Low-rank tensor recovery method and device based on weighted tensor tubular nuclear norm partial sum and storage medium

Abstract

The invention relates to the technical field of low-rank tensor recovery, in particular to a low-rank tensor recovery method and device based on a weighted tensor tubular nuclear norm partial sum and a storage medium. By introducing a weighted partial singular value threshold WPSVT, different weights are distributed in the singular value of the tensor, and a more accurate low-rank tensor recovery model is formed. Specifically, the method comprises the following steps: firstly, defining a weighted tensor tubular nuclear norm part and a WPSTNN model, ignoring first r singular values of a tensor sheet, and distributing decreasing weights for the residual singular values; and then, converting a minimization problem into a weighted truncated nuclear norm form, and carrying out step-by-step solution by using an alternating direction multiplier method ADMM. The method shows an excellent recovery effect in tasks of image completion, image denoising, video separation and the like, and can still realize higher peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) when data is incomplete or noise exists, so that the data recovery precision is remarkably improved.

Description

Technical Field

[0001] The present invention relates to the technical field of low-rank tensor recovery, and in particular to a low-rank tensor recovery method, device and storage medium based on the partial sum of tubular nuclear norms of weighted tensors. Background Art

[0002] As a natural representation format for a wide range of real-world data, tensors provide more accurate features and reveal the basic structural information behind high-dimensional data such as videos, hyperspectral data, three-dimensional distance data, and network traffic data. Tensors have received increasing attention and have been successful in applications such as hyperspectral image denoising, color images, recommendation systems, and grayscale video restoration. However, due to hardware limitations or some damage, the collected data often have incomplete or interfering items, which seriously damages the data quality and hinders subsequent applications. Therefore, it is important to recover unknown low-rank tensors from partial observations or noise-contaminated data. In general, recovering any original tensor without prior knowledge is an ill-posed problem. Some studies have shown that the low rank or approximate low rank in real data can be used to deal with this ill-posed problem. However, how to determine the rank of a tensor remains a thorny problem. In the past decade, the two most widely used tensor rank definitions are CANDECOMP / PARAFAC (CP)-rank and Tucker-rank, which correspond to CANDECOMP / PARAFAC decomposition and Tucker decomposition, respectively. In addition, Kilmer et al. proposed tensor tube rank and tensor tube multi-rank under the framework of tensor singular value decomposition (T-SVD), transforming the tensor recovery problem into the minimization problem of tensor tube rank and tensor tube multi-rank. The two ranks defined under the framework of tensor singular value decomposition overcome the shortcomings of CP rank and Tucker rank because T-SVD is computable and does not require matrixing of tensors.

[0003] Tensor singular value decomposition (T-SVD) performs Fourier transformation on a third-order tensor and decomposes the Fourier domain tensor into a diagonal tensor and two orthogonal tensors in a manner similar to matrix singular value decomposition. Under the framework of tensor singular value decomposition, the mainstream tensor rank approximation models can be divided into two categories, namely convex models and non-convex models. In recent years, researchers have proposed many convex models. Lu and Feng et al. proposed a tensor nuclear norm to approximate the tensor tube rank for image denoising. Lu et al. defined a new tensor nuclear norm under reversible linear transformation and proposed a new tensor robust principal analysis model based on it. Various outstanding studies have also been conducted on tensor completion. Tao et al. verified that the tensor completion problem under the framework of tensor singular value decomposition is applicable to image and video data. Hu et al. proposed the twisted tensor nuclear norm (t-TNN) and used it for video completion. In 2017, Zhang et al. found that there is a high probability of success in accurately recovering low-tube rank tensors by solving the tensor tube nuclear norm minimization problem.

