Tensor interpolation decomposition method and device based on Bayesian learning
Through the tensor interpolation decomposition method based on Bayesian learning, the weight matrix is optimized by clustering and probability modeling, and the problems of interpretability and computational cost in high-order tensor decomposition are solved, achieving efficient tensor decomposition effect.
Patent Information
- Application Number
- CN202510454795.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-04-11
- Publication Date
- 2025-07-25
AI Technical Summary
Existing tensor decomposition methods are insufficient in interpretability and computational cost, especially in poor performance when processing higher order tensor data, and excessive parameter space for matrix interpolation decomposition leads to high computational cost and possible overfitting.
The tensor interpolation decomposition method based on Bayesian learning is adopted. By expanding multi-order tensors into matrices and clustering, column vectors are extracted to form a skeleton matrix, combining Bayesian learning for probability modeling and Gibbs sampling, setting a priori distribution to optimize the weight matrix, and iteratively computing the tensor approximation.
It improves the interpretability and computational efficiency of tensor decomposition, reduces the number of updates of the factor matrix, reduces the calculation cost, and enhances the processing ability of high-order tensor data.
Smart Images
Figure CN120372247A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of tensor decomposition, and in particular to a tensor interpolation decomposition method and device based on Bayesian learning. Background Art
[0002] Tensor data is characterized by its multidimensional structure and is a powerful computational tool. For example, sensor data collected over time can be represented as a third-order tensor, where the dimensions correspond to spatial position, time, and sensor type. Tensor decomposition is a tensor analysis technique that captures the inherent structure in tensors. As a foundation, tensor decomposition has been widely used in various fields, including recommendation systems, computer vision, and computational biology. Currently, common tensor decomposition methods include CP decomposition, Tucker decomposition, and tensor ring decomposition. They provide differentiated decomposition frameworks and have their own advantages in different scenarios.
[0003] Although the above tensor decomposition methods are more generally applicable, they often ignore the intrinsic characteristics of the data, resulting in limited interpretability. In contrast, in the context of second-order tensors, i.e. matrices, matrix interpolation decomposition is a technique that approximates the original matrix with a smaller low-rank matrix by selecting a subset of columns or rows. Thanks to this, matrix interpolation decomposition is more interpretable than other methods. However, matrix interpolation decomposition still has several limitations. Specifically, it is mainly designed for two-dimensional matrices and cannot be directly applied to high-order tensors, while real-world data often produces high-order tensors due to complex multiple attributes. In addition, the parameter space of matrix interpolation decomposition can be large, which makes the computational cost high and may lead to overfitting.
[0004] On the other hand, the solution of the factor matrix in the tensor decomposition process is not limited to a specific algorithm. With the development of computing technology, methods based on Bayesian learning have been widely used and paid attention to the problem of solving factor matrices in recent years. The core idea of this method is to regard the elements in the factor matrix as latent variables, that is, those variables that are not directly observed but can be indirectly inferred through data. Specifically, the Bayesian method constructs a prior distribution and a likelihood function, and uses Bayes' theorem to update the belief in the latent variables to obtain the posterior distribution. This posterior distribution contains not only the expected value of the latent variables, but also their uncertainty information, which is crucial for making robust inferences. Therefore, tensor decomposition based on Bayesian learning allows researchers to design appropriate prior distributions according to the characteristics of specific problems, making the solution process more suitable for actual application scenarios, thereby enhancing the interpretability of tensor decomposition. Summary of the invention
[0005] In view of this, it is necessary to provide a tensor interpolation decomposition method and device based on Bayesian learning to effectively solve the technical problem of low interpretability of current tensor decomposition methods.
