩ, 0,1 < ρ < 2 of μ > 0, β >, 0,0 p≤1 < τ > 0, λ >；Wherein, Y, Z are multipliers
, μ, β are penalty term parameters, and ρ is the update coefficient of punishment parameter；λ is punishment parameter,
Secondly, iteration following steps are updated repeatedly, until reaching the number of iterations Iter or the front and back variable of iteration twice
It is a certain amount of that difference is less than certain:
1) fixed variable W and En, update the matrix X to be restored:
SimultaneouslyAndWherein
2) fixed X and W, updates error variance EΩ:
Wherein,
3) fixed X and En, update equivalence variable W:
Wherein
4) multiplier item z, Y are updatedΩWith punishment parameter μ and β:
YΩ:=YΩ-μ(XΩ-EΩ-DΩ)
Z:=z- β (W-X)
μ :=ρ μ
β :=ρ β
Specifically, passing through following steps solution formula(Schatten
Capped p norm is concave function, when p takes (0,1], so formulaIt is excellent
Change problem cannot be solved with conventional method):
Firstly, being initialized: enabling formulaIn, W=U ∑ VT, G=Q Δ
PT, δiFor i-th of singular value element of Δ, each singular value can be solved with following formula:
Wherein, setting and initiation parameter λ > 0；
Secondly, by the corresponding solution σ of the singular value element of each Δi, new matrix ∑ is formed, on the leading diagonal of ∑
Value is σi, remaining position is 0；
Finally solve formulaOptimal solution W=Q ∑ PT。
In the above method, the number of iterations can be rule of thumb arranged, and often just restrain for tens times in an experiment, can also be with
It is a certain amount of and stop iteration being less than certain with the difference of the front and back variable of iteration twice.
The present invention solves optimization problem, meter using the alternating direction multipliers method based on Schatten Capped p norm
It is small to calculate complexity, concurrent operation can be carried out very easily for large-scale data.
It is furthermore preferred that by following steps to formulaIt is solved:
Firstly, initializationv1=v+ λ pvp-1；
Secondly, solving optimal value x*: work as δiLess than v1When, x*Equal to 0；Work as δiEqual to v1When, x*Equal to v；Work as δiGreater than v1
When, x*, it determines by the following method:
(1), x is used(0)Initialize δi；
(2), it iterates to calculate:
x(i+1)=δi-λp(x(i))p-1
After convergence, optimal solution x is obtained*；
Finally, obtaining formulaOptimal solution: if τ≤τ*, then σ*=δ, if τ >
τ*, then σ*=x*；Wherein,
In the present invention, since Schatten Capped p norm is concave function, when p takes (0,1], to formulaOptimization problem cannot be solved with conventional method, therefore inventors herein propose above
Method for solving, where this is also innovation difficult point of the invention.The above method through the invention, not only realizes and asks formula
Solution, and its precision is very high.
In matrix complementing method above-mentioned based on Schatten Capped p norm, the value of p is small more than or equal to 0.6
In equal to 0.9.So as to faster obtain optimal solution, and then improve the efficiency and precision of matrix completion.
In matrix complementing method above-mentioned based on Schatten Capped p norm, the value of τ is more than or equal to 30.From
And the precision of matrix recovery can be improved.
In matrix complementing method above-mentioned based on Schatten Capped p norm,So as to
Information loss is effectively reduced, so that the accuracy rate that data are restored is higher.
Compared with prior art, the invention proposes a kind of new norms: Schatten p- τ norm, essentially by
Truncation Parameters are set, the singular value σ of the matrix X finally restored is choseni, so that carrying out matrix benefit by means of the present invention
Entirely, data matrix is low-rank, and can guarantee that main information does not lose, and improves the quality of data recovery, i.e., of the invention
There are better recovery effects for the incomplete matrix with low-rank property.
Technical difficulty of the invention is, due to joined τ, Schatten in new norm Schatten p- τ norm
P- τ norm is concave function, when p takes (0,1], to formulaOptimization problem cannot be with normal
The methods of rule solves, and more complicated, therefore the method for solving more than inventors herein proposing is calculated, so that data of the invention are restored
Effect is more preferable.
