Efficient low-rank tensor recovery method

Abstract

The invention discloses a high-efficiency low-rank tensor recovery method. The method comprises the following steps: constructing an equilibrium tensor; a reconstruction operation is given for an unbalanced tensor, and a more balanced tensor is obtained; a low-rank tensor recovery algorithm based on a non-convex Logdet function is constructed; a non-convex Logdet function is used for replacing a nuclear norm, and a better approximation matrix rank is achieved; introducing a matrix decomposition method; through matrix decomposition, a large-scale matrix is decomposed into a product of two small matrixes, and the size of the matrix is reduced; fusing the above methods to obtain an efficient low-rank tensor recovery algorithm; according to the scheme, the equilibrium tensor and the non-convex function are fused, so that high-precision low-rank tensor recovery is realized, and furthermore, the efficiency of a low-rank tensor recovery algorithm can be effectively improved by introducing a matrix decomposition method.

Description

Technical Field

[0001] The present invention relates to the technical field of low-rank tensor recovery, and specifically relates to an efficient low-rank tensor recovery method. Background Art

[0002] In practical applications, from structured product digital model data, to large-scale multi-domain simulation data, and then to multi-channel detection and sensing data in the manufacturing process, their scale dimensions are constantly increasing and the structural forms are becoming more and more complex. As a high-order generalization of matrices, tensors have better complex data representation capabilities. Therefore, data representation forms based on tensors have received more and more attention. Due to the complex engineering data acquisition environment, there is noise or some data is missing. Such complex and incomplete data is not conducive to data analysis;

[0003] As the most representative algorithm in low-rank data analysis, the low-rank low-rank tensor completion algorithm can mine the internal correlation relationships of data and can effectively recover the original data from data contaminated by noise or missing data. It has extensive applications in the fields of image processing, machine learning, computer vision, and pattern recognition. Currently, the low-rank tensor recovery algorithm based on Tucker rank is the most studied and widely used method. However, the low-rank tensor recovery algorithm based on Tucker rank has the following disadvantages:

[0004] 1. The unbalanced structure of the tensor is not considered. The sizes of different modes of the tensor are different, resulting in an unbalanced structure of the unfolded matrix of some modes, and even a high-rank or full-rank matrix, which will lead to accuracy loss;

[0005] 2. Minimizing the rank is an NP-hard problem. Using the "convex relaxation" method of replacing the rank with the nuclear norm will lead to accuracy loss;

[0006] 3. Non-convex functions can approximate the rank better than the nuclear norm, but it will increase the computational complexity and the algorithm time;

[0007] In view of the above, the present application provides an efficient low-rank tensor recovery method to solve the above problems. Summary of the Invention

[0008] In view of the above situation, to overcome the defects of the prior art, the present invention provides an efficient low-rank tensor recovery method. This solution realizes high-precision low-rank tensor recovery by fusing an equilibrium tensor and a non-convex function. Further, by introducing a matrix decomposition method, the efficiency of the low-rank tensor recovery algorithm can be effectively improved.

[0009] An efficient low-rank tensor recovery method, characterized by comprising the following steps:

[0010] S1: Construct an equilibrium tensor; perform a reconstruction operation on the non-equilibrium tensor to obtain a more balanced tensor;

[0011] S2: Construct a low-rank tensor recovery algorithm based on the non-convex Logdet function; use the non-convex Logdet function to replace the nuclear norm to achieve a better approximation of the matrix rank;

[0012] S3: Introduce the matrix decomposition method; through matrix decomposition, decompose a large-scale matrix into the product of two small matrices to reduce the size of the matrix;

[0013] S4: By fusing S1 - S3, an efficient low-rank tensor recovery algorithm is obtained.

[0014] The beneficial effects of the above technical solutions are as follows:

[0015] (1) Through the reconstruction operation, a more balanced tensor can be constructed to eliminate the accuracy loss caused by the full-rank or high-rank unfolded matrix resulting from the unbalanced structure;

[0016] (2) By introducing a non-convex function, the matrix rank can be better approximated than the nuclear norm, thus effectively improving the algorithm accuracy;

[0017] (3) By introducing matrix decomposition, the scale of the matrix to be calculated can be reduced, the increase in computational complexity caused by the non-convex function can be reduced, and the computational efficiency can be improved. Brief Description of the Drawings

[0018] Figure 1 It is a schematic diagram for expressing the technical solution of the present invention;

