CN109921799A - A kind of tensor compression method based on cumulative amount dictionary learning - Google Patents

A kind of tensor compression method based on cumulative amount dictionary learning Download PDF

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CN109921799A
CN109921799A CN201910126833.5A CN201910126833A CN109921799A CN 109921799 A CN109921799 A CN 109921799A CN 201910126833 A CN201910126833 A CN 201910126833A CN 109921799 A CN109921799 A CN 109921799A
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CN109921799B (en
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张祖凡
毛军伟
甘臣权
孙韶辉
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Jimoji Network Technology (Chongqing) Co.,Ltd.
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Chongqing University of Post and Telecommunications
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Abstract

A kind of tensor compression method based on cumulative amount dictionary learning is claimed in the present invention, belongs to field of signal processing.It the described method comprises the following steps: 1, tensor being subjected to Plutarch decomposition and rarefaction representation respectively, obtain dictionary, sparse coefficient and core tensor；2, by the relationship of the sparse coefficient of tensor and core tensor, the new rarefaction representation form about tensor is obtained；3, dimensionality reduction is carried out to the dictionary in mapping matrix using cumulative amount dictionary learning algorithm, to realize the compression of tensor.Tensor compression algorithm proposed by the present invention based on cumulative amount dictionary learning, realizes being effectively compressed for tensor, relative to other compression algorithms, can more effectively retain the information of original tensor, reaches preferably denoising effect.

Description

Tensor compression method based on energy-gathering dictionary learning
Technical Field
The invention belongs to the field of signal processing, and particularly relates to a tensor signal compression algorithm based on energy-gathering dictionary learning, which can realize effective compression of tensor signals.
Background
With the development of information technology, multidimensional signals play an increasingly important role in the field of signal processing. Meanwhile, multi-dimensional (MD) signals also impose a great burden on the transmission and storage process. To deal with the challenges of multi-dimensional signal processing, people pay attention to tensor representation of multi-dimensional signals, and the multi-dimensional signals are represented as tensors and processed, so that great convenience is brought to the processing of the multi-dimensional signals. Therefore, the compression of the multidimensional signal is essentially effective compression of the tensor, so that the tensor compression algorithm plays an increasingly important role in the field of multidimensional signal compression, and is a hot spot of current research.
In recent years, researchers have proposed many effective compression algorithms based on CP decomposition and tack decomposition algorithms for the problem of tensor compression, and one of them is to awaken vectorization operation of tensor data directly based on tensor decomposition for the tensor data itself. However, according to the well-known theorem of ugly ducklings, there is no optimal representation of patterns without any a priori knowledge, in other words, vectorization of tensor data is not always efficient. Specifically, this may cause the following problems: firstly, the inherent high-order structure and the inherent correlation in the original data are destroyed, and the information loses or masks the redundant information and the high-order dependency of the original data, so that a model representation which is more meaningful possibly cannot be obtained in the original data; second, vectorization operations can generate high-dimensional vectors, leading to "overfitting", "dimension disasters", and small sample problems.
On the other hand, the sparse representation of the tensor is applied to the compression of the tensor. Due to the equivalence of the tensor tack model and the Kronecker representation, a tensor can be defined as a representation of a given Kronecker dictionary with certain sparsity, such as multidirectional sparsity and block sparsity. With the occurrence of tensor sparsity, some sparse coding modes also occur, such as algorithms of Kroneker-OMP, N-way Block OMP and the like, and the algorithms bring great convenience to tensor compression; meanwhile, a plurality of dictionary learning algorithms are generated corresponding to various sparse coding algorithms, and tensors are subjected to sparse coding and dictionary learning in each dimension, so that the aim of sparse representation is fulfilled. However, in the above process of the tensor processing algorithm based on the sparse representation, some new noise is often introduced, which will have a certain influence on the accuracy of the data. In addition, in the sparse representation process, the determination of the sparsity also brings certain challenges to tensor processing.
Therefore, in processing a tensor including a large amount of data, the tensor is effectively compressed, useful information is extracted, and unprecedented challenges are faced in reducing transmission and storage costs.