[0006] The present invention provides a tensor interpolation decomposition method based on Bayesian learning, including the following steps:
[0007] Step S1: Organize the input data into a multi-order tensor according to set features, expand the multi-order tensor into a matrix in one dimension, cluster the matrix, and extract a certain number of column vectors from different clustering categories to form a skeleton matrix;
[0008] Step S2: Regard the skeleton matrix as one of the factor matrices in CP decomposition, and regard the remaining factor matrices as weight matrices; use Bayesian learning to perform probability modeling on tensor interpolation decomposition and add a prior to the tensor;
[0009] Step S3: Set a prior for the weight matrix, draw a probabilistic graphical model according to the weight matrix prior, and derive the posterior distribution of the model based on the probabilistic graphical model;
[0010] Step S4: Use the Gibbs sampling algorithm to perform posterior approximation calculation of the weight matrix to obtain a tensor approximation;
[0011] Step S5: Calculate the model error value based on the tensor approximation. If the model error value is less than the set threshold, output the weight matrix; otherwise, iterate step S4.
[0012] Preferably, step S1 is specifically:
[0013] Organize the input data into a multi-order tensor according to certain features, and expand the tensor into a matrix in a set mode;
[0014] Set the number of clusters. After clustering the matrix, extract the set total number of column vectors from different clustering categories. The set total number is the CP rank of the tensor, and the column vectors are the set mode fibers of the tensor. Obtain the index set of the column vectors in the matrix;
[0015] According to the index set, extract the column vectors from the matrix to form a skeleton matrix.
[0016] Preferably, in step S2, regarding the skeleton matrix as one of the factor matrices in CP decomposition and regarding the remaining factor matrices as weight matrices is specifically:
[0017] When decomposing an M-order tensor based on the CP decomposition form, there are M factor matrices. One of the factor matrices is the skeleton matrix, and the remaining M - 1 factor matrices are the weight matrices. The M-order tensor is expressed as:
[0018]
[0019] Among them, A represents the skeleton matrix, ⊙ represents the Kronecker product, and W (n) represents the nth weight matrix, and N is the number of weight matrices.
[0020] Preferably, in step S2, Bayesian learning is used to perform probabilistic modeling on tensor interpolation decomposition, and a prior is added to the tensor. Specifically:
[0021] Perform probabilistic modeling on tensor interpolation decomposition using Bayesian learning, and perform interpolation decomposition on the tensor based on the factor matrix:
[0022]
[0023] Among them, represents the tensor, represents the Kruskal operator, A represents the skeleton matrix, and W (n) represents the nth weight matrix, and N is the number of weight matrices;
[0024] Set a normal distribution prior with mean and variance for the tensor.
[0025] Preferably, step S2 further includes adding an inverse gamma distribution prior to the variance.
[0026] Preferably, in step S3, a prior is set for the weight matrix. Specifically:
[0027] Following the value limit requirements of the weight matrix in interpolation decomposition, set a truncated normal distribution as the prior for the weight matrix.
[0028] Preferably, in step S3, the posterior distribution of the model is derived based on the probabilistic graphical model. Specifically:
[0029] Obtain the posterior distributions of the weight matrix and the noise variance based on the probabilistic graphical model;
[0030] According to the probabilistic graphical model, the posterior distribution of the weight matrix is:
[0031]
[0032] Among them, represents fixing the rth column in the nth weight matrix, where r represents the rth column in the weight matrix, the other columns in the same matrix, A represents the skeleton matrix, and W (-n) represents all the remaining weight matrices except the fixed nth weight matrix, represents the tensor, and μw is the mean of the distribution, τ w is the precision, σ is the variance, represents the generalized truncated normal distribution, a and b are the value ranges of the truncated normal distribution, represents the mean parameter, represents the precision parameter;
[0033] The update rules for the mean parameter and the precision parameter are:
[0034]
[0035] where ⊙ represents the Kronecker product, represents the Khatri - Rao product after removing the k - th weight matrix, Ω represents the set of all observed element indices in the tensor, <, > represents the generalized inner product, i m r represents the vector index in the m - th mode, and T represents the transpose of the matrix;
[0036] According to the probability graph, the posterior distribution of the noise variance is:
[0037]
[0038] where, and are the shape parameter and the scale parameter of the inverse gamma distribution respectively;
[0039] The update rules for the posterior parameters are:
[0040]
[0041] where |Ω| represents the number of all elements in the tensor, is the Kruskal operator, and is the model error with the Frobenius norm.
