Tucker tensor decomposition-based missing value completion prediction method

Abstract

The invention discloses a Tucker tensor decomposition-based missing value completion prediction method, and relates to the field of second-hand electronic product recovery evaluation result missing value research and application. The method comprises the following steps: 1) removing abnormal data based on a box plot; 2) removing the feature attributes with relatively high feature attribute missing rate; 3) searching feature attributes highly related to the recovery evaluation result value based on a mutual information mode; and 4) constructing a Tucker tensor decomposition completion model based on the training set, verifying the accuracy of the model by the test set, and finally realizing completion prediction of the missing value of the second-hand electronic product recovery evaluation result. The specific construction prediction process is divided into four small steps: initializing a sparse tensor and a factor matrix, calculating a complemented prediction tensor, calculating the difference Loss between a real tensor and the prediction tensor, descending according to a gradient, and simultaneously circulating the last three small steps until convergence. After model training is completed, the model can be directly provided for related employees to use, and therefore efficient recycling is achieved.

Description

Technical Field

[0001] This invention relates to the field of research and application of missing values ​​in the evaluation results of second-hand electronic product recycling, specifically to a missing value completion prediction method based on Tucker tensor decomposition. Background Technology

[0002] The rapid development of technological productivity and the increasingly faster pace of electronic product upgrades have led to a massive generation of secondhand electronic products globally. However, only 20% of these products are recorded and recycled. These secondhand electronic products contain many reusable rare heavy metals and even some strategic resources. However, improper recycling and disposal of these products can generate large amounts of harmful substances, impacting the surrounding soil, groundwater, and air environment, and affecting human health. At the same time, secondhand electronic products also have high recycling value; increasing the recycling rate of secondhand electronic and electrical equipment can significantly improve economic and environmental value. Therefore, there is an urgent need for a method for assessing and predicting the recycling of secondhand electronic products to achieve more efficient recycling.

[0003] In recent years, to achieve accurate assessment and prediction of recycled second-hand electronic products, researchers in this field have proposed mathematical modeling-based approaches to model the correlation between various characteristic factors and assessment results, thereby achieving higher accuracy in predictions. These include smartphone assessment models using fuzzy comprehensive evaluation methods, which improve accuracy and practicality through objective weighting and weighted average operators. Game theory-based models analyze consumer and recycler behavior, establishing a comprehensive static information game theory model to build corresponding assessment and prediction models. Additionally, another group of scholars has proposed time-series-based prediction and assessment models. For example, some researchers have proposed a method integrating convolutional neural networks (CNN), bidirectional long short-term memory networks (BiLSTM), and attention mechanisms (AM) to predict stock market outcomes. Alternatively, variational mode decomposition (VMD) combined with LSTM is used to construct prediction models for various non-ferrous metals. They utilize VMD to first decompose the original sequence into several relatively stable subsequences, and then use LSTM to mine intrinsic dependencies within these subsequences to learn the changing trends of the results. Some researchers also employ classic models from traditional machine learning to achieve assessment and prediction in related fields.

[0004] However, these methods lack sufficient stability to accurately predict evaluation results when faced with the complex value-feature relationships and sparse data relationships in the used electronics field. For example, the performance of the RBF model is affected by the choice of kernel function and parameter settings; in used electronics evaluation prediction, due to a lack of sufficient domain knowledge, it may be difficult to determine the optimal parameter combination. LSTM, as an excellent time series model, can learn the time series features of predicted results to predict future evaluation results; however, used electronics data is relatively sparse, especially in certain models or categories, often requiring a large amount of continuous data to capture the dynamic features of the time series, making it difficult to apply to predicting missing values ​​in used electronics evaluation results. MLP and BPNN, as neural network models, perform well in fitting nonlinear relationships, but the lack of relevant datasets for used electronics recycling makes them difficult to apply to scenarios with sparse data like used electronics.

