Bayesian Tensor Completion Algorithm Based on Complex Noise

Abstract

The present invention provides a Bayesian tensor completion algorithm based on complex noise. For target data with missing values and complex noise, the target data is represented as a tensor, which is the sum of a tensor estimated value and noise. The CP decomposition is used to extract the low-rank information of the tensor, and Gibbs sampling is performed in combination with the framework of CP decomposition and Bayesian method. The tensor estimated value is obtained through iteration, and then the target data is simultaneously completed and denoised based on the tensor estimated value. Since the CP decomposition is used to fully exploit the low-rank information of the tensor, and the observed tensor information is fully utilized, and iterative sampling is performed, the completion algorithm can also achieve good completion and denoising for outliers and complex noise. It is a robust and effective tensor completion algorithm. Compared with the existing completion methods in the prior art, the completion algorithm of the present invention can obtain a more accurate tensor estimated value, so as to achieve more accurate completion and denoising of the target data.

Description

Technical Field

[0001] The present invention belongs to the technical field of tensor completion for data with noise, and particularly relates to a Bayesian tensor completion algorithm based on complex noise. Background Art

[0002] In the big data era, the data generated during the operation of human society is becoming increasingly complex and has a higher dimension. At the same time, many data in the real world often have data missing and noise, and how to process such data has also become one of the important issues in the fields of machine learning, data mining, computer vision, etc. A tensor is a high-dimensional array, as a generalization of vectors and matrices, which can be used to represent multi-dimensional data with complex internal structures. A tensor is a natural expression form of high-dimensional data in the real world. Therefore, the analysis of tensors has become an important tool in multi-dimensional data analysis and has applications in many fields. Aiming at the problems of data missing and noise in multi-dimensional data, the problems of tensor completion and denoising have become an active research field.

[0003] Most tensor completion models assume that there is no noise in the non-missing data (Liu et al., 2012), or only contain Gaussian noise (Chen et al., 2019). However, in the real world, there are not only Gaussian noises, but also other non-Gaussian noises including impulse noises. Therefore, some models have been proposed to adapt to non-Gaussian noises (Meng et al., 2015). The L1 norm and the L2 norm are used to depict the Laplace and Gaussian distributions. However, the noises in the real world are usually not a specific kind of noise, and the mixture Gaussian distribution can be used to describe a wider range of distributions, and the Laplace distribution and the Gaussian distribution can also be regarded as special cases of the mixture Gaussian distribution.

[0004] Chen et al. (2016) proposed an EM algorithm model to solve the problem of mixture Gaussian noise under the condition of tensor missing, but it may cause overfitting problems. Luo et al. (2017) proposed a Bayesian framework to solve the problem of mixture Gaussian noise of tensors, but Luo et al. (2017) did not consider the problem of missing data. Zhao et al. (2015c) assumed that the observed values of tensors are the sum of a low-rank tensor, outliers and Gaussian noise, and it only has good performance when the tensor data is mixed with outliers and small Gaussian noises, and it cannot handle complex noise problems. Summary of the Invention

[0005] The present invention is made to solve the above problems, and the purpose is to provide a tensor completion algorithm that can simultaneously perform tensor completion and denoising and can provide more accurate estimation results. The present invention adopts the following technical solutions:

[0006] The present invention provides a Bayesian tensor completion algorithm based on complex noise, which is characterized by including the following steps:

[0007] Step S1: Represent the target data as a tensor, where the observed value of the tensor is expressed as the sum of the tensor estimate and the noise;

[0008] Step S2: Decompose the tensor using CP decomposition to obtain multiple factor matrices, and calculate the outer product of the multiple factor matrices as the tensor estimate;

[0009] Step S3: Introduce hidden parameters and express the observed tensor information in the tensor as the likelihood function of a mixture Gaussian distribution;

[0010] Step S4: Set the prior distribution of each parameter used for Gibbs sampling as a conjugate prior distribution, and calculate the conditional posterior distribution of each parameter based on the prior distribution of each parameter;

[0011] Step S5: Use the Gibbs sampling method to sample from the conditional posterior distribution of each parameter in turn to obtain the joint posterior distribution of the multiple parameters;

[0012] Step S6: Determine whether the predetermined number of iterations is reached. If the determination is no, repeat Step S5;

[0013] Step S7: When the determination in Step S6 is yes, output the final tensor estimate, and interpolate the missing values of the target data based on the final tensor estimate, so as to complete and denoise the target data.