[0042] Preferably, the step S4 is specifically:
[0043] Based on the model posterior distribution and the posterior parameter update rules, iterative calculations are performed based on the Gibbs sampling algorithm to obtain the approximation of the tensor.
[0044] Preferably, the step S5 is specifically:
[0045] Calculate the difference between the tensor and the tensor approximation as the model error. If the model error is greater than the set threshold, the model has not converged, and the step S4 is repeated through Gibbs sampling. If the model error is not greater than the set threshold, the model has converged, and the weight matrix is output as the interpolation information of the skeleton matrix.
[0046] The present invention also provides a tensor interpolation decomposition device based on Bayesian learning, comprising a memory and a processor, wherein a computer program is stored in the memory, and when the computer program is executed by the processor, the tensor interpolation decomposition method based on Bayesian learning is implemented.
[0047] Compared with the prior art, the present invention has the following beneficial effects: the present invention combines Bayesian learning and interpolation decomposition, Bayesian learning provides appropriate priors in the field, and interpolation decomposition is used for subsets of the data itself to mine the potential features of the data, and the decomposition interpretability is stronger than other methods. At the same time, the skeleton matrix is used to replace a factor matrix in the CP decomposition, resulting in a reduction in the factor matrix that needs to be updated in the posterior, which makes the present invention more computationally efficient than other methods. BRIEF DESCRIPTION OF THE DRAWINGS
[0048] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of this application. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:
[0049] Figure 1 A flowchart of an embodiment of a tensor interpolation decomposition method based on Bayesian learning provided by the present invention;
[0050] Figure 2 yes Figure 1 A model diagram of an embodiment of a probability graph model in the illustrated embodiment;
[0051] Figure 3 yes Figure 1 The error convergence curve diagram of an embodiment of the interpolation model iteration process in the illustrated embodiment. DETAILED DESCRIPTION
[0052] The preferred embodiments of the present invention are described in detail below in conjunction with the accompanying drawings, wherein the accompanying drawings constitute a part of this application and are used together with the embodiments of the present invention to illustrate the principles of the present invention, but are not used to limit the scope of the present invention.
[0053] The following first explains and illustrates the technical terms of this embodiment:
[0054] Tensor CP decomposition: Tensor CP decomposition is a method to decompose a high-order tensor into the sum of multiple "factor tensors" of rank 1. Specifically, for a three-dimensional tensor The goal of CP decomposition is to represent it as the sum of R rank-1 tensors, where R is the rank of the tensor, and the mathematical expression of CP decomposition is:
[0055]
[0056] Among them, ar , b r and c r are factor vectors respectively, and denotes the outer product of vectors. If these factor vectors are further organized into factor matrices A, B, and C, then the CP decomposition can be further expressed as:
[0057]
[0058] Matrix interpolation decomposition: Matrix interpolation decomposition is a special low-rank matrix decomposition method. It approximates the original matrix by selecting several key columns or rows from the original matrix as the skeleton. The core of this method lies in selecting a smaller column set such that the original matrix can be approximately represented by the product of these columns and an interpolation matrix.
[0059] Gibbs sampling: Gibbs sampling is a commonly used statistical algorithm and belongs to a type of Markov chain Monte Carlo method. It is particularly suitable for sampling from complex multivariate probability distributions. Its core idea is to gradually approximate the target distribution by continuously updating each variable to conform to the conditional probability distribution given other variables.