[0004] However, although it can be solved accurately by the alternating direction method of multipliers (ADMM), the convex model is still a suboptimal relaxation of the tensor rank minimization problem. Therefore, a large number of non-convex attempts have been proposed to improve the tensor rank approximation. An intuitive scheme is to enforce the low rank of the tensor by using non-convex constraints, such as t-gamma, MCP and SCAD penalties, and Schatten-p. However, most of these functions treat all singular values ​​of the f-diagonal tensor obtained after the tensor singular value decomposition equivalently, and are therefore not effective enough in many practical applications where different tensor rank components have different contributions. Therefore, Mu et al. used the weighted tensor nuclear norm (WTNN) to approximate the tensor tube rank, which adds a weight w to each singular value in the f-diagonal tensor. Although WTNN is more flexible in retaining the main data components by penalizing larger singular values ​​rather than smaller singular values, it does not fully consider the tensor prior rank information of the problem encountered. To address this limitation, Jiang et al. invented the partial sum of tensor tube nuclear norms (PSTNN). PSTNN only minimizes the minimum (n1,n2)-r singular values ​​of each slice of the Fourier domain tensor, so it is a tighter envelope of the tensor tube multi-rank. However, since no PSTNN also considers the contribution of different singular values, the soft threshold is not ideal. Summary of the invention

[0005] To solve the above technical problems, the present invention provides a low-rank tensor recovery method based on the partial sum of tubular nuclear norms of weighted tensors, comprising the following steps:

[0006] S1. Construct the weighted tensor tubular nuclear norm part and model, assign weights to the singular values ​​of the tensor slices, ignore the first r singular values ​​of each tensor slice, and assign weights that decrease as the singular values ​​increase to the remaining singular values;

[0007] S2. construct the obtained weighted tensor tubular nuclear norm part and model as the minimization objective function of low-rank tensor recovery;

[0008] S3. Converting the minimization objective function into a truncated nuclear norm weighted problem of n3 complex domains;

[0009] S4. The partial singular value threshold is expanded to a weighted partial singular value threshold, a weighted partial singular value threshold operator is constructed, and the alternating direction multiplier method is used to solve the low-rank tensor recovery minimization objective function, and the tensor variables and Lagrange multipliers are iteratively updated respectively until the function converges and the tensor recovery is completed.

[0010] Furthermore, the weighted tensor tubular nuclear norm part and model can be expressed as:

[0011]

[0012] in, For tensors The partial sum of the weighted tensor tubular nuclear norm of ; r represents the tensor polyhedral rank; w j Represents the weight of each singular value; σ j represents the jth largest singular value; the setting range of the weight of each singular value is 0 to 1, the weight of the first r singular values ​​is zero, and the weight value decreases as the singular value increases.

[0013] Furthermore, the minimization objective function is constructed by Lagrangian function, which can be expressed as:

[0014]

[0015] Among them, the tensor and is a third-order tensor with the same three-dimensional space, μ is the penalty parameter, and F is the F-norm.

[0016] Furthermore, in step S3, the low-rank tensor recovery minimization objective function is converted into a truncated nuclear norm weighted problem in n3 complex domains and then solved. The converted function can be expressed as:

[0017]

[0018] Among them, w is the weight.

[0019] Further, the alternating multiplier method in step S4 includes the following steps:

[0020] a. Fix auxiliary variables and Lagrange multipliers, and update tensor variables;

[0021] b. Fix tensor variables and Lagrange multipliers, and update auxiliary variables;

[0022] c. Fix the tensor variables and auxiliary variables, and update the Lagrange multipliers until the convergence conditions are met.

[0023] Furthermore, the tensor recovery includes tensor completion and tensor robust principal component analysis, which are used to recover low-rank components in incomplete observation data and noise-contaminated data, respectively.

[0024] The present invention also provides a computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, the steps of a low-rank tensor recovery method based on the partial sum of tubular nuclear norms of weighted tensors as described in any of the preceding items are implemented.

[0025] The present invention also provides a storage medium having a computer program stored thereon, characterized in that when the computer program is executed by a processor, the steps of a low-rank tensor recovery method based on the partial sum of tubular nuclear norms of weighted tensors as described in any of the preceding items are implemented.