[0014] The Bayesian tensor completion algorithm based on complex noise provided by the present invention may further have the following technical feature: in Step S1, the observed value x of the tensor is expressed as:

[0015]

[0016] where is the tensor estimate, ε is the noise, and x has missing values.

[0017] In Step S2, the calculated tensor estimate is:

[0018]

[0019] where are the d-th columns of the factor matrices respectively, and the symbol represents the outer product.

[0020] The elements of the tensor estimate are expressed as:

[0021]

[0022] In the formula, u id , v jd , t kd are the (i, d), (j, d), and (k, d) elements of the factor matrices U, V, and T, respectively.

[0023] The Bayesian tensor completion algorithm based on complex noise provided by the present invention may also have the following technical feature: in step S3, the observed tensor information is expressed as:

[0024]

[0025] In the formula, (i, j, k) ∈ Ω, where Ω is the index set of non-missing values in the tensor observation x, and ∈ ijk obeys a mixture Gaussian model:

[0026]

[0027] In the formula, π n is the probability that the observed data belongs to the nth sub-model, and π n ≥ 0, N is the number of sub-models in the Gaussian mixture model, represents a Gaussian distribution with a mean of μ n and a precision of τ n .

[0028] Therefore, each x ijk obeys a mixture Gaussian distribution:

[0029]

[0030] In the formula,

[0031] In step S3, the likelihood function is expressed as:

[0032]

[0033] In the formula, z ijkn is the introduced binary hidden function, z ijkn ∈ {0, 1} and

[0034] The Bayesian tensor completion algorithm based on complex noise provided by the present invention may also have the following technical feature: in step S4, the parameters include:

[0035] Hyperparameters The prior distribution is set to Gaussian-Wishart(μ0, β0, W0, v0), that is, p(μ (k) , Λ (k) | -) = N(μ (k) |μ (0) , (β0Λ (k) ) -1 ) × Wishart(Λ (k) |W0, μ0);

[0036] The row vectors of the factor matrices U, V, and T. Among them, the row vector u i ~N(μ (1) , (Λ (1) ) -1 ), the row vector v j ~N(μ (2) , (Λ (2) ) -1 ), and the row vector t k ~N(μ (3) , (Λ (3) ) -1 );

[0037] The parameters μ n , τ n represent the mean and precision respectively. The prior distribution is μ n , τ n ~N(μ n |μ0, (θ0τ n ) -1 ) Gam(τ n |c0, d0);

[0038] The parameter π. The prior distribution is the Dirichlet distribution of the parameter α0, π~Dir(π|α0), where α0 = (α 01 , α 02 ,..., α 0N ); and

[0039] The hidden parameter z ijk . The prior distribution is the multinomial distribution of the parameter π, z ijk ~Multinomial(z ijk |π).

[0040] The Bayesian tensor completion algorithm based on complex noise provided by the present invention may also have the following technical features. Among them, step S5 includes the following sub-steps:

[0041] Step S5-1, when k = 1, sample in the conditional posterior distribution of the hyperparameters;

[0042] Step S5-2, sample in the conditional posterior distribution of the row vectors of the factor matrix. Specifically, sample the row vectors of the factor matrix one by one until the entire factor matrix is updated;

[0043] Step S5-3, when k = 2, sample in the conditional posterior distribution of the hyperparameters;

[0044] Step S5-4, sample in the conditional posterior distribution of the row vectors of the factor matrix;

[0045] Step S5-5, when k = 3, sample in the conditional posterior distribution of the hyperparameters;

[0046] Step S5-6, sample in the conditional posterior distribution of the row vectors of the factor matrix;

[0047] Step S5-7, sample in the conditional posterior distribution of the parameters;

[0048] Step S5-8, sample in the conditional posterior distribution of the parameters;

[0049] Step S5-9, sample in the conditional posterior distribution of the implicit function.