[0060] Embodiment 1
[0061] Please refer to Figure 1 , a tensor interpolation decomposition method based on Bayesian learning in this embodiment specifically includes the following steps:
[0062] Step S1: Organize the input data into a multi-order tensor according to the set features, expand the multi-order tensor into a matrix in one dimension, cluster the matrix, and extract a certain number of column vectors from different clustering categories to form a skeleton matrix;
[0063] Step S2: Regard the skeleton matrix as a factor matrix in the CP decomposition, and regard the remaining factor matrices as weight matrices; use Bayesian learning to perform probability modeling on the tensor interpolation decomposition and add a prior to the tensor;
[0064] Step S3: Set a prior for the weight matrix, draw a probabilistic graphical model according to the weight matrix prior, and derive the posterior distribution of the model based on the probabilistic graphical model;
[0065] Step S4: Use the Gibbs sampling algorithm to perform posterior approximation calculation of the weight matrix to obtain a tensor approximation;
[0066] Step S5: Calculate the model error value based on the tensor approximation. If the model error value is less than the set threshold, output the weight matrix; otherwise, iterate step S4.
[0067] This embodiment introduces a clustering strategy for skeleton matrix selection. Specifically, the data is clustered into certain categories, and some elements in these categories will be selected as the interpolation factors of the model. In addition, the weight matrix is solved using the Bayesian learning framework. In the probability model, the prior is used to restrict the value distribution in the weight matrix. Finally, Gibbs sampling is used for posterior parameter update. The advantage of the tensor decomposition method based on Bayesian learning is that it can handle data incompleteness and noise while providing uncertainty quantification of parameter estimation, which is of great significance for decision-making and model selection. The concept of matrix interpolation decomposition is extended to high-order tensors, retaining the characteristics of interpolation. The tensor is represented as a subset of the data itself and other weight matrices learned from the data, enabling the analysis of complex multi-dimensional tensors originating from real-world data and enhancing the interpretability of the decomposition.
[0068] Specifically, step S1 is as follows:
[0069] Organize the input data into a multi-order tensor according to certain characteristics, and expand the tensor into a matrix according to mode m;
[0070] Set the number of clusters. After clustering the matrix, extract a set number of column vectors from different cluster categories. The set number is the CP rank of the tensor, and the column vectors are the m-mode fibers of the tensor. Obtain the index set of the column vectors in the matrix;
[0071] According to the index set, extract the column vectors from the matrix and organize them into a skeleton matrix.
[0072] Organize the given data into an M-order tensor according to certain characteristics Let the tensor have a dimension size of I1×…×I M , for example, sensor data collected over a certain number of days can be organized into a 3-order tensor according to the sensor ID, number of days, and sampling time. When M = 2, the tensor interpolation decomposition method degenerates into matrix interpolation decomposition.
[0073] Based on the principle that elements within the same category after clustering are more likely to represent other elements within the category, this embodiment introduces a clustering strategy to improve the interpretability of the skeleton matrix, specifically including: expanding the tensor into a matrix according to mode m Specify the number of clusters C, and cluster the matrix . The clustering method in this step is not restricted and can be flexibly selected according to the characteristics of the data itself, as long as the requirements of unsupervised learning are met. After clustering, select a total of R column vectors from different cluster categories. R is the CP rank of the tensor. In this step, the extracted column vectors are also called the m-mode fibers of the tensor. Let the extracted column vectors be in The index set is col. According to the index col, these column vectors are extracted from the matrix and organized into the skeleton matrix A. Specifically, A can be expressed as:
[0074]
[0075] Among them, [col] represents extracting the column vectors with indices in col from the matrix and organizing them into a matrix in descending order. In this step, the size of the obtained skeleton matrix A is I m ×R.
[0076] In step S2, the skeleton matrix is regarded as one of the factor matrices in CP decomposition, and the remaining factor matrices are regarded as weight matrices. Specifically:
[0077] When decomposing the M-order tensor based on the CP decomposition form, there are M factor matrices. One of the factor matrices is the skeleton matrix, and the remaining M - 1 factor matrices are the weight matrices. The M-order tensor is expressed as:
[0078]
[0079] Among them, represents the tensor unfolding according to mode m, A represents the skeleton matrix, ⊙ represents the Kronecker product, and W (n) represents the nth weight matrix, and N is the number of weight matrices.