[0026] The beneficial effects of the present invention are:

[0027] (1) By introducing the weighted partial singular value threshold (WPSVT) into the objective function and assigning different weights to singular values, the key components of the data are retained, making low-rank recovery more accurate, especially when the data is incomplete or noisy.

[0028] (2) By using an adjustable weight mechanism and adjusting the threshold according to the size of the singular value, the model can be adapted to different data dimensions and data quality, providing high flexibility for a wide range of application scenarios.

[0029] (3) Compared with the traditional convex optimization method, this scheme adopts non-convex weighted processing, which has better optimization effect, reduces unnecessary data loss, and improves the performance of the model in high-noise environments.

[0030] (4) The model is solved step by step through the alternating direction multiplier method (ADMM), which reduces the computational complexity of direct solution, improves the efficiency of data recovery, and is suitable for large-scale data processing.

[0031] (5) The WPSTNN model has demonstrated superior performance in tasks such as image completion, image denoising, and video separation, improving the quality of restored data and providing effective technical support for the field of image and video processing. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 , a flowchart of a low-rank tensor recovery method based on the partial sum of tubular nuclear norms of weighted tensors of the present invention.

[0033] Figure 2, a schematic diagram of the algorithm for weighted partial singular value threshold according to an embodiment of the present invention.

[0034] Figure 3 , a schematic diagram of a tensor completion model solving algorithm based on the weighted tubular nuclear norm partial sum according to an embodiment of the present invention.

[0035] Figure 4 , and numerical result diagram of the tensor completion model experiment of an embodiment of the present invention.

[0036] Figure 5 , a schematic diagram of the completion results of a random image using the tensor completion model of an embodiment of the present invention.

[0037] Figure 6 , a schematic diagram of the block completion results of a random image using the tensor completion model of an embodiment of the present invention.

[0038] Figure 7 , a schematic diagram of a tensor robust principal component analysis model solving algorithm based on the partial sum of weighted tensor tubular nuclear norms according to an embodiment of the present invention.

[0039] Figure 8 , a diagram of numerical results of the tensor robust principal component analysis model experiment of an embodiment of the present invention.

[0040] Fig. 9 , a schematic diagram of the denoising results of random noise in an image using the tensor robust principal component analysis model according to an embodiment of the present invention.

[0041] Fig.10 , a schematic diagram of the denoising results of salt and pepper noise in an image using the tensor robust principal component analysis model of an embodiment of the present invention.

[0042] Fig.11 , a schematic diagram of the separation results of dynamic video using the tensor robust principal component analysis model of an embodiment of the present invention.

[0043] Fig.12 , a schematic diagram of the internal structure of a computer device of the present invention.

[0044] Fig.13 , a schematic diagram of a storage medium of the present invention. DETAILED DESCRIPTION

[0045] In order to enable those skilled in the art to better understand the content of the present invention and make the purpose, technical solution and advantages of the present invention clearer, the present invention is further described in detail below in conjunction with the embodiments and the accompanying drawings. The illustrative embodiments of the present invention and their description are only used to explain the present invention and are not intended to further limit the present invention.

[0046] Embodiment 1:

[0047] like Figure 1As shown, an embodiment of the present invention provides a low-rank tensor recovery method based on the partial sum of tubular nuclear norms of weighted tensors, comprising the following steps:

[0048] S1. Construct the weighted tensor tubular nuclear norm part and the model WPSTNN, which can be expressed as:

[0049]

[0050] in, For tensors The partial sum of the weighted tensor tubular nuclear norm of , r represents the tensor polyhedral rank, w j represents the weight of each singular value, σ j represents the jth largest singular value. This model ignores the first r singular values ​​of each slice and assigns different weights to its remaining singular values. It is worth mentioning that the weights increase and decrease with the increase of singular values, but these weights are between 0 and 1.

[0051] S2. The obtained weighted tensor tubular nuclear norm part and model are constructed as the minimization objective function of low-rank tensor recovery, which can be expressed as:

[0052]

[0053] Among them, the tensor and is a third-order tensor with the same three-dimensional space, μ is the penalty parameter, and F is the F-norm.