[0050] For the Bayesian tensor completion algorithm based on complex noise provided by the present invention, it may further have the following technical feature. In step S7, the final tensor estimation value is the average of the M tensor estimation values in M iterations.

[0051] Functions and Effects of the Invention

[0052] For the Bayesian tensor completion algorithm based on complex noise according to the present invention, for target data with missing values and complex noise, by representing the target data as a tensor, which is the sum of a tensor estimation value and noise, and using CP decomposition to extract the low-rank information of the tensor, combining CP decomposition and the framework of the Bayesian method for Gibbs sampling, the tensor estimation value is obtained through iteration, and then the target data is simultaneously completed and denoised based on the tensor estimation value. Since CP decomposition is used to fully mine the low-rank information of the tensor, and the observed tensor information is fully utilized, and iterative sampling is performed, this completion algorithm can also achieve good completion and denoising for outliers and complex noise. It is a robust and effective tensor completion algorithm. Compared with the existing completion methods in the prior art, the completion algorithm of the present invention can obtain a more accurate tensor estimation value, thereby achieving more accurate completion and denoising of the target data. Description of the Drawings

[0053] Figure 1It is the flowchart of the Bayesian tensor completion algorithm based on complex noise in the embodiment of the present invention;

[0054] Figure 2 It is the schematic diagram of CP decomposition in the embodiment of the present invention;

[0055] Figure 3 It is the schematic diagram of the Bayesian network in the embodiment of the present invention;

[0056] Figure 4 It is the schematic diagram of the Gibbs sampling process in the embodiment of the present invention;

[0057] Figure 5 It is the comparison chart of the completion and denoising effects of the completion method in the embodiment of the present invention and the method in the prior art under the addition of mixed noise to the multi-spectral image. Detailed implementation manners

[0058] In order to make the technical means, creative features, achieved purposes and effects of the present invention easy to understand, the Bayesian tensor completion algorithm based on complex noise of the present invention will be specifically described below in conjunction with embodiments and drawings.

[0059] <Embodiment>

[0060] This embodiment provides a Bayesian tensor completion algorithm based on complex noise (MoG BTC-CP), which is used to represent the target data with missing measurement values as a corresponding tensor, and at the same time complete and denoise the tensor to obtain a more accurate tensor estimation value, and interpolate the target data based on the tensor estimation value, so as to realize the completion and denoising of the target data. Figure 2 It is the schematic diagram of the Bayesian network. As Figure 2 shown, the completion method of this embodiment combines the frameworks of CP decomposition and Bayesian methods for Gibbs sampling to output the estimated value of the tensor.

[0061] In this embodiment, the completion algorithm of this embodiment is implemented in Matlab for experimental analysis. That is, the completion algorithm of this embodiment runs on the computing server as a pre-written program.

[0062] Figure 1 It is the flowchart of the Bayesian tensor completion algorithm based on complex noise in this embodiment.

[0063] As Figure 1 shown, in this embodiment, the Bayesian tensor completion algorithm based on complex noise specifically includes the following steps:

[0064] Step S1, represent the target data to be completed as a tensor, and the observed value of the tensor is represented as the sum of the tensor estimation value and the noise.