[0019] Figure 2 It is a schematic diagram for expressing the ranks of different functions of the present invention;

[0020] Figure 3 It is a schematic diagram for the calculation error and time of different methods of the present invention;

[0021] Figure 4 It is a schematic diagram for comparing the image restoration results of different methods adopted by the present invention;

[0022] Figure 5 It is a schematic diagram for comparing the PSNR values of different methods of the present invention. Detailed Embodiments

[0023] Regarding the foregoing and other technical contents, features and effects of the present invention, they will be clearly presented in the following detailed description of the embodiments with reference to the accompanying drawings of the present application. The structural contents mentioned in the following embodiments are all based on the accompanying drawings of the specification.

[0024] Current research on low-rank tensor recovery algorithms is mainly based on minimizing the tensor rank. The definition of tensor rank is the first core issue, mainly including CP rank, Tucker rank, and tensor tree rank, etc. Among them, CP rank is based on the CP decomposition of tensors. However, the form of CP decomposition is not unique and is an NP-hard problem. There is currently no good method to directly obtain the CP rank. Tucker rank is based on Tucker decomposition. The main principle is to expand the tensor according to different modes to obtain a set of expansion matrices, and then perform low-rank optimization on them respectively. This method is relatively easy to solve and can reflect the differences in different structural directions of the tensor. Therefore, it has received the most extensive attention and there are also the most relevant studies. Tensor tree rank is a newly emerging rank definition method, which can be divided into binary tree, tensor train, and tensor ring forms. This rank definition is relatively complex, and its decomposition process usually requires some constraints. Currently, it is used less in practical applications;

[0025] The low-rank tensor recovery algorithm based on Tucker rank is currently the most studied and widely used. Considering that low-rank tensor recovery algorithms are mostly used for data recovery, prediction, and decision support in engineering practice, and usually have high requirements for accuracy and efficiency. Therefore, in such research, how to further improve the accuracy and efficiency of the algorithm still faces challenges;

[0026] Disadvantages of existing low-rank tensor recovery algorithms based on Tucker rank:

[0027] The unbalanced structure of the tensor is not considered. The sizes of different modes of the tensor are different, resulting in unbalanced structures of the expansion matrices of some modes, and even high-rank or full-rank matrices, which will lead to accuracy loss.

[0028] Minimizing the rank is an NP-hard problem. Using the "convex relaxation" method of replacing the rank with the nuclear norm will lead to accuracy loss;

[0029] Non-convex functions can approximate the rank better than the nuclear norm, but they will increase the computational complexity and the algorithm time.

[0030] In the above context, aiming at the problems of insufficient recovery accuracy and low efficiency caused by non-convex functions in existing research, this application proposes an efficient low-rank tensor recovery method, which is specifically as follows:

[0031] I. Construct an equilibrium tensor

[0032] The minimum number of known element values required for accurate tensor recovery is jointly determined by the structure of the tensor itself, that is, the expansion matrices of each mode. When the number m of known element values in the tensor satisfies m ≥ Crn K-1When the conditions are met, there is a high probability of accurately recovering the tensor x by minimizing the sum of the nuclear norms of each mode, where C is a constant, r is the rank of the unfolded matrix, and n is the dimension size of the tensor mode. It can be seen that for an unbalanced tensor, its unfolded matrix will be full rank or high rank, and more known elements are required for accurate recovery;

[0033] Therefore, the present solution gives the following reconstruction operation to construct a more balanced tensor;

[0034] Given an unbalanced tensor of order K where n i is the dimension size of its unbalanced mode, the reconstruction operation function is defined to fuse the i-th mode of the tensor with the -th mode (the reconstruction operation is achieved through the reshape function):

[0035]

[0036] The newly obtained tensor of order (K - 1) after reconstruction can be expressed as:

[0037] where

[0038] II. Low-rank Tensor Recovery Algorithm Model Based on Non-convex Logdet Function

[0039] The non-convex logDet function can achieve higher completion accuracy because the logDet function is a tighter rank approximation function than the nuclear norm, as Figure 2 shown;

[0040] To prove this, first express the logDet function and the nuclear norm respectively as:

[0041]

[0042] Secondly, construct the function f(X) = log(σ i (X) + 1) - σ i (X) and take its first derivative. It can be known that when σ i (X) > 0 holds, where σ i (X) represents the i-th singular value of the matrix X. Thus, it can be seen that f(X) is a decreasing function, that is, f(X) < f(0) = 0. Finally, a conclusion can be drawn: logdet(X + I) < ||X|| * ;

