CN109658362B - Data recovery method based on supplied core norm - Google Patents
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Abstract
The invention discloses a data recovery method based on a supplied core norm, which comprises the following steps: s1, setting input original incomplete data into a form X of a three-dimensional tensor; s2, defining a supported kernel norm of tensor X on a three-dimensional layer θ ：Wherein θ is a supplied parameter, σ i (X) is the singular value of the tensor, I X I * Is the kernel norm of the tensor; s3, minimizing the supported kernel norm until X converges; the reduced tensor is output. According to the invention, the missing data in the tensor is complemented through the rank of the three-dimensional tensor, specifically, the supported core norm of the tensor on the three-dimensional layer is defined, and the supported core norm is solved iteratively so as to be minimized, so that the low-rank tensor is obtained, and the quick and accurate complement of the three-dimensional tensor is realized.
Description
Technical Field
The invention relates to a data recovery method based on a supplied core norm, and belongs to the technical field of data recovery.
Background
With the continuous development of information technology, the scale of people to acquire data is getting larger and larger. Among the many information collected, most of the information is incomplete. For example, image or video streaming data obtained by low-power wireless sensors may be insufficient due to bandwidth and power limitations, and large-scale user information may have minor drawbacks. Thus, large-scale information is not used directly in most cases.
At present, the method for adopting the nuclear norms and cutting off the nuclear norms in the three-dimensional tensor completion cannot well approximate the rank of the tensor, and the convergence rate is low, so that the three-dimensional tensor cannot be rapidly and accurately completed. Thus, improvements are still needed.
Disclosure of Invention
The invention aims to provide a data recovery method based on a supported kernel norm, which can realize rapid and accurate completion of a three-dimensional tensor.
In order to solve the technical problems, the invention adopts the following technical scheme: the data recovery method based on the supplied core norm comprises the following steps:
s1, setting input original incomplete data (such as data of pictures, video streams and the like) into a form X of a three-dimensional tensor;
s2, defining a supported kernel norm (namely cap kernel norm) of tensor X on a three-dimensional layer θ ：Wherein θ is a supplied parameter (i.e., cap parameter), σ i (X) is the singular value of the tensor, I X I * Is the kernel norm of the tensor; n is n 1 ，n 2 The lengths of tensors along the first dimension and the second dimension respectively;
s3, minimizing the supported kernel norm until X converges; the reduced tensor is output.
Preferably, minimizing the capped kernel norm in step S3 includes the steps of: first, data initialization is performed: setting and initializing a supplied parameter theta, a Lagrangian multiplier lambda, a tensor U, a penalty parameter rho (rho can be initialized to be about 6.5 e-3), rho max (the maximum upper limit of the penalty parameter may be set to 1e10 in general) and the update step parameter μ (e.g., the update step parameter may be initialized to 1.25); initializing Lagrangian multiplier Λ to be equal to X;
secondly, the following steps are cycled until X converges:
1) Updating auxiliary tensor S
Wherein D is α (. Cndot.) is a singular value thresholding function, α represents a singular value thresholding threshold parameter (i.e., a shrinkage parameter); k represents the current iteration number (how many iterations are optionalAnd) k is 0;
2) Updating X
3) Updating Λ
Λ k+1 ＝Λ k +ρ k (X k+1 -S k+1 )
4) Updating ρ
ρ k+1 ＝min(ρ k *μ，ρ max )；
When X k+1 -X k || 2 And if the temperature is less than or equal to 0.1, converging, and ending the iteration.
The method can realize rapid and accurate completion of the three-dimensional tensor, the speed can be thirty seconds faster than that of the prior art in the background technology, and the precision is generally high by 2-point PSNR values.