[0054] At the same time, because the relationship between the f-norm of the original tensor and the Fourier transformed tensor can be expressed as:

[0055]

[0056] in, It is represented as a tensor after Fourier transformation. The Fourier transformation keeps the tensor dimension unchanged and each real number element in the tensor becomes a complex number. Then, the minimization objective function of the low-rank tensor recovery can be transformed into:

[0057]

[0058] S3. The minimization objective function is converted into a truncated nuclear norm weighted problem of n3 complex domains, which can be expressed as:

[0059]

[0060] Among them, w is the weight.

[0061] S4. Expand the partial singular value threshold to the weighted partial singular value threshold and obtain the weighted partial singular value threshold WPSVT: Two complex matrices X, Y∈C n1×n2 .

[0062] The complex matrix Y is represented as m by:

[0063] Among them, Y1 and Y2 are matrices corresponding to the first r largest singular values ​​and the last min(n1,n2)-r singular values ​​respectively; U Y1 and are two orthogonal tensors obtained by singular value decomposition of Y1; U Y2 and They are two orthogonal tensors obtained by singular value decomposition of Y2.

[0064] When two complex matrices satisfy the following formula:

[0065]

[0066] Then the complex field matrix X is:

[0067]

[0068] in, is a weight vector, the first term is 0, and the second term is a number that gradually increases between 0 and 1, which means that the weights of the first r singular values ​​are set to 0, and the weights of the last min(n1,n2)-r singular values ​​are related to the size of the singular value itself. The smaller the singular value, the greater the weight. In general, it can be expressed as:

[0069]

[0070] S5. Proof: Defining the weighted cutoff singular value threshold WPSVT includes the following sub-steps:

[0071] S5.1. According to von Neumann's theorem:

[0072] Singular values ​​of two complex matrices yes and but:

[0073]

[0074] When matrices X and Y have the same left and right singular value vectors, the equality of the above formula holds. It can be expressed as:

[0075] X=Udiag(σ(X))V H and Y=Udiag(σ(Y))V H;

[0076] S5.2. Reiterate the WPSVT model, which can be expressed as:

[0077]

[0078] S5.3. Where the matrices X,Y∈C n1×n2 , Z = min(n1, n2), first take the second half of the formula described in S5.2 and transform it to obtain:

[0079]

[0080] S5.4. According to von Neumann's theorem, the above model can be transformed into:

[0081]

[0082] Where σ(X) and σ(Y) represent the singular values ​​of the two matrices respectively. And, It can be considered as a constant.

[0083] S5.5. Therefore, the minimization objective function of the model in S5.2 can be expressed as:

[0084]

[0085] S5.6. Convert the minimization objective function in S5.5 into:

[0086]

[0087] The model in step S5.6 can be regarded as a σ i The quadratic equation of (X) yields:

[0088]

[0089] in, We thus prove the WPSVT operator.

[0090] S6. For different singular values, we will set different weights, and the weights are set as follows:

[0091]

[0092] Among them, v is a very small number. The purpose of this parameter is to prevent the influence of the denominator 0, and r represents the multi-rank of the tensor tube.

[0093] Thus, we can get Figure 2 The algorithm of the weighted partial singular value threshold WPSVT is shown.

[0094] S7. Repeat the above steps until the minimization function converges and tensor recovery is completed.

[0095] Furthermore, the tensor recovery includes tensor completion and tensor robust principal component analysis, which are used to recover low-rank components in incomplete observation data and noise-contaminated data, respectively.