[0080] Based on CP decomposition, the M-order tensor generates M factor matrices after CP decomposition. In this embodiment, the skeleton matrix A is one of the fixed factor matrices, and the remaining M - 1 factor matrices are regarded as weight matrices. Let N be M - 1.
[0081] For the sake of simplicity in representation, assuming that the tensor is always unfolded according to a certain mode, the interpolation decomposition of the tensor based on the factor matrices is expressed as:
[0082]
[0083] Among them, represents the tensor, represents the Kruskal operator, A represents the skeleton matrix, and W (n) represents the nth weight matrix, and N is the number of weight matrices.
[0084] In step S2, Bayesian learning is used to perform probabilistic modeling on the tensor interpolation decomposition and add a prior to the tensor. Specifically:
[0085] In this embodiment, Bayesian learning is used to model the tensor interpolation decomposition for the tensor Set the prior of the normal distribution. In addition, for the sake of simplicity in representation, let the tensor always expand according to Pattern 1 and select the skeleton matrix, and set the normal distribution prior with mean and variance for the said tensor:
[0086]
[0087] where, W (n) represents the nth weight matrix, N is the number of weight matrices, and is the set of indices i = i1, i2,...i N and σ is the variance.
[0088] The said step S2 further includes adding an inverse gamma distribution prior to the said variance:
[0089]
[0090] where, σ is the variance, and c0 and d0 are hyperparameters, generally 10 -6 .
[0091] To overcome the influence brought by the noise in the real data and uniformly put the model into the Bayesian framework, an inverse gamma distribution prior is also added to the variance σ. By modeling the noise variance σ, the model can not only provide the point estimate of the weight matrix, but also provide the uncertainty estimate of the weight matrix.
[0092] In the said step S3, setting the prior for the said weight matrix specifically is:
[0093] Following the value limit requirement of the weight matrix in the interpolation decomposition, set the truncated normal distribution as the prior for the said weight matrix:
[0094] In the matrix interpolation decomposition, the value of the weight matrix is restricted within -1 to 1. To meet this numerical limit, the weight matrix W (n) is set with the prior of the truncated normal distribution,
[0095] Set the prior of the truncated normal distribution with the range from -1 to 1 for the said weight matrix:
[0096]
[0097] where, W (n) represents the nth weight matrix, represents the generalized truncated normal distribution, μ w is the mean, τ w is the precision, and a and b are the value ranges of the generalized truncated normal distribution, which are -1 and 1 respectively in this embodiment.
[0098] In the said step S3, deriving the posterior distribution of the model based on the said probabilistic graphical model specifically is:
[0099] According to the above prior model, a probabilistic graphical model can be obtained as Figure 2 shown. Finally, through the conjugate relationship, the posterior distributions of the weight matrix and the noise variance can be obtained based on the probabilistic graphical model.
[0100] According to the probabilistic graphical model, the posterior distribution of the weight matrix is:
[0101]
[0102] where denotes fixing the r-th column in the n-th weight matrix, :r represents the r-th column in the weight matrix, the other columns in the same matrix, A represents the skeleton matrix, W (-n) represents all the other weight matrices except the fixed n-th weight matrix, denotes the tensor, μ w is the mean of the distribution, τ w is the precision, σ is the variance, denotes the generalized truncated normal distribution, a and b are the value ranges of the truncated normal distribution, denotes the mean parameter, denotes the precision parameter;
[0103] The update rules for the mean parameter and the precision parameter are:
[0104]
[0105] where ⊙ represents the Kronecker product, denotes the Khatri-Rao product after removing the k-th weight matrix, Ω represents the set of all observed element indices in the tensor, <,> represents the generalized inner product, i m r represents the vector index in the m-th mode, and T represents the transpose of the matrix;
[0106] According to the probabilistic graph, the posterior distribution of the noise variance is:
[0107]
[0108] where and are the shape parameter and the scale parameter of the inverse gamma distribution respectively;
[0109] The update rule for the posterior parameter is:
[0110]
[0111] where |Ω| represents the number of all elements in the tensor, is the Kruskal operator, and is the model error with the Frobenius norm. c0 and d0 are respectively and initial values.