Disclosure of Invention
The present invention is directed to solving the above problems of the prior art. The tensor compression method based on the energy-gathering dictionary learning can improve the storage capacity of original data and enhance the denoising capacity. The technical scheme of the invention is as follows:
a tensor compression method based on energy-gathered dictionary learning comprises the following steps:
step 1): acquiring a multi-dimensional signal, expressing the multi-dimensional signal as a tensor, inputting the tensor, and performing sparse expression and tack decomposition;
step 2) obtaining a new tensor sparse representation form related to the original tensor by utilizing the approximate relation between the sparse coefficient tensor in the sparse representation and the core tensor obtained by the Tak decomposition; (ii) a
Step 3) converting the new tensor sparse representation form in the step 2) into a mapping matrix form related to dictionary representation according to tensor operation properties;
and 4) reducing the dimension of the dictionary in the mapping matrix by using the idea of reducing the dimension of the energy-gathering dictionary learning algorithm, thereby realizing the compression of the tensor.
Further, the step 1) obtains the form of the product of the core tensor and the mapping matrix in each dimension through the Tack decomposition, and the following expression is obtained in the Tack decomposition process
A∈RI×P,B∈RJ×Q,C∈RK×RThe orthogonal matrix is also called a factor matrix, reflects principal components in each dimension, and Z belongs to RP×Q×RFor the core tensor, reflecting the correlation of each dimension, P, Q and R correspond to the number of columns of factor matrices A, B and C, I, J and K represent the size of each dimension of the original tensor, and if P, Q, R is smaller than I, J, K, the core tensor can beCompression, considered as the original tensor;
for an N-order tensor, the Tak decomposition form is
χ＝Z×1A1×2A2...×NAN
χ denotes an input tensor signal, Z denotes a core tensor, AiThe decomposition matrix in each dimension is represented as an orthogonal matrix.
Further, the expression form of the sparse representation of the tensor of the step 1) is
Representing the signal after sparse representation, S representing the sparse coefficient tensor, N representing the order of the self-tensor, DiRepresenting dictionaries in various dimensions.
Further, the step 2) finds that the two expressions are similar by observing the sparse representation of the tensor and the Tak decomposition form, and the representation of the nuclear tensor obtained by the Tak decomposition is shown as
Z＝χ×1A1 T×2A2 T...×NAN T
Substituting the expression of the nuclear tensor into tensor sparse representation by using the approximate relation between the sparse coefficient tensor and the nuclear tensor to obtain
Further, the step 3) converts the new tensor sparse representation form in the step 2) into a mapping matrix form related to dictionary representation according to tensor operational properties, and specifically includes:
in the operation of the tensor,
when m is not equal to n, the number of the first and second groups,
Ψ×mA×nB＝Ψ×n(BA)
wherein Ψ represents an N-order tensor, andmm-mode product of expression tensor and matrixnRepresenting the n-modulo product of the tensor and matrix.
When m is not equal to n, the number of the first and second groups,
Ψ×mA×nB＝Ψ×nB×mA
applying the above two properties to a new sparse representation can result in
Further, in the step 4), the dictionary in the mapping matrix is subjected to dimension reduction by using the idea of dimension reduction of the energy-gathered dictionary learning algorithm, so that the compression of the tensor is realized, and the specific steps include:
inputting: tensor composed of T training samplesSparsity k, maximum number of iterations Itermax, termination threshold ε
1. Initializing a dictionaryIs a Gaussian matrix and normalizes columns of each dictionary
2. And (3) decomposing tack: chi ═ Z curve1A1×2A2...×NAN；
3. Initializing a dictionary and performing a gamma (D) process in an energy-gathered dictionary learning algorithm;
4. calculating S when the update time i is 0, and usingCalculating the absolute error E without any iterative update0；
5. The ith dictionary update
For k＝1:IN
By usingCalculating an updated value of the ith dictionary
End
6. Normalizing the N dictionaries obtained in the last step;
7. updating sparse coefficient tensor S by using ith updated dictionaryi；
8. Calculating the absolute error E after the ith updateiAnd relative error Er；
9. Judging the termination condition, if ErIf the iteration number is less than epsilon or exceeds the maximum limit, the loop is terminated, otherwise, the step 5-8 is continued;
10. get a dictionary
11. According to the formulaTo obtain UdThen dictionary
12. Bring dictionary P in
And (3) outputting: compression tensor
Further, in order to make the dictionary P after dimension reduction retain the principal components in the original dictionary D, the dictionary D needs to be preprocessed, and the processing procedure is as follows
DTD＝uΛvT
u is left singular matrix of singular value decomposition, v is right singular matrix, Λ is singular value matrix, and the expression form isSecondly, updating singular values
Where k denotes the number of dictionary columns, tdWhich is indicative of a principal component threshold value,representing the first d singular values after updating,representing the updated last r singular values.