Preferably, the tensor U is determined by the following method: carrying out singular value decomposition on input data, namely an original tensor X; in the calculation process, the singular value on each diagonal matrix is subjected to three-valued: if the value is larger than the cap parameter theta, the value is changed to 1; if the value is smaller than the cap parameter theta, changing the value to 0; if the cap parameter theta is equal to the cap parameter theta, changing to 0.5; then, performing inverse Fourier transform to restore the diagonal tensor; and multiplying the tensor by the left singular tensor and the right singular tensor to obtain a tensor divided by the supported parameter theta to obtain a tensor U. Thereby, the accuracy of complementing the three-dimensional tensor can be further improved.
In the above-mentioned data recovery method based on the supported kernel norm, the singular value decomposition is performed on the input data, i.e. the original tensor X, i.e. the decomposition target is x=u×s×v.
In the foregoing data recovery method based on the supported kernel norm, the singular value decomposition is specifically performed on the input data, that is, the original tensor X, by the following method:
first, fourier transforming the tensorGet the changeThe tensor after the conversion;
next, the transformed tensor is subjected to singular value decomposition along a third dimension to form a two-dimensional matrix of the tensorThe lower case letters are matrices, the columns of u are left singular vectors, s are diagonal matrices, the diagonal is singular value, and the columns of v are right singular vectors;
again, (because of) Respectively performing inverse Fourier transform on matrixes obtained by singular value decomposition, namelyThe three tensors are tensor singular values of the original tensor, namely the decomposition result X=U×S×V * 。
The method has high calculation speed, so that the speed of three-dimensional tensor complement can be further improved.
Preferably, in the method, the supplied parameter θ is initialized to 0.01, and the penalty parameter ρ is initialized to 6.5e-3; the update step size parameter μ is initialized to 1.25. Therefore, the three-dimensional tensor complement accuracy is higher and the speed is higher.
Preferably, singular value thresholding of the tensors is performed by: decomposing the tensor singular value to obtain singular values on the diagonal matrix and subtracting the contraction parameter; if the difference is negative, setting 0; the original diagonal matrix is replaced by the obtained matrix, and finally, the diagonal matrix is restored to tensor.
Compared with the prior art, the invention complements the lost data in the tensor through the rank of the three-dimensional tensor, specifically defines the supported core norm of the tensor on the three-dimensional layer, and solves the supported core norm iteratively so as to minimize the supported core norm, thereby obtaining the low-rank tensor and realizing rapid and accurate complement of the three-dimensional tensor. The method can be applied to video recovery, image reconstruction and recovery of high-dimensional tensor loss information in a recommendation system. The inventors have further found that: the existing capped kernel norms are limited on a two-dimensional matrix, and are not defined on a three-dimensional layer, so that partial operation rules cannot be expanded to a three-dimensional tensor, and the requirements of increasingly high rapid and accurate completion on the three-dimensional tensor cannot be met. By utilizing the method provided by the invention, the input data, namely the original tensor X, is subjected to singular value decomposition and singular value thresholding of the tensor, and the cap kernel norms of the tensor are restrained and minimized, so that the quick and accurate completion of the three-dimensional tensor is finally realized. This is also a technical difficulty of the present invention.
Drawings
FIG. 1 is a workflow diagram of one embodiment of the present invention;
fig. 2 is a schematic diagram of an initial picture in an experimental example;
fig. 3 is a schematic diagram of a picture restoration result in an experimental example.
The invention is further described below with reference to the drawings and the detailed description.
Detailed Description
Embodiments of the invention: the data recovery method based on the supplied core norm, as shown in fig. 1, comprises the following steps:
s1, setting input original incomplete data (such as data of pictures, video streams and the like) into a form X of a three-dimensional tensor;
s2, defining a supported kernel norm (namely cap kernel norm) of tensor X on a three-dimensional layer θ ：Wherein θ is a capped parameter (i.e., cap parameter, typically θ can be set to 0.01, and can be adjusted appropriately according to practical effect), σ i (X) is the singular value of the tensor, I X I * Is the kernel norm of the tensor; n is n 1 ，n 2 The values of n1 and n2 of the tensor can be automatically obtained for the length of the tensor along the first and second dimensions (size function in matlab), respectively;
s3, minimizing the supported kernel norm until X converges; the reduced tensor is output.