[0096] Embodiment 2:

[0097] Based on a low-rank tensor recovery method based on a weighted tensor tubular nuclear norm partial sum and a weighted tensor tubular nuclear norm partial sum model WPSTNN proposed in Example 1 of the present invention, Example 2 proposes a tensor completion model and a solution method based on a weighted tensor tubular nuclear norm partial sum WPSTNN, which is used to implement the tensor completion described in Example 1, including the following steps:

[0098] A1. A tensor completion formula based on the weighted partial tubular nuclear norm of the tensor is proposed, which can be expressed as:

[0099]

[0100] in, is the original tensor, To recover the tensor, Ω is the observation term, The part equal to 0 is the observation item, and the rest is 0A2. Introducing the solution conditions, it can be expressed as:

[0101]

[0102] in,

[0103] A3. Introduce the auxiliary variable M, and according to the introduced solution conditions, use the alternating direction multiplier method to solve the tensor completion formula model, then its calculation model can be expressed as:

[0104]

[0105] A4. Expand the calculation model in step A3 by Lagrangian to obtain:

[0106]

[0107] in represents the Lagrange multiplier, μ is the penalty parameter, Represents a constant.

[0108] A5. Solve iteratively and Lagrange multipliers set up It is a 0 tensor and includes the following sub-steps:

[0109] A5.1. Fixation and renew

[0110]

[0111] A5.2. Fixation and renew

[0112]

[0113] Available The solution is:

[0114]

[0115] Where Ω represents the observation set, c represents the unobserved set,

[0116] A5.3. Fixation and renew

[0117]

[0118] A6. Figure 3 The algorithm for solving the tensor completion model based on the weighted tubular nuclear norm partial sum is shown.

[0119] A7. The tensor completion model based on the partial sum of weighted tubular nuclear norm is applied to image completion and compared with the existing tensor recovery model. Both quantitative and qualitative evaluation methods are adopted.

[0120] In the quantitative evaluation, we use the peak signal-to-noise ratio (PSNR) and the structural similarity index metric (SSIM) to measure the effect of tensor completion. The larger the PSNR value and the SSIM value, the closer the two images are and the better the restoration effect. First, we compare the WPSTNN algorithm with TNN, HALRTC, SILRTC, T-TNNR, t-log, WTNN and PSTNN. In order to more comprehensively demonstrate the advantages of WPSTNN, we conduct experiments and analysis on 15 pictures and set two sampling rates: 0.6 and 0.5. The experimental numerical results are shown in the figure. Figure 4 As shown in the figure, the SSIM value and PSNR value of the image obtained by the WPSTNN algorithm are larger, and the completion effect is the best.

[0121] In the qualitative evaluation, the same part of the same image restored by different algorithms is magnified by the same factor, and the restoration effect can be observed with the naked eye.

[0122] First, a picture is randomly sampled with a sampling rate of 0.35. Different algorithms and PSNR values ​​are used to complete the image and partially enlarge it to obtain the following: Figure 5 The completion results, PSNR values, and magnification results shown all indicate that the WPSTNN model performs best in random sampling completion.

[0123] Then, different algorithms and PSNR values ​​are used to complete the block of an image and locally enlarge its block part to obtain the following Figure 6 The completion results, PSNR values, and magnification results shown all indicate that the WPSTNN model performs best in block completion.

[0124] Embodiment 3:

[0125] Based on the low-rank tensor recovery method based on the partial sum of the tubular nuclear norm of a weighted tensor and the partial sum model of the tubular nuclear norm of a weighted tensor WPSTNN proposed in Example 1 of the present invention, Example 3 proposes a tensor robust principal component analysis model and a solution method based on the tubular nuclear norm partial sum of a weighted tensor WPSTNN, which is used to implement the tensor robust principal component analysis described in Example 1, including the following steps:

[0126] B1. Tensor robust principal component analysis (TRPCA) is to denoise the noise-contaminated tensor to obtain a low-rank tensor and a sparse tensor, where the low-rank tensor is close to the clean original tensor and the sparse tensor is the separated noise tensor. The tensor robust principal component analysis model based on WPSTNN is:

[0127]

[0128] in, represents the observation tensor, low-rank tensor and sparse tensor, λ is a positive parameter, Representing a tensor , where the l0 norm is a tensor of the number of non-zero values ​​of , and the elements of the sparse tensor are composed of 0 and 1, the noisy ones are 1, and the noise-free ones are 0.