[0043] The low-rank tensor recovery algorithm model based on the non-convex Logdet function can be expressed as:

[0044]

[0045] Among them, represents known elements, and Ω is the index set of known elements. is a sampling function that keeps the elements in the set Ω unchanged, and α i is a constant parameter and satisfies

[0046] III. Matrix Decomposition

[0047] Through matrix decomposition, a large-scale matrix can be decomposed into the product of two small matrices, where the scale of each is much smaller than the original matrix. However, when applying it to tensor decomposition, first, a problem needs to be faced, that is, how to ensure that the rank of the decomposed matrix is the same as that of the original matrix. To solve this problem, an orthogonal constraint will be added during matrix decomposition;

[0048] Given a tensor which is a low-rank tensor ( is a commonly used symbol in mathematics, representing the set of real numbers, and the superscript represents the dimensional size of the numbers in the set), and its Tucker rank is rank(r1, r2,..., r N ). The unfolding matrix of the n-th mode of this tensor can be expressed as: Through matrix decomposition, this unfolding matrix can be expressed as the product of two matrices, that is where the matrix matrix U n satisfies the orthogonal constraint, U n ∈ St(I n , s n ), representing the orthogonal constraint, and there is s n > r n . Therefore, the following conclusion can be drawn:

[0049]

[0050] IV. Hybrid Model

[0051] Fusing the above methods, first, the tensor is reconstructed, where is a more balanced tensor obtained after reconstruction. Secondly, the non-convex Logdet function is combined with the matrix decomposition method. The combination point is that for each unfolding matrix there can be a matrix V n corresponding to it with the same rank as it, and the matrix V nThe scale is much smaller than the unfolded matrix. Therefore, only the matrix V needs to be optimized for low rank based on the non-convex Logdet function, and the low-rank tensor recovery algorithm hybrid model of this paper is given: n Among them, x is the more balanced tensor obtained after reconstruction. For the convenience of expression, a function L(V

[0052]

[0053] ) is redefined. This function is the objective function in the reconstruction algorithm: n ) is redefined. This function is the objective function in the reconstruction algorithm:

[0054]

[0055] In addition, for the matrix V n , variable separation is performed in this paper, that is, a new auxiliary variable P n = V n , n = 1, 2,..., N. At the same time, it is noted that in actual applications, most of the elements of the tensor are non-negative. Therefore, the algorithm model proposed in this paper can be expressed in the following form:

[0056]

[0057] Among them, the constant parameter α n satisfies the conditions α n ≥ 0 and The above hybrid model is optimized and solved by the alternating direction method of multipliers.

[0058] V. Experimental Verification

[0059] This scheme takes the publicly available color video dataset as the experimental object for experimental verification, and compares this method with methods such as HaLRTC and Logdet. The results are as follows:

[0060] Table 1 Comparison of Experimental Results

[0061]

[0062] In Table 1, BT represents building a balanced tensor, and MF represents matrix decomposition. The following conclusions can be obtained from Table 1 and Figure 3 :

[0063] 1) By combining the balanced tensor and the Logdet function, the Logdet-BT method can obtain more accurate recovery results than other methods at all sampling rates;

[0064] 2) Due to the introduction of non-convex functions and the increase in the dimension of the tensor unfolded matrix caused by reconstructing the balanced tensor, the time consumed by the Logdet-BT method increases;

[0065] 3) By further introducing the matrix factorization method, the computational efficiency of the Logdet-BT-MF method is significantly improved. Except for the case with a sampling rate of 10%, at other sampling rates, the time taken by this method is close to that of the HaLRTC method and much lower than that of the other two methods based on non-convex functions.

[0066] 4) While improving the efficiency, the Logdet-BT-MF method maintains a relatively high recovery accuracy, slightly lower than that of the Logdet-BT method and almost on par with the Logdet method.

[0067] From the above conclusions, it can be seen that constructing the balanced tensor and introducing non-convex functions can effectively improve the algorithm accuracy, while introducing the matrix factorization method can significantly improve the computational efficiency. The combination of the three can achieve good performance in terms of both accuracy and efficiency.