[0005] In summary, existing methods for predicting missing values ​​in evaluation results have the following problems:

[0006] 1. Traditional mathematical modeling methods, which involve cumbersome steps to establish the relationship between the characteristic attributes of second-hand electronic products and the evaluation results, suffer from low evaluation and prediction accuracy and are difficult to extend.

[0007] 2. Machine learning-based methods and deep learning approaches that learn patterns from large amounts of data struggle to learn sufficient patterns when dealing with sparse scenarios involving secondhand electronic products, thus affecting the accuracy of the final model's predictions. Furthermore, these models remain in a black-box research environment, making it impossible to determine the interaction between the predicted values ​​and feature attributes, thus hindering accurate understanding and improvement of predictions.

[0008] Tensor decomposition completion aims to find a low-rank structure that approximates the original data tensor. Specifically, it involves combining simple, low-dimensional, linearly dependent tensors into a complex, high-dimensional, linearly independent tensor. Tensor decomposition completion is based on the assumption that tensors have a low-rank structure, predicting missing terms by using a minority of the tensor's sampled terms. Some major tensor decomposition methods include CP decomposition and Tucker decomposition. CP decomposition represents a tensor as the sum of multiple rank tensors, while Tucker decomposition decomposes a tensor into a core tensor and the product of multiple factor matrices. This allows for explicit modeling of interactions, ultimately achieving higher prediction accuracy than CP decomposition. Furthermore, Tucker decomposition allows for control of model complexity by adjusting the size and shape of the core tensor, resulting in stronger resistance to overfitting. In addition, compared to models in the machine learning field, Tucker decomposition completion has a clear mathematical foundation, leading to a better understanding of model behavior and performance. It also provides interpretability for predictions by analyzing the components of the decomposed tensor to understand the characteristics and relationships of the data. Summary of the Invention

[0009] To address the limitations and problems of existing methods for predicting missing values ​​in relevant assessment results, this invention proposes a missing value completion prediction method based on Tucker tensor decomposition. The aim is to provide a more accurate and efficient method for predicting missing values ​​in the assessment results of used electronic product recycling, thereby further promoting the development of the used electronic product recycling industry, supporting the development of the recycling industry, and contributing to the realization of carbon neutrality.

[0010] To achieve the above objectives, this invention uses a real dataset of second-hand mobile phone transactions from a certain company as the method for verification. This company is a professional internet high-tech company that buys back electronic products. The dataset contains 1,996 original second-hand mobile phone data samples that are actually usable.

[0011] The missing value completion prediction method based on Tucker tensor decomposition of the present invention adopts the following technical solution steps:

[0012] Step 1: Outlier Removal Based on Boxplots. The secondhand mobile phone dataset exhibits a significant imbalance. Directly applying this data to train the Tucker Tensor Decomposition Completion model may affect the accuracy of the model evaluation due to the uneven distribution of the recycling evaluation results. While normalization techniques may aid in data preprocessing, they cannot completely overcome the challenges posed by the uneven sample distribution. Therefore, the boxplot data removal method, systematically promoted by Tucker, is adopted. The maximum (Max), minimum (Min), lower quartile (QL), and upper quartile (QU) of the boxplot are calculated, yielding the interquartile range (IQR), calculated as IQR = QU - QL. Finally, the lower limit value can be calculated as max{QL - 1.5IQR, Min}, and the upper limit value as min{QU + 1.5IQR, Max}, thus removing outlier data exceeding the upper and lower limits.

[0013] Step Two: Feature Removal with High Missing Attribute Rates. Modeling the relationship between feature attributes and recycling assessment results is crucial for accurately predicting missing values ​​in the recycling assessment of used electronic products. Therefore, it's necessary to remove sample data with significant missing information to avoid affecting the accuracy of the predicted recycling assessment results. To ensure optimal experimental results, maintain the integrity of the training samples, and ensure the rationality of the experiment, samples with a missing attribute information rate exceeding 50% in the original data are selectively removed.