[0129] B2. Using l1 norm instead of l0 norm, the tensor robust principal component analysis model can be converted to:

[0130]

[0131] B3. Solve the tensor robust principal component analysis model in B2 by the alternating direction multiplier method ADMM, and expand the tensor robust principal component analysis model Lagrangian as:

[0132]

[0133] in represents the Lagrange multiplier, is the penalty parameter. All are 0 tensors, and are solved iteratively and Lagrange multipliers It includes the following sub-steps:

[0134] B3.1. Fixation and renew

[0135]

[0136] B3.2. Fixation and renew

[0137]

[0138] Solving by the soft threshold operator, we can get:

[0139]

[0140] B3.3. Update Lagrange multipliers The calculation formula can be expressed as:

[0141]

[0142] B4. Update and solve O, L and S respectively, and get the following Figure 7 The tensor robust principal component analysis model solution algorithm based on the partial sum of the weighted tensor tubular nuclear norm is shown.

[0143] B5. The tensor robust principal component analysis model based on WPSTNN is applied to image denoising and compared with the existing tensor recovery model, using quantitative and qualitative evaluation methods respectively.

[0144] In the quantitative evaluation, the peak signal-to-noise ratio (PSNR) and the structural similarity index metric (SSIM) are used to measure the effect of tensor robust principal component analysis. The larger the PSNR value and the SSIM value, the closer the two images are and the better the restoration effect. In the experimental part of image denoising, the WPSTNN algorithm is compared with SN, TNN, tf-TRPCA, schattenp and PSTNN. In order to more fully demonstrate the advantages of WPSTNN, experiments were conducted on 15 images and two noise ratios were set: 0.15 and 0.2. The numerical results of the experimental results are shown in Figure 8 As shown in the figure, the SSIM value and PSNR value of the image obtained by the WPSTNN algorithm are larger, and the denoising effect is the best.

[0145] In the qualitative evaluation, the same part of the same image restored by different algorithms is magnified by the same factor, and the restoration effect can be observed with the naked eye.

[0146] First, a picture is randomly added with noise, the noise rate is 0.2, and the image is denoised using different algorithms and PSNR values, and the image is partially enlarged to obtain the following Fig. 9 The denoising results, PSNR values ​​and magnification results shown all indicate that the WPSTNN algorithm has the best random noise denoising effect.

[0147] Then, a picture is randomly added with salt and pepper noise, with a noise rate of 0.2, and the image is denoised using different algorithm PSNR values, and the image is locally amplified to obtain the following: Fig.10 The denoising results, PSNR values ​​and magnification results shown all indicate that the WPSTNN algorithm has the best effect in salt and pepper noise denoising.

[0148] B6. The WPSTNN-based tensor robust principal component analysis model is applied to video separation and compared with the existing tensor recovery model.

[0149] In the video separation part: Tensor principal component analysis can be used to separate the background of the video. For a dynamic video, it can be regarded as a color image of n frames, and each frame can be regarded as an n1×n2×3 tensor. In the video background image separation, we convert the N tensors of n1×n2×3 into a three-dimensional tensor of n1·n2×N×3, and then process it. During the processing, the tubular multi-rank of the tensor is set to [1,1,1], and the following is obtained: Fig.11 The video separation results shown in the figure demonstrate the superiority of WPSTNN in video separation.

[0150] Embodiment 4:

[0151] Further, as a preferred embodiment of the present application, a terminal device is proposed based on a low-rank tensor recovery method of a weighted tensor tubular nuclear norm partial sum, such as Fig.12 The terminal device 200 includes at least one memory 210, at least one processor 220, and a bus 230 connecting different platform systems.

[0152] The memory 210 may include a readable medium in the form of a volatile memory, such as a random access memory (RAM) 211 and / or a cache memory 212 , and may further include a read only memory (ROM) 213 .