[0068] Furthermore, to more intuitively see the differences in the recovery results of different methods in terms of accuracy, this paper also selects a certain frame of the video for detailed display, and the results are as Figure 4 shown. It can be seen from the figure that the three methods containing non-convex functions can all recover more details than the HaLRTC method, and among them, the Logdet-BT method performs the best, and the Logdet method and the Logdet-BT-MF method perform relatively close. Among them Figure 4 is the comparison of the recovery results of the video frame. From left to right, they are: the original picture, the sampled picture, the HaLRTC recovery picture, the Logdet recovery picture, the Logdet-BT recovery picture, and the Logdet-BT-MF recovery picture. From top to bottom, the sampling rates are 10% and 20% respectively.

[0069] In addition to a specific frame of the picture, this scheme also calculates the PSNR values of all 150 frames of images to evaluate its recovery effect, and the specific results are as Figure 5 shown;

[0070] On the one hand, this scheme improves the algorithm accuracy by constructing the balanced tensor, and on the other hand, it uses the non-convex logDet function to replace the nuclear norm, and further improves the algorithm accuracy through a more compact rank approximation. Finally, aiming at the problem of increased computational complexity caused by the introduction of non-convex functions, the matrix factorization method is introduced to reduce the size of the matrix and improve the efficiency of the algorithm.

[0071] The content of this invention is supported by the Key Research and Development Project (Science and Technology Research) Fund of Henan Province, project number: 222102210113.

[0072] The above is only for the purpose of illustrating the present invention. It should be understood that the present invention is not limited to the above embodiments, and all kinds of variations conforming to the idea of the present invention are within the protection scope of the present invention.

Claims

1. An efficient low-rank tensor recovery method, characterized in that, Including the following steps: S1: Construct an equilibrium tensor; give a reconstruction operation for the non-equilibrium tensor to obtain a more balanced tensor; S2: Construct a low-rank tensor recovery algorithm based on the non-convex Logdet function; use the non-convex Logdet function to replace the nuclear norm to achieve a better approximation of the matrix rank; S3: Introduce a matrix factorization method; through matrix factorization, decompose a large-scale matrix into the product of two small matrices to reduce the size of the matrix; S4: By fusing S1 - S3, an efficient low-rank tensor recovery algorithm is obtained.

2. An efficient low-rank tensor recovery method according to claim 1, characterized in that The process of constructing the equilibrium tensor in S1 is as follows: S1-1: Given an unbalanced tensor of order K where n i represents the dimension size of the unbalanced norm; S1-2: Reconstruction operation function is defined to fuse the i-th norm of a tensor with the -th norm: S1-3: The new (K - 1)-order tensor obtained after reconstruction is expressed as: Among them 3. An efficient low-rank tensor recovery method according to claim 1, characterized in that, The process of constructing the low-rank tensor recovery algorithm based on the non-convex Logdet function in S2 is as follows: S2-1: The logDet function and the nuclear norm are respectively expressed as: S2-2: Construct the function f(X) = log(σ i (X) + 1) - σ i (X) and take its first derivative. When σ i (X) > 0 then f(X) is a decreasing function, that is, f(X) < f(0) = 0, and further it can be obtained that: logdet(X + I) < ||X|| * ; S2-3: The low-rank tensor recovery algorithm model based on the non-convex Logdet function can be expressed as: Among them, represents known elements, and Ω is the index set of known elements, is a sampling function such that elements within the set Ω remain unchanged, and α i is a constant parameter and satisfies 4. An efficient low-rank tensor recovery method according to claim 2, characterized in that While introducing the matrix factorization, add an orthogonal constraint, and the process is as follows: S1: Given a low-rank tensor and its Tucker rank is rank(r1, r2, …, r N ), the unfolding matrix of the n-th mode is expressed as: S2: Represent the above-expanded matrix as the product of two matrices, i.e., where matrix matrix U n satisfies the orthogonal constraint, and s n > r n , and finally obtain:

5. An efficient low-rank tensor recovery method according to claim 4, characterized in that The low-rank tensor recovery algorithm model in S4 is: Among them, is a more balanced tensor obtained after reconstruction. For the sake of convenience of expression, a function L(V n ) is redefined, and its expression is as follows:

6. An efficient low-rank tensor recovery method according to claim 5, characterized in that For matrix V n Perform variable separation and introduce a new auxiliary variable P n = V n , n = 1, 2, …, N, then the low-rank tensor recovery algorithm model is expressed in the following form: V n = P n , n = 1, 2, …, N, Among them, the constant parameter α n satisfies the condition α n ≥ 0 and

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