Optionally, minimizing the supported kernel norm in step S3 includes the following steps: first, data initialization is performed: setting and initializing a supplied parameter theta, a Lagrangian multiplier lambda, a tensor U, a penalty parameter rho (rho can be initialized to be about 6.5 e-3), rho max (the maximum upper limit of the penalty parameter may be set to 1e10 in general) and the update step parameter μ (e.g., the update step parameter may be initialized to 1.25); initializing Lagrangian multiplier Λ to be equal to X;
next, the following steps are cycled until X k+1 -X k || 2 And (3) convergence:
1) Updating auxiliary tensor S
Wherein D is α (. Cndot.) is a singular value thresholding function, α represents a contraction parameter (i.e., a singular value thresholded threshold parameter); k represents the current iteration number (how many iterations are needed), and the initial value of k is 0;
2) Updating X
3) Updating Λ
Λ k+1 ＝Λ k +ρ k (X k+1 -S k+1 )
4) Updating ρ
When X k+1 -X k || 2 And if the temperature is less than or equal to 0.1, converging, and ending the iteration.
Optionally, the tensor U is determined by the following method: carrying out singular value decomposition on input data, namely an original tensor X; in the calculation process, the singular value on each diagonal matrix is subjected to three-valued: if the value is larger than the cap parameter theta, the value is changed to 1; if the value is smaller than the cap parameter theta, changing the value to 0; if the cap parameter theta is equal to the cap parameter theta, changing to 0.5; then, performing inverse Fourier transform to restore the diagonal tensor; and multiplying the tensor by the left singular tensor and the right singular tensor to obtain a tensor divided by the supported parameter theta to obtain a tensor U.
Optionally, the singular value decomposition is performed on the input data, i.e. the original tensor X, i.e. the decomposition target is x=u×s×v * 。
The input data, namely the original tensor X, is subjected to singular value decomposition by the following method:
first, fourier transforming the tensorObtaining a transformed tensor;
next, the transformed tensor is subjected to singular value decomposition along a third dimension to form a two-dimensional matrix of the tensorThe lower case letters are matrices, the columns of u are left singular vectors, s are diagonal matrices, the diagonal is singular value, and the columns of v are right singular vectors;
again, (because of) Respectively performing inverse Fourier transform on matrixes obtained by singular value decomposition, namelyThe three tensors are tensor singular values of the original tensor, namely the decomposition result X=U×S×V * 。
Optionally, singular value thresholding of the tensors is performed by: decomposing the tensor singular value to obtain singular values on the diagonal matrix and subtracting the contraction parameter; if the difference is negative, setting 0; replacing the original diagonal matrix with the obtained matrix, and finally recovering the diagonal matrix into tensors;
namely, the following treatment is carried out:
where r is the rank of the matrix and τ is the contraction parameter.