[0153] Among them, the memory 210 also stores a computer program, and the computer program can be executed by the processor 220, so that the processor 220 executes any of the above-mentioned low-rank tensor recovery methods based on the partial sum of the tubular nuclear norm of the weighted tensor in the embodiments of the present application, and its specific implementation method is consistent with the implementation method and the technical effect achieved in the above-mentioned embodiments, and some contents are not repeated. The memory 210 may also include a program / utility 214 having a set (at least one) of program modules 215, such program modules include but are not limited to: an operating system, one or more applications, other program modules and program data, each of these examples or some combination may include the implementation of a network environment.

[0154] Accordingly, the processor 220 may execute the above-mentioned computer programs, and may execute the program / utility 214 .

[0155] The bus 230 may be a bus representing one or more of several types of bus structures, including a memory bus or memory controller, a peripheral bus, an accelerated graphics port, a processor, or a local bus using any of a variety of bus architectures.

[0156] The terminal device 200 may also communicate with one or more external devices 240 such as keyboards, pointing devices, Bluetooth devices, etc., and may also communicate with one or more devices that can interact with the terminal device 200, and / or communicate with any device (such as a router, a modem, etc.) that enables the terminal device 200 to communicate with one or more other computing devices. Such communication may be performed through an input / output (I / O) interface 250. In addition, the terminal device 200 may also communicate with one or more networks (such as a local area network (LAN), a wide area network (WAN) and / or a public network, such as the Internet) through a network adapter 260. The network adapter 260 may communicate with other modules of the terminal device 200 through a bus 230. It should be understood that, although not shown in the figure, other hardware and / or software modules may be used in conjunction with the terminal device 200, including but not limited to: microcode, device drivers, redundant processors, external disk drive arrays, RAID systems, tape drives, and data backup storage platforms, etc.

[0157] Further, as a preferred implementation scheme of this embodiment, a computer-readable storage medium is proposed for a low-rank tensor recovery method based on a partial sum of a weighted tensor tubular nuclear norm, on which instructions are stored, and when the instructions are executed by a processor, any of the above-mentioned low-rank tensor recovery methods based on a partial sum of a weighted tensor tubular nuclear norm is implemented. Its specific implementation method is consistent with the implementation method and the technical effect achieved in the above-mentioned embodiment, and some contents are not repeated here.

[0158] like Fig.13As shown, the program product 300 provided in this embodiment for implementing the above-mentioned low-rank tensor recovery method based on the partial sum of the tubular nuclear norm of the weighted tensor can be a portable compact disk read-only memory (CD-ROM) and include program code, and can be run on a terminal device, such as a personal computer. However, the program product 300 of the present invention is not limited to this. In this document, the readable storage medium can be any tangible medium containing or storing a program, which can be used by or in combination with an instruction execution system, device or device. The program product 300 can use any combination of one or more readable media. The readable medium can be a readable signal medium or a readable storage medium. The readable storage medium can be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, device or device, or any combination of the above. More specific examples of readable storage media (a non-exhaustive list) include: an electrical connection having one or more conductors, a portable disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), optical fibers, a portable compact disk-read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing.

[0159] Computer-readable storage media may include data signals propagated in baseband or as part of a carrier wave, wherein readable program codes are carried. Such propagated data signals may take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination of the above. The readable storage medium may also be any readable medium other than the readable storage medium, which may send, propagate, or transmit a program used by or in combination with an instruction execution system, an apparatus, or a device. The program code contained on the readable storage medium may be transmitted with any appropriate medium, including but not limited to wireless, wired, optical cable, RF, etc., or any suitable combination of the above. The program code for performing the operation of the present invention may be written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Java, C++, etc., and also conventional procedural programming languages ​​such as "C" language or similar programming languages. The program code may be executed entirely on a user computing device, partially on a user device, as an independent software package, partially on a user computing device, partially on a remote computing device, or entirely on a remote computing device or server. Where a remote computing device is involved, the remote computing device may be connected to the user computing device through any type of network, including a local area network (LAN) or a wide area network (WAN), or may be connected to an external computing device (e.g., through the Internet using an Internet service provider).