Detailed description of the invention
Fig. 1 is a kind of work flow diagram of embodiment of the invention；
Fig. 2 is effect diagram (the different ratios of the first row picture expression that picture recovery is carried out using method of the invention
Random site pixel missing, the second row indicate use the corresponding restoration result of the method for the present invention)；
For the effect diagram using algorithms of different to four different pictures progress completions, (each row is pair respectively to Fig. 3
Four different pictures carry out the completion of different shortage of data, and same row is same algorithm)；
Fig. 4 is the 50% random site pixel loss for retaining different singular values, the RE and PSNR changed with p；
Fig. 5 is that algorithms of different carries out corresponding PSNR schematic diagram when picture completion in table 1；
Fig. 6 is that algorithms of different carries out corresponding RE schematic diagram when picture completion in table 1.
To better understand the objects, features and advantages of the present invention, with reference to the accompanying drawing and specific real
Applying mode, the present invention is further described in detail.It should be noted that in the absence of conflict, the implementation of the application
Feature in example and embodiment can be combined with each other.
In the following description, numerous specific details are set forth in order to facilitate a full understanding of the present invention, still, the present invention may be used also
To be implemented using other than the one described here other modes, therefore, protection scope of the present invention is not by described below
Specific embodiment limitation.
Specific embodiment
The embodiment of the present invention: a kind of matrix complementing method based on Schatten Capped p norm, as shown in Figure 1,
The following steps are included:
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 collection of empty position
It closes (similarlyIndicate that data matrix D respective items are the set of empty position, so Ωc+ Ω=
Eyes (m, n))；Indicate the matrix after restoring；
S2 defines the Schatten Capped p norm of matrix
WhereinIndicate 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 X of completion:
s.t.EΩ=XΩ-DΩ, X=W
Wherein, W is equivalence variable, and γ is punishment parameter.
XΩIndicate that Ω with X corresponding element is multiplied, and Ω can be understood as the matrix and there was only 0 and 1 composition, 0 indicates that element lacks
The position of mistake, 1 indicates the position that element retains.
In preferably step S3, the alternating direction multipliers method based on Schatten Capped p norm can be used
Schatten Capped p norm-ADMM solves optimization problem (can also optimize using existing other methods),
Specifically includes the following steps:
Firstly, setting and initiation parameter:
Enable DΩ=W=Y=Z=XΩ, 0,1 < ρ < 2 of μ > 0, β >, 0,0 p≤1 < τ > 0, λ >；Wherein, Y, Z are multipliers
, μ, β are penalty term parameters, and ρ is the update coefficient of punishment parameter；λ is punishment parameter,
Secondly, iteration following steps are updated repeatedly, until reaching the number of iterations Iter or the front and back variable of iteration twice
It is a certain amount of that difference is less than certain:
1) fixed variable W and EΩ, update the matrix X to be restored:
SimultaneouslyAndWherein
2) fixed X and W, updates error variance EΩ:
Wherein,
3) fixed X and EΩ, update equivalence variable W:
Wherein
4) multiplier item z, Y are updatedΩWith punishment parameter μ and β:
YΩ:=YΩ-μ(XΩ-EΩ-DΩ)
Z:=z- β (W-X)
μ :=ρ μ
β :=ρ β
Specifically, passing through following steps solution formula(Schatten
Capped p norm is concave function, when p takes (0,1], so formulaIt is excellent
Change problem cannot be solved with conventional method):
Firstly, being initialized: enabling formulaIn, W=U Σ VT, G=Q Δ
Pτ, δiFor i-th of singular value element of Δ, each singular value can be solved with following formula:
Wherein, setting and initiation parameter λ > 0；
Secondly, by the corresponding solution σ of the singular value element of each Δi, new matrix ∑ is formed, on the leading diagonal of ∑
Value is σi, remaining position is 0；
Finally solve formulaOptimal solution W=Q ∑ PT。
In the above method, the number of iterations can be rule of thumb arranged, and often just restrain for tens times in an experiment, can also be with
It is a certain amount of and stop iteration being less than certain with the difference of the front and back variable of iteration twice.