In the calculation process, tensor multiplication is involved, tensorWherein, the block cyclic operation of tensor is: expanding the matrix on the third dimension of the tensor to the plane in a cyclic formTensor unfold operation: vertically spreading out to a plane along a third dimensionTherefore, fold (& gt) is operated as +.>
Experimental example: the scheme of the invention can better approximate the tensor rank and restrict the tensor rank, and is suitable for image complement with low rank characteristic. The method specifically comprises the following steps:
s1, firstly (the technology of matlab immread () and the like can be adopted) setting a picture to be restored (shown in figure 2) into a form of a three-dimensional tensor, namely X;
s2, defining a supported kernel norm (namely cap kernel norm) of tensor X on a three-dimensional layer θ ：Wherein θ is a supplied parameter (i.e., cap parameter), σ i (X) is the singular value of the tensor, I X I * Is the kernel norm of the tensor; n is n 1 ，n 2 The values of n1 and n2 of the tensor can be automatically obtained for the length of the tensor along the first and second dimensions (size function in matlab), respectively;
s3, minimizing the supported kernel norm until X converges; outputting a reduced tensor; the method specifically comprises the following steps:
first, data initialization is performed: setting and initializing a supplied parameter theta, a Lagrangian multiplier lambda, a tensor U, penalty parameters rho, rho max Updating the step size parameter mu, and initializing the Lagrangian multiplier lambda to be equal to X; setting the supported parameter θ to 0.01, ρ is initialized to 6.5e-3; mu is initialized to 1.25; ρ max Set to 1e10, ρ is not greater than ρ max The method comprises the steps of carrying out a first treatment on the surface of the The tensor U is determined by the following method: carrying out singular value decomposition on input data, namely an original tensor X; in the calculation process, the singular value on each diagonal matrix is subjected to three-valued: if the value is larger than the cap parameter theta, the value is changed to 1; if the value is smaller than the cap parameter theta, changing the value to 0; if the cap parameter theta is equal to the cap parameter theta, changing to 0.5; then, performing inverse Fourier transform to restore the diagonal tensor; multiplying the tensor by the left singular tensor and the right singular tensor to obtain tensor divided by the supported parameter theta to obtain tensor U; setting the maximum iteration number of 200, and ending the iteration if the cycle exceeds 200 times. Let err= ||x k+1 -X k || 2 Ending the iteration when err is less than or equal to 0.1.
Next, the following steps are cycled until X k+1 -X k || 2 And (3) convergence:
let k=0; turning to 1);
1) Updating the auxiliary tensor S and turning to 2);
wherein D is α (. Cndot.) is a singular value thresholding function, α represents a singular value thresholding threshold parameter, i.e., a shrinkage parameter; k represents the current iteration numberThe initial value of k is 0; wherein singular value thresholding of the tensors is performed by: decomposing the tensor singular value to obtain singular values on the diagonal matrix and subtracting the contraction parameter; if the difference is negative, setting 0; replacing the original diagonal matrix with the obtained matrix, and finally recovering the diagonal matrix into tensors;
2) Update X and go to 3);
3) Updating Λ and turning to 4);
Λ k+1 ＝Λ k +ρk ( X k+1 -S k+1 )
4) Update ρ and go to 5);
ρ k+1 ＝min(ρ k *μ，ρ max )；
5) Judging X k+1 -X k || 2 Whether or not the value of (2) convergence? X k+1 -X k || 2 Convergence at less than or equal to 0.1)? If the convergence is carried out, ending the iteration and leading out the picture; let k=k+1, go to 1).
Wherein the input data, i.e. the original tensor X, is subjected to singular value decomposition, i.e. the decomposition target is x=u×s×v * 。
The input data, namely the original tensor X, is subjected to singular value decomposition by the following method:
first, fourier transforming the tensorObtaining a transformed tensor;
next, the transformed tensor is subjected to singular value decomposition along a third dimension to form a two-dimensional matrix of the tensorThe lower case letters are matrices, the columns of u are left singular vectors, s are diagonal matrices, the diagonal is singular value, and the columns of v are right singular vectors;
again, (because of) Respectively performing inverse Fourier transform on matrixes obtained by singular value decomposition, namelyThe three tensors are tensor singular values of the original tensor, namely the decomposition result X=U×S×V * 。
The final precision and time consumption of three-dimensional tensor completion by adopting the method of the invention are related to theta parameters, rho and mu, and the parameters of the experimental example can be adjusted up and down according to the practical effect.