[0014] Step 3: Identifying Feature Attributes Highly Correlated with Recycling Assessment Results Based on Mutual Information. A Tucker tensor model is used to construct a multi-dimensional Tucker tensor representing the relationship between feature attributes and assessment results, aiming for more accurate predictions of recycling assessment results for used electronic products. However, an excessive number of feature attributes in the model can drastically increase its complexity, thus affecting the accuracy of the assessment predictions. Therefore, before formally constructing the Tucker tensor model, a mutual information feature attribute selection strategy is employed. This strategy comprehensively considers the correlation between each feature attribute and the assessment results to select the feature attributes that contribute most to the model. Mutual information (MI) is a measure of statistical independence between two random variables. It is a more general method for measuring nonlinear relationships and can be used to identify relationships between datasets that are not detected by commonly used linear correlation measures. For random variables... The formula for mutual information between T and T is described as follows:

[0015]

[0016] in It is the marginal distribution of the random variable Z. It is the marginal distribution of the random variable T, and It is the joint distribution of (Z, T). This represents the mutual information value between two random variables Z and T. In practical terms, random variable Z represents the characteristic attributes of used electronic products, and random variable T represents the predicted value of the used electronic product recycling assessment to be completed.

[0017] To better select feature attributes based on mutual information values ​​and avoid the influence of different units, the value range of mutual information (MI) is mapped to between 0 and 1, resulting in more representative standardized mutual information (NMI). The final standardized mutual information value is then used to determine the NMI for random variables. The formula for mutual information between T and T is as follows:

[0018]

[0019] in, Represents random variables entropy, Represents random variables entropy, This represents the mutual information value between two random variables Z and T. This represents the standardized mutual information value between two random variables Z and T. In practical terms, random variable Z represents the characteristic attributes of used electronic products, and random variable T represents the predicted value of the used electronic product recycling assessment to be completed.

[0020] Step Four: Before proceeding to Step Four, let's introduce the definition of Tucker decomposition completion. The Tucker tensor can be decomposed into the product of a core tensor and factor matrices along each dimension. Using an n-order tensor... For example, its Tucker decomposition is as follows:

[0021]

[0022] in, Known as the core tensor, This is called the factor matrix. This represents the pattern product of the tensor in the nth dimension.

[0023] After the first three steps, the five feature attributes with the highest standardized mutual information values ​​are retained: ID lock, function options, color, storage capacity, and memory. We define the entire tensor constructed based on the training dataset as a sparse tensor. Define the core tensor. Representing the core interaction of 5 characteristic attributes, The symbol representing the set of real numbers indicates that... It is ( , , , , A set of real tensors of size ). This represents the dimensional length of the interaction with the ID lock's feature attributes. Based on continuous experimentation, it has been found that when... The effect is best when the value is 3, so we choose 3. =3; This represents the length of the dimension interacting with the feature attributes of the function options. Based on continuous experimentation, it has been found that when... The effect is best when the value is 3, so we choose 3. =3; This represents the dimensional length of the interaction with color feature attributes. Based on continuous experimentation, it has been found that when... The effect is best when the value is 3, so choose 3. =3; This represents the length of the dimension that interacts with the storage capacity characteristic attribute. Based on continuous experiments, it has been found that when... The effect is best when the value is 3, so we choose 3. =3; This represents the dimensional length of the interaction with memory feature attributes. Based on continuous experiments, it has been found that when... The effect is best when the value is 3, so choose 3. =3. Define the factor matrix. Represents the ID lock characteristic attribute, The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of ID lock feature attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents the feature attributes of the function options. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of feature attribute values ​​for the function option. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents color characteristic attributes. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of color feature attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents storage capacity characteristics. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of color feature attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents memory characteristics and attributes. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of memory characteristic attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3. Define error disturbance. Representing the difference before and after the loss function, based on continuous experiments, it has been found that when... At this value, the best results are achieved; exceeding or falling below this value will lead to poorer prediction results. Definition The learning rate, representing gradient descent, has been found through continuous experimentation to be... At this value, the best results are achieved; exceeding or falling below this value will lead to poorer prediction results. Definition The contribution of the regularization penalty term is represented by the value of the regularization penalty term. Based on continuous experiments, it has been found that when... At this value, the best results are achieved; exceeding or falling below this value will lead to poorer prediction results. Definition Represents the complete tensor after Tucker tensor decomposition to complete the predicted missing evaluation values. Definition This represents the specific position of a non-zero element in the tensor, therefore it is defined as follows: Representative is sparse tensor A non-zero element at a given position. (Definition) Represents the complete tensor after Tucker decomposition and completion. The element that completes the position. Definition This represents the pattern product of the tensor in the nth dimension, therefore The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in a given dimension. Definition The first element representing the ID lock feature attribute matrix Row vectors of a row; definition The first of the feature attribute matrix representing the functional options Row vectors of a row; definition The first element representing the color feature attribute matrix Row vectors of a row; definition The first of the storage capacity feature attribute matrix Row vectors of a row; definition The first element representing the memory feature attribute matrix The row vector of a row. This represents the Kronecker product of matrices.