[0065] In this embodiment, the target data used in the experiment is multi-spectral images (Jelly beans, Paints, Flowers, Egyptian statue, Chart and Stuffed toy, Beers, Glass tiles, and Strawberries respectively). Each series of multi-spectral images includes 31 bands, and each band is a grayscale image with a size of 512×512 pixels. Each multi-spectral image can be regarded as a three-dimensional tensor x of 512×512×31. Before tensor completion, the multi-spectral images are preprocessed as follows: the multi-spectral images are resized to half size (i.e., 256×256×31), then the resized multi-spectral images are normalized and converted into data on [0, 1]. Then, mixed noise is added to the normalized multi-spectral images: 20% of the data in the image is set as missing values, 20% of the data is mixed with noise from a uniform distribution [-5, 5], 20% of the data is mixed with noise from a Gaussian distribution N(0, 0.2), and 40% of the data is mixed with noise from a Gaussian distribution N(0, 0.01). Through the above preprocessing, the obtained multi-spectral images are images with missing values and mixed noise, which is convenient for verifying the method of this embodiment of the present invention.

[0066] According to the mixture Gaussian model, the tensor observation value x corresponding to such a multi-spectral image can be expressed as the sum of a tensor estimate value and noise:

[0067]

[0068] In the above formula, x has missing values.

[0069] Step S2: Use CP decomposition to decompose the tensor to obtain multiple factor matrices, and calculate the outer product of the multiple factor matrices as the estimated value of the tensor.

[0070] Figure 2 is a schematic diagram of CP decomposition in this embodiment.

[0071] CP decomposition is used to extract the low-rank structure of the tensor. Its basic principle is as Figure 2 shown. The multi-dimensional tensor is decomposed into multiple one-dimensional factor matrices. In this embodiment, CP decomposition is applied to the tensor estimated value to calculate the outer product of the multiple factor matrices as the estimated value of the tensor:

[0072]

[0073] In the above formula, are respectively the d-th columns of the factor matrices , and the symbol represents the outer product.

[0074] Equivalently, the tensor estimate elements can be expressed as:

[0075]

[0076] where u id , v jd , t kd are the (i, d), (j, d), (k, d) elements of the factor matrices U, V, T, respectively.

[0077] Step S3, introduce hidden parameters and express the observed tensor information in the tensor as the likelihood function of a mixture Gaussian distribution.

[0078] In this embodiment, define Ω as the set of indicators of non-missing values in the tensor observation value x. Therefore, the non-missing observation values ((i, j, k) ∈ Ω), that is, the observed tensor information, can be expressed as:

[0079]

[0080] In this embodiment, assume that ∈ ijk obeys a mixture Gaussian model:

[0081]

[0082] In the above formula, π n is the probability that the observed data belongs to the nth sub-model, and π n ≥ 0, N is the number of sub-models in the Gaussian mixture model, represents a Gaussian distribution with mean μ n and precision τ n .

[0083] Therefore, each x ijk obeys a mixture Gaussian distribution:

[0084]

[0085] In the above formula,

[0086] The likelihood function of x can be defined as:

[0087]

[0088] where π = (π1, π2,..., π N ), λ = (λ1, λ2,..., λ N ), τ1 = (τ1, τ2,..., τ N ).

[0089] Introduce a binary hidden parameter z ijkn , z ijkn ∈ {0, 1} and Define z ijk = (z ijk1 , z ijk2 ,..., z ijkn ), then z ijk follows a multinomial distribution with probability π n (n = 1,..., N). Let Then the (conditional) likelihood function can be expressed as:

[0090]

[0091] Define the elements of the indicator tensor as b ijk . When (i, j, k) ∈ Ω, b ijk = 1; otherwise, b ijk = 0. Therefore, the likelihood function can be reformulated as:

[0092]

[0093] Step S4, set the prior distribution of each parameter used for Gibbs sampling to a conjugate prior distribution, and calculate the conditional posterior distribution of each parameter based on the prior distribution of each parameter.

[0094] In this embodiment, to calculate the explicit expression of the conditional posterior distribution of the parameters for Gibbs sampling, the parameters are all set to follow conjugate prior distributions. The parameters include the row vectors of the above factor matrices U, V, T, the hyperparameters μ (k) , Λ (k) , the mean μ n , the precision τ n , the parameter π, and the hidden function z ijk .