Compared with the prior art, the most advanced three-dimensional tensor complement method based on the truncated nuclear norm has the accuracy of 27 and the time consumption of 40s. In the experimental example, by using the iterative method based on the cap kernel norm, the PSNR value (PSNR is a value describing the quality of the picture and can be understood as precision) applied to the picture can reach 28.3, which is 1 point higher than the precision of the current most advanced method based on the truncated kernel norm; whereas the iteration time is 13.1s, which is 32.5% of the current most advanced truncated kernel norm-based method.
Claims (6)
1. The data recovery method based on the supplied core norm is characterized by comprising the following steps:
s1, setting a picture to be restored into a three-dimensional tensor form, namely X, by adopting an imread () function in matlab;
s2, defining a supported kernel norm II of the tensor X on the three-dimensional layer θ ： Wherein θ is a supplied parameter, σ i (X) is tensor singular value, |X|| * Is the kernel norm of the tensor; n is n 1 ，n 2 Respectively are of sheet typeMeasuring the length along the first dimension and the second dimension;
s3, minimizing the supported kernel norm until X converges; outputting a restored tensor, and converting the restored tensor into a picture, namely realizing restoration of the picture to be restored; wherein said minimizing said supported core norms comprises the steps of:
first, data initialization is performed: setting and initializing a supplied parameter theta, a Lagrangian multiplier lambda, a tensor U, penalty parameters rho, rho max Updating the step size parameter mu; initializing Lagrangian multiplier Λ to be equal to X; secondly, the following steps are cycled until X converges:
1) Updating auxiliary tensor S
Wherein D is α (. Cndot.) is a singular value thresholding function, α representing the shrinkage parameter; k represents the current iteration number, and the initial value of k is 0;
2) Updating X
3) Updating Λ
Λ k+1 ＝Λ k +ρ k (X k+1 -S k+1 )
4) Updating ρ
ρ k+1 ＝min(ρ k *μ，ρ max )；
When X k+1 -X k || 2 And when the temperature is less than or equal to 0.1, X converges, and iteration is finished.
2. The method for recovering data based on the supplied core norm as recited in claim 1, wherein the tensor U is determined by: carrying out singular value decomposition on input data, namely an original tensor X; in the calculation process, the singular value on each diagonal matrix is subjected to three-valued: if the value is larger than the cap parameter theta, the value is changed to 1; if the value is smaller than the cap parameter theta, changing the value to 0; if the cap parameter theta is equal to the cap parameter theta, changing to 0.5; then, performing inverse Fourier transform to restore the diagonal tensor; and multiplying the tensor by the left singular tensor and the right singular tensor to obtain a tensor divided by the supported parameter theta to obtain a tensor U.
3. The method for recovering data based on the weighted kernel norm as recited in claim 2, wherein the singular value decomposition is performed on the input data, i.e., the original tensor X, wherein the decomposition target is x=u×s×v * 。
4. A data recovery method based on a supplied kernel norm as defined in claim 3, characterized by performing singular value decomposition of the input data, i.e. the original tensor X, in particular by:
first, fourier transforming the tensorObtaining a transformed tensor;
next, the transformed tensor is subjected to singular value decomposition along a third dimension to form a two-dimensional matrix of the tensorThe lower case letters are matrices, the columns of u are left singular vectors, s are diagonal matrices, the diagonal is singular value, and the columns of v are right singular vectors;
again, the matrices resulting from the singular value decomposition are respectively subjected to an inverse fourier transform, i.eThe three tensors are tensor singular values of the original tensor, namely the decomposition result X=U×S×V * 。
5. The data recovery method based on the supplied core norm according to claim 1, wherein the supplied parameter θ is initialized to 0.01 and the penalty parameter ρ is initialized to 6.5e-3; the update step size parameter μ is initialized to 1.25.
6. The method of claim 1, wherein the singular value thresholding of the tensor is performed by: decomposing the tensor singular value to obtain singular values on the diagonal matrix and subtracting the contraction parameter; if the difference is negative, setting 0; the original diagonal matrix is replaced by the obtained matrix, and finally, the diagonal matrix is restored to tensor.
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