[0160] The present invention is explained from the viewpoints of purpose of use, efficiency, progress and novelty, and the practical progress it has is in line with the functional enhancement and use requirements emphasized by the Patent Law. The above description and drawings of this application are only preferred embodiments of this application, and are not intended to limit this application. Therefore, all structures, devices, features, etc. that are similar or identical to those of this application, that is, all equivalent replacements or modifications made in accordance with the scope of the patent application of this application, should fall within the scope of protection of the patent application of this application.

[0161] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A low-rank tensor recovery method based on partial sum of tubular nuclear norm of weighted tensor, characterized in that: The following steps are involved: S1. Construct the weighted tensor tubular nuclear norm part and model, assign weights to the singular values ​​of the tensor slices, ignore the first r singular values ​​of each tensor slice, and assign weights that decrease as the singular values ​​increase to the remaining singular values; S2. construct the obtained weighted tensor tubular nuclear norm part and model as the minimization objective function of low-rank tensor recovery; S3. Converting the minimization objective function into a truncated nuclear norm weighted problem of n3 complex domains; S4. The partial singular value threshold is expanded to a weighted partial singular value threshold, a weighted partial singular value threshold operator is constructed, and the alternating direction multiplier method is used to solve the low-rank tensor recovery minimization objective function, and the tensor variables and Lagrange multipliers are iteratively updated respectively until the function converges and the tensor recovery is completed.

2. The low-rank tensor recovery method based on the partial sum of tubular nuclear norms of weighted tensors according to claim 1, characterized in that: The weighted tensor tubular nuclear norm part and model can be expressed as: in, is the partial sum of the weighted tensor tubular nuclear norm of the tensor x; r represents the tensor polyhedral rank; w j Represents the weight of each singular value; σ j represents the jth largest singular value; the setting range of the weight of each singular value is 0 to 1, the weight of the first r singular values ​​is zero, and the weight value decreases as the singular value increases.

3. The low-rank tensor recovery method based on partial sum of weighted tensor tubular nuclear norm according to claim 1, characterized in that: The minimization objective function is constructed by the Lagrangian function and can be expressed as: Among them, tensors x and y are third-order tensors with the same three-dimensional space, μ is the penalty parameter, and F is the F-norm.

4. The low-rank tensor recovery method based on the partial sum of tubular nuclear norms of weighted tensors according to claim 1, characterized in that: In step S3, the low-rank tensor recovery minimization objective function is converted into a truncated nuclear norm weighted problem in n3 complex domains and then solved. The converted function can be expressed as: Among them, w is the weight.

5. The low-rank tensor recovery method based on partial sum of tubular nuclear norm of weighted tensor according to claim 1, characterized in that: The alternating multiplier method in step S4 comprises the following steps: a. Fix auxiliary variables and Lagrange multipliers, and update tensor variables; b. Fix tensor variables and Lagrange multipliers, and update auxiliary variables; c. Fix the tensor variables and auxiliary variables, and update the Lagrange multipliers until the convergence conditions are met.

6. The low-rank tensor recovery method based on partial sum of weighted tensor tubular nuclear norm according to claim 1, characterized in that: The tensor recovery includes tensor completion and tensor robust principal component analysis, which are used to recover low-rank components in incomplete observation data and noise-contaminated data respectively.

7. A computer device comprising a memory, a processor and a computer program stored in the memory and executable on the processor, characterized in that: When the processor executes the computer program, the steps of a low-rank tensor recovery method based on the partial sum of tubular nuclear norms of weighted tensors as described in any one of claims 1 to 6 are implemented.

8. A storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of a low-rank tensor recovery method based on the partial sum of tubular nuclear norms of weighted tensors as described in any one of claims 1 to 6 are implemented.

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