Thereby obtaining new singular values
Form a new dictionary with the initial left and right singular matrices, i.e.
Further, after preprocessing the dictionary, the dictionary needs to be updated, a tensor-based multidimensional dictionary learning algorithm TKSVD is adopted in the updating process to complete the dictionary updating process of the high-dimensional tensor signal, and the TKSVD algorithm is different from the K-SVD dictionary updating algorithm in that the TKSVD algorithm specifically includes:
(1) when carrying out tensor dictionary study, different from two-dimensional signal learning mode, according to the definition of tensor norm, can obtain:
wherein,DNan N-dimensional dictionary, D1Denotes a 1 st-dimensional dictionary, ITExpressing a T-order unit array, solving the above formula by a least square method to obtain a dictionary DiIs updated to
Wherein, yiThe i-mode expansion matrix representing the tensor,representing a pseudo inverse, i.e.WhereinPseudo-inverse matrix representing matrix M, MTRepresenting the transpose of matrix M. On the other hand, after one iteration is completed, the calculation can be performed under the current dictionary and the sparse coefficientThe absolute and relative errors between the recovered data and the original training data, the absolute error after the ith iteration still being defined by the Frobenius norm of the tensor
Wherein S represents a coefficient tensor,Representing the error of the true signal from the approximated signal after removing G atoms.
After the dictionary is updated, the dictionary is subjected to dimensionality reduction, and singular value decomposition is performed on the updated dictionary
Wherein U isdThe first d columns, U, representing the left singular matrixrThe rear r column, Θ, representing the left singular matrixdThe first d singular values, Θ, representing a matrix of singular valuesrRepresenting the last r singular values, V, of the matrix of singular valuesdThe first d columns, V, representing the right singular matrixrThe rear r columns of the right singular matrix are represented. The reduced-dimension dictionary is represented asAnd substituting the dictionary P subjected to the dimension reduction into the mapping matrix T, thereby completing tensor compression.
The invention has the following advantages and beneficial effects:
the invention provides a tensor compression algorithm based on energy-gathering dictionary learning. The specific innovative steps comprise: 1) the sparse representation of the tensor is applied to tensor compression, and the denoising capability is superior to other algorithms; 2) the compression of the tensor is completed through the dimensionality reduction of the mapping matrix, so that the damage of vectorization operation to the internal data structure of the tensor is avoided; 3) dimension reduction is performed on the dictionary of the tensor by adopting an energy-gathered dictionary learning algorithm, so that a data structure between data in each dimension can be reserved, and the data retention capacity is improved.
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Figure 1 is an exploded view of a third order tensor tack as used in the preferred embodiment of the present invention;
FIG. 2 is a schematic diagram of a tensor compression algorithm based on energy-gathered dictionary learning;
fig. 3 is a flow diagram of a tensor compression algorithm based on energy-gathered dictionary learning.
Detailed Description
The technical solutions in the embodiments of the present invention will be described in detail and clearly with reference to the accompanying drawings. The described embodiments are only some of the embodiments of the present invention.
The technical scheme for solving the technical problems is as follows:
the method mainly solves the problems that in the traditional tensor compression algorithm, a data structure is damaged, information is lost, and new noise is introduced. The method mainly includes the steps that a dictionary, a sparse coefficient tensor and a nuclear tensor are obtained through Tak decomposition and sparse representation, then a new sparse representation form is formed through the approximate relation between the sparse coefficient tensor and the nuclear tensor, finally dimension reduction is conducted on the dictionary in the sparse representation through an energy-gathering dictionary learning algorithm, and therefore tensor compression is achieved.
FIG. 2 is a general flow chart of the present invention, which is described below with reference to the accompanying drawings, and includes the following steps:
the method comprises the following steps: simultaneously carrying out sparse representation and tack decomposition on the input tensor, and obtaining the following representation in the sparse representation process
After the tensor is subjected to sparse representation, a product form of the sparse coefficient tensor in each dimension is obtained, in other words, in the sparse representation, the sparse coefficient tensor takes the dictionary D as a mapping matrix and is projected in each dimension, and therefore the sparse tensor is obtained.
Giving a third order tensor x ∈ RI×J×KThe Take decomposition process is shown in FIG. 1. In the course of the Take decomposition, the following expression is obtained
A∈RI×P,B∈RJ×Q,C∈RK×RThe orthogonal matrix is also called a factor matrix, reflects principal components in each dimension, and Z belongs to RP×Q×RThe core tensor is the reflection of the relevant situation of each dimension. P, Q and R correspond to the number of columns of factor matrices A, B and C, respectively, I, J and K represent the size of each dimension of the original tensor, if P, Q, R is smaller than I, J, K, then the core tensor can be regarded as the compression of the original tensor.