[0112] The specific steps of step S4 are as follows:
[0113] According to the posterior distribution of the model and the posterior parameter update rule, based on the Gibbs sampling algorithm, iterative calculations are performed to obtain the approximate value of the tensor.
[0114] According to the weight matrix posterior parameter update rule and the specific posterior distribution given above, the Gibbs sampling algorithm is used to perform iterative calculations to obtain the posterior distribution value of the variable. The weight matrix represents the importance degree of the components in the skeleton matrix, and the noise variance is the uncertainty estimation information.
[0115] The specific steps of step S5 are as follows:
[0116] Calculate the difference between the tensor and the approximate value of the tensor as the model error. If the model error is greater than the set threshold, the model has not converged. Repeat step S4 through Gibbs sampling. If the model error is not greater than the set threshold, the model has converged, and output the weight matrix as the interpolation information of the skeleton matrix.
[0117] Through the formula Calculate the error between the model output tensor and the original tensor. If the model error is greater than the set threshold, the model has not converged and needs to repeat step S4 through Gibbs sampling. If the model error is less than the set threshold, the model has converged, and the output weight matrix represents the interpolation information of the skeleton matrix.
[0118] Specifically, Figure 3 is the model error convergence curve graph in the iterative process of Gibbs sampling of the interpolation model in this embodiment. The model error is calculated by the formula obtained, Figure 3 The horizontal axis in is the number of iterations, and the vertical axis is the error. It can be seen from the figure that the convergence speed of the interpolation model error in this embodiment is very fast.
[0119] Embodiment 2
[0120] This embodiment provides a tensor interpolation decomposition device based on Bayesian learning, including a memory and a processor. A computer program is stored on the memory, and when the computer program is executed by the processor, it implements the tensor interpolation decomposition method based on Bayesian learning described in Embodiment 1.
[0121] The tensor interpolation decomposition device based on Bayesian learning provided in this embodiment is used to implement the tensor interpolation decomposition method based on Bayesian learning. Therefore, the technical effects possessed by the tensor interpolation decomposition method based on Bayesian learning are also possessed by the tensor interpolation decomposition device based on Bayesian learning, which will not be elaborated here.
[0122] As mentioned above, the above are only the preferred specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed by the present invention should be covered by the present invention.
Claims
1. A tensor interpolation decomposition method based on Bayesian learning, characterized in that It includes the following steps: Step S1: Organize the input data into a multi-order tensor according to the set features, expand the multi-order tensor into a matrix in one dimension, cluster the matrix, and extract a certain number of column vectors from different clustering categories to organize into a skeleton matrix; Step S2: Regard the skeleton matrix as a factor matrix in CP decomposition, and regard the remaining factor matrices as weight matrices; use Bayesian learning to perform probability modeling on tensor interpolation decomposition and add a prior for the tensor; Step S3: Set a prior for the weight matrix, draw a probabilistic graphical model according to the weight matrix prior, and derive the posterior distribution of the model based on the probabilistic graphical model; Step S4: Use the Gibbs sampling algorithm to perform posterior approximation calculation of the weight matrix to obtain a tensor approximation; Step S5: Calculate the model error value based on the tensor approximation. If the model error value is less than the set threshold, output the weight matrix; otherwise, iterate step S4.
2. The tensor interpolation decomposition method based on Bayesian learning according to claim 1, wherein The specific content of step S1 is: Organize the input data into a multi-order tensor according to certain features, and expand the tensor into a matrix in a set mode; Set the number of clusters. After clustering the matrix, extract the set total number of column vectors from different clustering categories. The set total number is the CP rank of the tensor, and the column vectors are the set mode fibers of the tensor, and obtain the index set of the column vectors in the matrix; According to the index set, extract the column vectors from the matrix and organize them into a skeleton matrix.