[0095] Among them, for the hyperparameters μ (k) , Λ (k) :

[0096] The prior distribution of the hyperparameter is set to a conjugate prior distribution: Gaussian-Wishart(μ0, β0, W0, v0)

[0097] Taking k = 1 as an example, the conditional posterior distribution of μ (1) , Λ (1) is:

[0098]

[0099] In the above formula:

[0100]

[0101]

[0102]

[0103]

[0104] Variable and S (1) are as follows:

[0105]

[0106]

[0107] According to the above formula, similarly, the hyperparameters μ (k) , Λ (k) 's conditional posterior distribution can be obtained in the cases of k = 2, 3.

[0108] For the row vectors of the factor matrices U, V, T:

[0109] The row vector u i of the factor matrix U follows a multivariate normal distribution, u i ~N(μ (1) , (Λ (1) ) -1 ), the row vector v j of the factor matrix V ~ N(μ (2) , (Λ (2) ) -1 ), and the row vector t k of the factor matrix T ~ N(μ (3) , (Λ (3) ) -1 ).

[0110] Among them, taking the factor matrix U as an example, the row vector of the factor matrix is defined as u i (i = 1,..., I), and μ = (μ1, μ2,..., μ N ), and its likelihood function can be written as:

[0111]

[0112] In the above formula, represents the Hadamard product.

[0113] The conditional posterior distribution of the row vector u i of the factor matrix U can be written as:

[0114]

[0115] In the formula,

[0116]

[0117] The factor matrices V and T can be deduced in the same way and will not be elaborated here.

[0118] For the parameter μ n , τ n :

[0119] The parameter μ n , τ n ~ N(μ n |μ0, (θ0τ n )) -1 ) Gam(τ n |c0, d0), and its likelihood function can be written as:

[0120]

[0121] The conditional posterior distribution of the parameters μ n , τ n can be written as:

[0122]

[0123] In the above formula:

[0124]

[0125]

[0126]

[0127]

[0128] For the parameter π:

[0129] The prior distribution of the parameter π is the Dirichlet distribution of the parameter α0, π ~ Dir(π|α0), where α0 = (α 01 , α 02 ,..., α 0N ).

[0130] The conditional posterior distribution of the parameter π is the Dirichlet distribution of the hyperparameter α, where α = (α1, α2,..., α N ): In the formula,

[0131] For the latent parameter z ijk :

[0132] The prior distribution of the latent parameter z ijk is the multinomial distribution of the parameter π, zijk ~Multinomial(z ijk |π), its likelihood function can be written as:

[0133]

[0134] z ijkn The prior distribution of is the multinomial distribution with parameter π:

[0135]

[0136] z ijk The conditional posterior distribution of follows the multinomial distribution with parameter r ijk where r ijk =(r ijk1 , r ijk2 ,..., r ijkN ):

[0137]

[0138] In the above formula, and there is:

[0139]

[0140] Step S5, using the Gibbs sampling method to sample from the conditional posterior distributions of each parameter in turn to obtain the joint posterior distribution of multiple parameters.

[0141] Figure 4 is the schematic diagram of the Gibbs sampling process in this embodiment.

[0142] As Figure 4 shown, in this embodiment, step S5 specifically includes the following sub-steps:

[0143] Step S5-1, when k = 1, sample in the conditional posterior distribution of the hyperparameters μ (k) , Λ (k) ;

[0144] Step S5-2, sample in the conditional posterior distribution of the row vectors of the factor matrix U. Specifically, sample u i (i = 1,..., I) one by one until the factor matrix U is fully updated;

[0145] Step S5-3, when k = 2, sample in the conditional posterior distribution of the hyperparameters μ (k) , Λ (k) ;

[0146] Step S5-4, sample in the conditional posterior distribution of the row vectors of the factor matrix V. Specifically, as in step S5-2, it will not be elaborated here;

[0147] Step S5-5, when k = 3, sample from the conditional posterior distribution of the hyperparameters μ (k) , Λ (k) .