More generally, for an order-N tensor, the Tak decomposition is in the form of
χ＝Z×1A1×2A2...×NAN
And step two) obtaining a new tensor sparse representation form related to the original tensor by utilizing the approximate relation between the sparse coefficient tensor in the sparse representation and the core tensor obtained by the Tak decomposition.
By observing the sparse representation of the tensor and the form of the Tak decomposition, it is found that the two expressions are similar, and the representation of the nuclear tensor obtained by the Tak decomposition is
Z＝χ×1A1 T×2A2 T...×NAN T
Substituting the expression of the nuclear tensor into tensor sparse representation by using the approximate relation between the sparse coefficient tensor and the nuclear tensor to obtain
Step three: converting the new tensor sparse representation form into a mapping matrix form related to dictionary representation according to tensor operational properties, and the specific steps comprise:
in the tensor operation, when m is not equal to n,
Ψ×mA×nB＝Ψ×n(BA)
wherein Ψ represents an N-order tensor, andmm-mode product of expression tensor and matrixnRepresenting the n-modulo product of the tensor and matrix.
When m is not equal to n, the number of the first and second groups,
Ψ×mA×nB＝Ψ×nB×mA
applying the above two properties to a new sparse representation can result in
Let Ti＝DiAi TWill TiBrought into the above formula to obtain
From the above equation, the sparse representation is in the form of a projection on one dimension with respect to the original matrix. In the MPCA algorithm, the high-dimensional tensor realizes tensor compression through processing of a projection matrix, namely, the main component of the projection matrix is reserved to achieve the purpose of compressing the projection matrix, so that the tensor compression is realized. Elicitation of the MPCA concept, here for matrix TiDimension reduction is performed, thereby realizing compression of the original tensor χ. To complete the matrix pair TiDimension reduction of (2), here considering sparse TiThe dictionary D is subjected to dimension reduction, so that the mapping matrix T is realizediThe dimensionality reduction of (1).
Step four: and (3) realizing the compression of the tensor by using the thought of dictionary dimension reduction in the energy-gathered dictionary learning algorithm. In the algorithm, in order to enable the dictionary P after dimensionality reduction to keep the principal components in the original dictionary D, the dictionary D needs to be preprocessed, and the processing process is as follows
DTD＝uΛvT
u is left singular matrix of singular value decomposition, v is right singular matrix, Λ is singular value matrix, and the expression form isSecondly, updating singular values
Where k denotes the number of dictionary columns, tdWhich is indicative of a principal component threshold value,representing the first d singular values after updating,representing the updated last r singular values.
Thereby obtaining new singular values
Form a new dictionary with the initial left and right singular matrices, i.e.
After preprocessing the dictionary, the dictionary needs to be updated, and a multidimensional dictionary learning algorithm (TKSVD) based on tensor is adopted in the updating process to complete the dictionary updating process of the high-dimensional tensor signals. Unlike the updating algorithm of the K-SVD dictionary, the TKSVD algorithm needs to be noticed by two points:
(2) when carrying out tensor dictionary study, different from two-dimensional signal learning mode, according to the definition of tensor norm, can obtain:
wherein,ITrepresenting a unit matrix of order T, DNAn N-dimensional dictionary, D1Representing a 1 st dimensional dictionary. Obviously, the above formula can be solved by the least square method, and the dictionary D can be obtainediIs updated to
Wherein, yiThe i-mode expansion matrix representing the tensor,representing a pseudo inverse, i.e.WhereinPseudo-inverse matrix representing matrix M, MTRepresenting the transpose of matrix M. On the other hand, after one iteration is completed, the absolute error and the relative error between the data which can be recovered under the current dictionary and the sparse coefficient and the original training data can be calculated, and the absolute error after the ith iteration is still defined by the Frobenius norm of the tensor
Wherein S represents a coefficient tensor,Representing the error of the true signal from the approximated signal after removing G atoms.