3. The tensor interpolation decomposition method based on Bayesian learning according to claim 1, characterized in that In step S2, regarding the skeleton matrix as a factor matrix in CP decomposition and the remaining factor matrices as weight matrices, specifically: When decomposing an M-order tensor based on the CP decomposition form, there are M factor matrices. One of the factor matrices is the skeleton matrix, and the remaining M - 1 factor matrices are the weight matrices. The M-order tensor is expressed as: Among them, A represents the skeleton matrix, ⊙ represents the Kronecker product, and W (n) represents the nth weight matrix, and N is the number of weight matrices.
4. The tensor interpolation decomposition method based on Bayesian learning according to claim 1, wherein In step S2, using Bayesian learning to perform probability modeling on tensor interpolation decomposition and adding a prior for the tensor, specifically: Use Bayesian learning to perform probability modeling on tensor interpolation decomposition and perform interpolation decomposition on the tensor based on the factor matrix; Among them, represents the said tensor, represents the Kruskal operator, A represents the skeleton matrix, W (n) represents the nth weight matrix, and N is the number of weight matrices; Set a normal distribution prior with a mean and variance for the tensor.
5. The tensor interpolation decomposition method based on Bayesian learning according to claim 4, characterized in that Step S2 further includes adding an inverse gamma distribution prior for the variance.
6. The tensor interpolation decomposition method based on Bayesian learning according to claim 1, characterized in that In step S3, setting a prior for the weight matrix, specifically: Follow the value limit requirements of the weight matrix in interpolation decomposition and set a truncated normal distribution as the prior for the weight matrix.
7. The tensor interpolation decomposition method based on Bayesian learning according to claim 1, wherein In step S3, deriving the posterior distribution of the model based on the probabilistic graphical model, specifically: Obtain the posterior distributions of the weight matrix and the noise variance based on the probabilistic graphical model; According to the probabilistic graphical model, the posterior distribution of the weight matrix is: Among them, represents fixing the r-th column in the n-th weight matrix, where r represents the r-th column in the weight matrix, other columns in the same matrix, A represents the skeleton matrix, W (-n) represents all the remaining weight matrices except the fixed n-th weight matrix, represents the said tensor, μ w is the mean of the distribution, τ w is the precision, σ is the variance, represents the generalized truncated normal distribution, a and b are the value ranges of the truncated normal distribution, represents the mean parameter, represents the precision parameter; The update rules for the mean parameter and the precision parameter are: where, ⊙ represents the Kronecker product, represents the Khatri-Rao product after removing the k-th weight matrix, Ω represents the set of all observed element indices in the tensor, <, > represents the generalized inner product, i m r represents the vector index in the m-th mode, and T represents the transpose of a matrix; According to the probabilistic graph, the posterior distribution of the noise variance is: wherein, and are the shape parameter and the scale parameter of the inverse gamma distribution, respectively; The update rule for the posterior parameter is: where \(|\Omega|\) represents the number of all elements in the tensor, is the Kruskal operator, and is the model error with the Frobenius norm.
8. The tensor interpolation decomposition method based on Bayesian learning according to claim 1, wherein The specific content of step S4 is: According to the posterior distribution of the model and the posterior parameter update rule, based on the Gibbs sampling algorithm, perform iterative calculations to obtain the approximation of the tensor.
9. The tensor interpolation decomposition method based on Bayesian learning according to claim 1, characterized in that The specific content of step S5 is: Calculate the difference between the tensor and the tensor approximation as the model error. If the model error is greater than the set threshold, the model has not converged. Repeat step S4 through Gibbs sampling. If the model error is not greater than the set threshold, the model has converged, and output the weight matrix as the interpolation information of the skeleton matrix.
10. A tensor interpolation decomposition device based on Bayesian learning, characterized in that, It includes a memory and a processor. A computer program is stored on the memory. When the computer program is executed by the processor, it implements the Bayesian learning-based tensor interpolation decomposition method according to any one of claims 1-9.