It preferably, can be by following steps to formulaIt is solved:
Firstly, initializationv1=v+ λ pvp-1；
Secondly, solving optimal value x*: work as δiLess than v1When, x*Equal to 0；Work as δiEqual to v1When, xw is equal to v；Work as δiGreater than v1
When, x*, it determines by the following method:
(1), x is used(0)Initialize δi；
(2), it iterates to calculate:
x(i+1)=δi-λp(x(i))p-1
After convergence, optimal solution x is obtained*；
Finally, obtaining formulaOptimal solution: if τ≤τ*, then σ*=δ, if τ >
τ*, then σ*=x*；Wherein,
In the present invention, since Schatten Capped p norm is concave function, when p takes (0,1], to formulaOptimization problem cannot be solved with conventional method, therefore inventors herein propose above
Method for solving, where this is also innovation difficult point of the invention.The above method through the invention, not only realizes and asks formula
Solution, and its precision is very high.
In matrix complementing method above-mentioned based on Schatten Capped p norm, the value of p is small more than or equal to 0.6
In equal to 0.9.So as to faster obtain optimal solution, and then improve the efficiency and precision of matrix completion.
In matrix complementing method above-mentioned based on Schatten Capped p norm, the value of τ is more than or equal to 30.From
And the precision of matrix recovery can be improved.
In matrix complementing method above-mentioned based on Schatten Capped p norm,
In addition, inventor also carries out matrix fill-in to different data to verify the effective of method proposed by the invention
Property.
Different matrix completion algorithms include:
Algorithm Schatten Capped p norm-ADMM of the invention: it is based on Schatten proposed by the present invention
The ADMM method of Capped p norm；
TNNR-APGL algorithm: the APGL matrix completion algorithm based on truncation nuclear norm punishment；
Schatten p-ADMM algorithm: Schatten p norm is that a kind of pair of rank function is effectively approximate；
Logarithm-ADMM algorithm: the ADMM algorithm based on Logarithm punishment；
Logarithm-IRNN algorithm: the IRNN algorithm based on Logarithm punishment；
Capped-L 1-IRNN: having non-convex property, and the IRNN algorithm of optimization is solved with subdifferential.
This experiment is to use i5-6500CPU, what the desktop computer of 4G memory emulated on matlab.
One, random site missing data
Compare the case where algorithms of different restores the picture of different random miss ratio.It is the present invention as shown in Figure 2
The restoration result of algorithm.Pixel deletion sites are randomly provided by ratio, and three channels are set as the same deletion sites.And
The recovery situation of algorithms of different is then as shown in table 1, as can be seen from Table 1, in addition to the TNNR- when retaining 20% random data
The recovery effects of APGL are slightly good compared with the method for the present invention, other situations method of the invention is optimal effect；Algorithms of different is extensive
Multiple PSNR and RE is as shown in Figure 5 and Figure 6.PSNR (Peak Singal-to-Noise Ratio) and SNR is common image
The discriminant criterion of Quality of recovery indicates signal power and influences the ratio of the destructive noise power of its expression precision, value
Bigger Quality of recovery is better, and inventive algorithm is more calculated using the psnr function that matlab is embedded.RE
(RelativeError) ratio of absolute error and initial data is indicatedIt is worth smaller expression
Quality of recovery is better.
Recovery situation of the different matrix completion algorithms of table 1 under different data loss rate
Two, blocky topagnosis data
In practical applications, image is usually a kind of data matrix with low-rank effect, and the main information of image is concentrated
In forward several biggish singular values, so being a kind of common experimental method, this reality for the matrix completion of image
It tests by the way that common RGB triple channel image is handled each channel individually come completion, does not consider the correlation between channel.
Algorithm of the invention and other five kinds of algorithms are all non-convex penalty term to be added in through objective function come approximate low
Rank function, as shown in figure 3, (h) column are that inventive algorithm recovers as a result, (a) column are original without pixel missing
Image, (b) column are pixel missings of different shapes: there is the delta-shaped region of large area to lack, there is the blocky missing of different zones,
There are also texts to block missing.Find that algorithm of the invention can be with preferable in several scenes by comparing different algorithms
Recovery effects restore (b) picture arranged.The specific relatively data PSNR that see Table 2 for details provides compares, and can be seen by table 2
Out, in addition to the volcano picture that text blocks be Capped-IRNN recovery effects it is slightly good other than, remaining is present invention side
Method recovery effects are best.
Recovery situation of the different matrix completion algorithms of table 2 under different blocky topagnosis
In addition, the selection for p value in Schatten p- τ norm, inventor have also carried out following experiment and have screened:
During experiment, inventors have found that the p value for penalty term norm is not the smaller the better, but take
(0,1] to reach Quality of recovery best for intermediate a certain value.