And after the dictionary is updated, performing dimension reduction processing on the dictionary. Singular value decomposition of updated dictionary
Wherein U isdThe first d columns, U, representing the left singular matrixrThe rear r column, Θ, representing the left singular matrixdThe first d singular values, Θ, representing a matrix of singular valuesrRepresenting the last r singular values, V, of the matrix of singular valuesdThe first d columns, V, representing the right singular matrixrThe rear r columns of the right singular matrix are represented. The reduced-dimension dictionary is represented asAnd substituting the dictionary P subjected to the dimension reduction into the mapping matrix T, thereby completing tensor compression.
The tensor compression algorithm based on the energy-gathered dictionary learning specifically comprises the following steps:
inputting: tensor composed of T training samplesSparsity k, maximum number of iterations Itermax, termination threshold ε
1. Initializing a dictionaryIs a Gaussian matrix and normalizes columns of each dictionary
2. And (3) decomposing tack: chi ═ Z curve1A1×2A2...×NAN
3. Initializing dictionaries through the gamma (D) process in a cumulative energy dictionary learning algorithm
4. Calculating S when the update time i is 0, and usingCalculating the absolute error E without any iterative update0
5. The ith dictionary update
For k＝1:IN
By usingCalculating an updated value of the ith dictionary
End
6. Normalizing the N dictionaries obtained in the last step
7. Updating sparse coefficient tensor S by using ith updated dictionaryi
8. Calculating the absolute error E after the ith updateiAnd relative error Er
9. Judging the termination condition, if Er< epsilon or the number of iterations exceeds the maximumLimiting, terminating the loop, otherwise continuing to step 5-8
10. Get a dictionary
11. According to the formulaTo obtain UdThen dictionary
12. Bring dictionary P in
And (3) outputting: compression tensor
The above examples are to be construed as merely illustrative and not limitative of the remainder of the disclosure. After reading the description of the invention, the skilled person can make various changes or modifications to the invention, and these equivalent changes and modifications also fall into the scope of the invention defined by the claims.

Claims (8)

1. A tensor compression method based on energy-gathered dictionary learning is characterized by comprising the following steps:
step 1): acquiring a multi-dimensional signal, expressing the multi-dimensional signal as a tensor, inputting the tensor, and performing sparse expression and tack decomposition;
step 2) obtaining a new tensor sparse representation form related to the original tensor by utilizing the approximate relation between the sparse coefficient tensor in the sparse representation and the core tensor obtained by the Tak decomposition; (ii) a
Step 3) converting the new tensor sparse representation form in the step 2) into a mapping matrix form related to dictionary representation according to tensor operation properties;
and 4) reducing the dimension of the dictionary in the mapping matrix by using the idea of reducing the dimension of the energy-gathering dictionary learning algorithm, thereby realizing the compression of the tensor.
2. The tensor compression method based on energy-gathered dictionary learning as claimed in claim 1, wherein the step 1) obtains the product form of the core tensor and the mapping matrix in each dimension through the Tak decomposition, and the following expression is obtained in the Tak decomposition process
A∈RI×P,B∈RJ×Q,C∈RK×RThe orthogonal matrix is also called a factor matrix, reflects principal components in each dimension, and Z belongs to RP×Q×RThe core tensor is the reflection of the relevant situation of each dimension. P, Q and R correspond to the number of columns of the factor matrix A, B and C, respectively, I, J and K represent the size of each dimension of the original tensor, if P, Q, R is smaller than I, J, K, the core tensor can be regarded as the compression of the original tensor;
for an N-order tensor, the Tak decomposition form is
χ＝Z×1A1×2A2...×NAN
χ denotes an input tensor signal, Z denotes a core tensor, AiThe decomposition matrix in each dimension is represented as an orthogonal matrix.
3. The tensor compression method based on energy-gathered dictionary learning as claimed in claim 2, wherein the expression form of the sparse representation of the tensor in the step 1) is represented by
Representing the signal after sparse representation, S representing the sparse coefficient tensor, N representing the order of the self-tensor, DiRepresenting dictionaries in various dimensions.