[0148] Step S5-6, sample from the conditional posterior distribution of the row vectors of the factor matrix T;

[0149] Step S5-7, sample from the conditional posterior distribution of the parameters μ n , τ n .

[0150] Step S5-8, sample from the conditional posterior distribution of the parameter π;

[0151] Step S5-9, sample from the conditional posterior distribution of the latent function z ijk .

[0152] Step S6, determine whether the predetermined number of iterations is reached. When the determination is no, repeat Step S5. When the determination is yes, proceed to Step S7.

[0153] Step S7, when the determination in Step S6 is yes, stop the iteration, output the final tensor estimate value, and interpolate the missing values of the target data based on the final tensor estimate value, thereby completing the complementation and denoising of the target data.

[0154] As Figure 4 shown, in this embodiment, the number of iterations is set to M times. When the number of iterations reaches M times, the iteration stops, and the estimated values of the tensors in the last M iterations are output (i.e., the CP product of the factor matrix estimate values), and the final tensor estimate value is the average of the M tensor estimate values obtained in the M iterations.

[0155] Figure 5 is a comparison graph of the complementation and denoising effects of the complementation method in this embodiment and the methods in the prior art under the addition of mixed noise to the multi-spectral image. Among them, column (a) shows multiple original images; column (b) shows the image after mixed noise; column (c) shows the processing result of the BGCP method in the prior art; column (d) shows the processing result of the MoG GWLRTF-CP method in the prior art; column (e) shows the processing result of the BRTF method in the prior art; column (f) shows the processing result of the complementation method (MoG BTCCP) in this embodiment.

[0156] As Figure 5 shown, from the processing results of the multi-spectral image, the complementation and denoising effects of the method in this embodiment are better than those of other existing methods both in the numerical evaluation system and in terms of image vision.

[0157] In addition, in this embodiment, the parts not described in detail are common general knowledge in the art.

[0158] Functions and effects of the embodiment

[0159] According to the Bayesian tensor completion algorithm based on complex noise provided in this embodiment, for target data with missing values and complex noise, by representing the target data as a tensor, which is the sum of a tensor estimate value and noise, and using CP decomposition to extract the low-rank information of the tensor, combining the framework of CP decomposition and the Bayesian method for Gibbs sampling, the tensor estimate value is obtained through iteration, and then the target data is simultaneously completed and denoised based on the tensor estimate value. Since CP decomposition is used to fully mine the low-rank information of the tensor, and the observed tensor information is fully utilized, and iterative sampling is performed, this completion algorithm can also achieve good completion and denoising for outliers and complex noise, and it is a robust and effective tensor completion algorithm. Compared with the completion methods in the prior art, the completion algorithm of this embodiment can obtain a more accurate tensor estimate value, so as to achieve more accurate completion and denoising of the target data.

[0160] The above embodiments are only used to illustrate the specific implementation manners of the present invention, and the present invention is not limited to the description scope of the above embodiments.

[0161] In the above embodiment, the Bayesian tensor completion algorithm based on complex noise is applied to a multi-spectral image with added mixed noise for simultaneously completing and denoising the data of the image. In fact, the completion algorithm of the embodiment can also be applied to other data with missing values and noise.