As shown in figure 4, the present invention selects image lena (beauty's image) and algorithm of the invention, (a) are in original image
The pixel for removing 50% random position, PSNR and RE using algorithm of the invention in different p values, (b) (c) (d)
It is then to retain preceding 30,20,10 singular values after respectively decomposing original image SVD, obtained image is removing random 50%
Position pixel, equally, using algorithm of the invention different p values PSNR and RE.
As seen from Figure 4, the quality that the picture of stringenter low-rank is restored using method of the invention is better.Restore simultaneously
Top-quality p value is not but optimal p to be obtained in interlude at the both ends of domain, i.e. p be not take close to 0 or
Close to 1 value, but take the value effect between [0.6,0.9] more preferable.
Claims (6)
1. a kind of matrix complementing method based on Schatten Capped p norm, which comprises 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 collection of empty position
It closes；Indicate the matrix after restoring；
S2 defines the Schatten Capped p norm of matrix WhereinIndicate 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 X of completion:
s.t.EΩ=XΩ-DΩ, X=W
Wherein, W is equivalence variable, and γ is punishment parameter.
2. the matrix complementing method according to claim 1 based on Schatten Capped p norm, which is characterized in that
In step S3, the alternating direction multipliers method Schatten Capped p norm- based on Schatten Capped p norm is used
ADMM solves optimization problem, specifically includes the following steps:
Firstly, setting and initiation parameter:
Enable DΩ=W=Y=Z=XΩ, 0,1 < ρ < 2 of μ > 0, β >, 0,0 p≤1 < τ > 0, λ >；Wherein, Y, Z are multiplier items, μ,
β is penalty term parameter, and ρ is the update coefficient of punishment parameter；λ is punishment parameter,
Secondly, repeatedly update iteration following steps, until reach the number of iterations Iter or front and back twice the variable of iteration difference it is small
Mr. Yu is a certain amount of:
1) fixed variable W and EΩ, update the matrix X to be restored:
SimultaneouslyAndWherein
2) fixed X and W, updates error variance EΩ:
Wherein,
3) fixed X and EΩ, update equivalence variable W:
Wherein
4) multiplier item Z, Y are updatedΩWith punishment parameter μ and β:
YΩ:=YΩ-μ(XΩ-EΩ-DΩ)
Z:=Z- β (W-X)
μ :=ρ μ
β :=ρ β
Specifically, passing through following steps solution formula
Firstly, being initialized: enabling formulaIn, W=U ∑ VT, G=Q Δ PT, δi
For i-th of singular value element of Δ, each singular value can be solved with following formula:
Wherein, setting and initiation parameter λ > 0；
Secondly, by the corresponding solution σ of the singular value element of each Δi, new matrix ∑ is formed, the value on the leading diagonal of ∑ is
σi, remaining position is 0；
Finally solve formulaOptimal solution W=Q ∑ PT。
3. the matrix complementing method according to claim 2 based on Schatten Capped p norm, which is characterized in that
By following steps to formulaIt is solved:
Firstly, initializationv1=v+ λ pvp-1；
Secondly, solving optimal value x*: work as δiLess than v1When, x*Equal to 0；Work as δiEqual to v1When, x*Equal to v；Work as δiGreater than v1When, x*,
It determines by the following method:
(1), x is used(0)Initialize δi；
(2), it iterates to calculate:
x(i+1)=δi-λp(x(i))p-1
After convergence, optimal solution x is obtained*；
Finally, obtaining formulaOptimal solution: if τ≤τ*, then σ*=δ, if τ > τ*, then
σ*=x*；Wherein,
4. described in any item matrix complementing methods based on Schatten Capped p norm according to claim 1~3,
It is characterized in that, the value of p is more than or equal to 0.6, is less than or equal to 0.9.
5. described in any item matrix complementing methods based on Schatten Capped p norm according to claim 1~3,
It is characterized in that, the value of τ is more than or equal to 30.
6. described in any item matrix complementing methods based on Schatten Capped p norm according to claim 1~3,
It is characterized in that,
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN111028162A (en) * | 2019-11-26 | 2020-04-17 | 广东石油化工学院 | Image missing recovery method based on truncated Schatten p-norm |
| 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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