4. The tensor compression method based on energy-gathered dictionary learning as claimed in claim 3, wherein the step 2) finds that the two expressions are similar by observing the sparse representation of the tensor and the Tak decomposition form, and the representation of the nuclear tensor obtained by the Tak decomposition is represented as
Z＝χ×1A1 T×2A2 T...×NAN T
Substituting the expression of the nuclear tensor into tensor sparse representation by using the approximate relation between the sparse coefficient tensor and the nuclear tensor to obtain
5. The tensor compression method based on energy-gathered dictionary learning as claimed in claim 4, wherein the step 3) converts the new tensor sparse representation form of the step 2) into a mapping matrix form related to dictionary representation according to tensor operation properties, and specifically comprises the following steps:
in the operation of the tensor,
when m is not equal to n, the number of the first and second groups,
Ψ×mA×nB＝Ψ×n(BA)
wherein Ψ represents an N-order tensor, andmm-mode product of expression tensor and matrixnRepresenting the n-modulo product of the tensor and matrix;
when m is not equal to n, the number of the first and second groups,
Ψ×mA×nB＝Ψ×nB×mA
applying the above two properties to a new sparse representation can result in
6. The tensor compression method based on energy-gathered dictionary learning according to claim 5, wherein the step 4) uses the idea of dimension reduction of the energy-gathered dictionary learning algorithm to perform dimension reduction on the dictionary in the mapping matrix, so as to realize tensor compression, and the specific steps include:
inputting: tensor composed of T training samplesSparsity k, maximum number of iterations Itermax, termination threshold ε
1. Initializing a dictionaryIs a Gaussian matrix and normalizes columns of each dictionary
2. And (3) decomposing tack: chi ═ Z curve1A1×2A2...×NAN；
3. Initializing a dictionary and performing a gamma (D) process in an energy-gathered dictionary learning algorithm;
4. calculating S when the update time i is 0, and usingCalculating the absolute error E without any iterative update0；
5. The ith dictionary update
For k＝1:IN
By usingCalculating an updated value of the ith dictionary
End
6. Normalizing the N dictionaries obtained in the last step;
7. updating sparse coefficient tensor S by using ith updated dictionaryi；
8. Calculating the absolute error E after the ith updateiAnd relative error Er；
9. Judging the termination condition, if ErIf the iteration number is less than epsilon or exceeds the maximum limit, the loop is terminated, otherwise, the step 5-8 is continued;
10. get a dictionary
11. According to the formulaTo obtain UdThen dictionary
12. Bring dictionary P in
And (3) outputting: compression tensor
7. The tensor compression method based on energy-gathered dictionary learning as claimed in claim 6, wherein the dictionary D needs to be preprocessed in order to make the dictionary P after dimension reduction retain the principal components in the original dictionary D, and the processing procedure is as follows
DTD＝uΛvT
u is left singular matrix of singular value decomposition, v is right singular matrix, Λ is singular value matrix, and the expression form isSecondly, updating singular values
Where k denotes the number of dictionary columns, tdWhich is indicative of a principal component threshold value,representing the first d singular values after updating,representing the updated last r singular values;
thereby obtaining new singular values
Form a new dictionary with the initial left and right singular matrices, i.e.
8. The tensor compression method based on energy-gathered dictionary learning of claim 7, wherein after the dictionary is preprocessed, the dictionary needs to be updated, a tensor-based multidimensional dictionary learning algorithm TKSVD is adopted in the updating process, the dictionary updating process of the high-dimensional tensor signal is completed, and the TKSVD algorithm is different from the K-SVD dictionary updating algorithm in that the TKSVD algorithm specifically comprises:
(1) when carrying out tensor dictionary study, different from two-dimensional signal learning mode, according to the definition of tensor norm, can obtain:
wherein,DNan N-dimensional dictionary, D1Denotes a 1 st-dimensional dictionary, ITExpressing a T-order unit array, solving the above formula by a least square method to obtain a dictionary DiIs updated to
Wherein, yiThe i-mode expansion matrix representing the tensor,representing a pseudo inverse, i.e.WhereinPseudo-inverse matrix representing matrix M, MTA transposed matrix representing the matrix M; on the other hand, after one iteration is finished, the absolute error and the relative error between the data which can be recovered under the current dictionary and the sparse coefficient and the original training data are calculated, and the absolute error after the ith iteration is still defined by the Frobenius norm of the tensor
Wherein S represents a coefficient tensor,Representing the error of the real signal and the approximation signal after removing G atoms;
after the dictionary is updated, the dictionary is subjected to dimensionality reduction, and singular value decomposition is performed on the updated dictionary
Wherein U isdThe first d columns, U, representing the left singular matrixrThe rear r column, Θ, representing the left singular matrixdThe first d singular values, Θ, representing a matrix of singular valuesrRepresenting the last r singular values, V, of the matrix of singular valuesdThe first d columns, V, representing the right singular matrixrRepresenting the rear r column of the right singular matrix, the dictionary after dimensionality reduction is represented asAnd substituting the dictionary P subjected to the dimension reduction into the mapping matrix T, thereby completing tensor compression.
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