Claims

1. A Bayesian tensor completion algorithm based on complex noise, which is used to simultaneously complete and denoise target data. The target data is a multi-spectral image, and it is characterized in that It includes the following steps: Step S1: Represent the target data as a tensor, and the observed value of this tensor is represented as the sum of a tensor estimate and noise; Step S2: Decompose the tensor using CP decomposition to obtain multiple factor matrices, and calculate the outer product of the multiple factor matrices as the tensor estimate; Step S3: Introduce hidden parameters, and express the observed tensor information in the tensor as the likelihood function of a mixture Gaussian distribution; Step S4: Set the prior distribution of each parameter used for Gibbs sampling as a conjugate prior distribution, and calculate the conditional posterior distribution of each parameter based on the prior distribution of each parameter; Step S5: Use the Gibbs sampling method to sample successively from the conditional posterior distribution of each parameter to obtain the joint posterior distribution of the multiple parameters; Step S6: Determine whether the predetermined number of iterations is reached. If the determination is no, repeat Step S5; Step S7: When the determination in Step S6 is yes, output the final tensor estimate, and interpolate the missing values of the target data based on the final tensor estimate, so as to complete and denoise the target data; Among them, in step S1, the observed value of the tensor is expressed as: In the formula, is the tensor estimation value, and ε is the noise, has missing values, In step S2, the calculated tensor estimate value is as follows: In the formula, are the d-th columns of the factor matrices respectively, and the symbol represents the outer product. The estimated tensor value is represented by elements as follows: where u id , v jd , t kd are the (i, d)-th, (j, d)-th, and (k, d)-th elements of the factor matrices U, V, and T, respectively. In Step S3, the observed tensor information is expressed as: where (i, j, k) ∈ Ω, and Ω is the set of indicators of non-missing values in the tensor observations and ∈ ijk obeys a mixture Gaussian model: where, π n is the probability that the observed data belongs to the n-th sub-model, and π n ≥ 0, N is the number of sub-models in the Gaussian mixture model, represents a Gaussian distribution with a mean of μ n , and a precision of τ n ​ Therefore, each x ijk obeys a Gaussian mixture distribution: In the formula, In Step S3, the likelihood function is expressed as: In the formula, z ijkn is the introduced binary hidden parameter, and z ijkn ∈ {0, 1} and In Step S4, the parameters include: Hyperparameter The prior distribution is set to Gaussian-Wishart(μ0, β0, W0, v0), That is, p(μ (k) , Λ (k) | - ) = N(μ (k) | μ (0) , (β0Λ (k) )) -1 ) × Wishart(Λ (k) | W0, μ0); The row vectors of the factor matrices U, V, and T, where the row vector u of the factor matrix U i ~N(μ (1) , (Λ (1) )) -1 ), the row vector v of the factor matrix V j ~N(μ (2) , (Λ (2) )) -1 ), the row vector t of the factor matrix T k ~N(μ (3) , (Λ (3) )) -1 ); Parameter μ n , τ n , which represent the mean and precision respectively, and the prior distribution is μ n , τ n ~ N(μ n |μ0, (θ0τ n )) -1 ) Gam(τ n |c0, d0); The parameter π, whose prior distribution is the Dirichlet distribution of parameter α0, π ∼ Dir(π|α0), where α0 = (α 01 , α 02 ,..., α 0N ); and Latent parameter z ijk , where the prior distribution is the multinomial distribution of the parameter π, z ijk ~ Multinomial(z ijk |π).

2. The Bayesian tensor completion algorithm based on complex noise according to claim 1, It is characterized in that: Among them, Step S5 includes the following sub-steps: Step S5-1, when k = 1, sample from the conditional posterior distribution of the hyperparameters μ (k) , Λ (k) . Step S5-2, sample in the conditional posterior distribution of the row vectors of the factor matrix U. Specifically, sample each row vector u i (i = 1,..., I) of the factor matrix U one by one until the entire factor matrix U is updated; Step S5-3, when k = 2, sample in the conditional posterior distribution of the hyperparameters μ (k) , Λ (k) ; Step S5-4: Sample in the conditional posterior distribution of the row vectors of the factor matrix V; Step S5-5, when k = 3, sample from the conditional posterior distribution of the hyperparameters μ (k) , Λ (k) under the said conditions; Step S5-6: Sample in the conditional posterior distribution of the row vectors of the factor matrix T; Step S5-7, sample from the posterior distribution of the condition with respect to the parameters μ n , τ n . Step S5-8: Sample in the conditional posterior distribution of the parameter π; Step S5-9, sample in the conditional posterior distribution of the hidden parameter z ijk under the said conditions.

3. The Bayesian tensor completion algorithm based on complex noise according to claim 1, characterized in that: Among them, In Step S7, the final tensor estimate is the average of the M tensor estimates in M iterations.

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