[0024] Based on the above definitions, a Tucker tensor decomposition completion prediction model is constructed. The accuracy of the model is verified using a test set, ultimately achieving the completion prediction of missing values ​​in the evaluation results of used electronic product recycling. The specific prediction construction process is simplified into four steps, and the textual explanation can be found below.

[0025] 1. Based on the existing training dataset, initialize a 5-dimensional Tucker sparse tensor. Based on the initialized 5-dimensional Tucker sparse tensor Initialize a 5-dimensional core tensor and the corresponding 5-dimensional factor matrix ;

[0026] 2. Constructing a 5-dimensional Tucker tensor after missing value imputation prediction based on the gradient descent iterative algorithm of Tucker tensor decomposition and completion. ;

[0027] 3. Calculate and compare the prediction tensors With the real tensor The difference between them is lost;

[0028] 4. Update and adjust the core tensor based on the difference value using the gradient descent update algorithm. Factor matrix Repeat steps 2, 3, and 4 until convergence. At this point, the Tucker tensor can be considered complete after prediction. With the actual tensor Since they are the same tensor and approximately equal, the missing values ​​in the recovery evaluation results can be considered to have been filled in and predicted.

[0029]

[0030]

[0031] Algorithm 1 is explained as follows: First, input a sparse Tucker tensor constructed based on the training set. The values ​​of the lattice elements inside this tensor represent the evaluation results of the used electronic products, and then a minimum error value is input. It is used to determine the condition for the final gradient descent iteration to stop, that is, to represent the tensor after completing the missing values ​​of the evaluation result in the output. With the real sparse Tucker tensor The difference loss reached the convergence condition. Step 1, based on the definition of the Tucker tensor decomposition formula, defines a randomly initialized 5th-order core tensor. This represents the core interaction relationships across five characteristic attribute dimensions of secondhand electronic products, and can be randomly initialized. Then, five factor matrices are initialized separately. , , , , Their internal values ​​are derived from the basic attribute values ​​representing the five feature attribute dimensions. Step 2: Hyperparameters The degree of suppression of the regularization term in the loss function, representing the final error calculated in the gradient descent iterations, is a hyperparameter. The greater the penalty from the regularization term on the overall loss function, the better it can prevent overfitting. The step size in step 3... This represents the learning rate and step size of the Tucker decomposition completion learning formula during gradient descent. The larger the value, the greater the fluctuation. Gradient descent might reach the lower convergence minimum more quickly, but it could also skip the minimum, leading to convergence failure. Steps 4-16 can be considered a complete epoch iteration. Each complete iteration calculates the previous... With this The degree of decrease, when the degree of decrease is less than the minimum error value At this point, gradient descent has reached a stable state, and convergence is considered achieved. Steps 5-13 represent traversing the training set to construct the sparse Tucker tensor. Each element in the dataset (i.e., the training set sample data) is then backpropagated based on the gradient form of the Tucker decomposition formula to update its core tensor parameters and factor matrix parameters. This represents the pattern product of the tensor in the nth dimension. This represents the Kronecker product of matrices. Step 6 formula. This represents the core tensor based on the decomposition definition of the Tucker tensor. In respectively The product of the dimension and the corresponding factor matrix yields the completed new element. Step 7 uses the Tucker decomposition formula to analyze the ID lock feature attribute matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the ID lock feature attribute factor matrix. The The row vector of the row, where the ← symbol represents the update and replacement operation; Step 8 formula is based on the Tucker decomposition formula for the feature attribute factor matrix of the function options. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated by backpropagation to update the feature attribute matrix of the function options. The The row vector of the row, where the ← symbol indicates an update / replacement operation; step 9's formula is based on the Tucker decomposition formula for the color feature attribute factor matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated by backpropagation to the color feature attribute matrix. The The row vector of the row, where the ← symbol represents the update and replacement operation; Step 10 formula is based on the Tucker decomposition formula for the storage capacity feature attribute factor matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the storage capacity feature attribute matrix. The The row vector of rows, where the ← symbol represents the update and replacement operation; step 11 formula is based on the Tucker decomposition formula for the memory feature attribute factor matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the memory feature attribute matrix. The The row vector of the row, where the ← symbol represents the update / replacement operation; the formula in step 12 is based on the Tucker decomposition formula for the core tensor. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the entire core tensor. Here, the left arrow (←) sign indicates the update / replacement operation. The completion of step 13 indicates that the tensor is sparse. Every non-zero element in the array has been traversed. Step 14 then represents using... Update and replace The formula for calculating the loss function in step 15 is as follows:

[0032]

[0033] in, Represents a sparse Tucker tensor constructed based on the training set. . This represents the latest kernel tensor parameters after training. Represents the tensor The least squares result is calculated by combining all non-zero elements in the data with the elements after Tucker decomposition and completion. , , , , This represents the feature attribute value parameter matrix after training and updating. This represents the L2 regularization term of the core tensor. Represents the ID lock feature attribute factor matrix L2 regularization term, Represents the feature attribute factor matrix of functional options L2 regularization term, Represents the color feature attribute factor matrix L2 regularization term, Represents the storage capacity characteristic attribute factor matrix L2 regularization term, Represents the memory feature attribute factor matrix The L2 regularization term. The process up to step 16 can be considered a complete epoch iteration, meaning one complete iteration through the training data has been performed. Step 17 represents obtaining a complete, padded tensor. .

[0034] Compared with existing technical methods, the beneficial effects of the present invention are: the present invention, through the missing value completion prediction method of Tucker tensor decomposition, can more effectively make accurate predictions of the recycling evaluation results of second-hand electronic products, and can clearly understand the completion prediction process based on the formula principle of Tucker tensor decomposition, thus avoiding the existence of black box mode. Attached Figure Description

[0035] Figure 1 This is a flowchart of the core process of a missing value completion prediction method based on Tucker tensor decomposition of the present invention.

[0036] Figure 2 This is a result image after calculating the characteristics of the box plot data;

[0037] Figure 3 The flowchart describes how to construct a Tucker tensor decomposition completion model to predict and complete missing values ​​in the final evaluation results of used electronic products. Detailed Implementation

[0038] The present invention will be further illustrated below with reference to the accompanying drawings and related tables, but this does not limit the scope of the invention in any way.

[0039] like Figure 1 As shown, the missing value completion prediction method based on Tucker tensor decomposition of the present invention adopts the following technical solution steps. At the same time, in order to achieve the above objectives, the present invention uses a dataset of second-hand mobile phone transactions from a certain company as the method verification. The company is a professional Internet high-tech company that buys back electronic products. The actual usable original second-hand mobile phone data in the dataset consists of 1996 sample data, with a total of 16 feature attributes.

[0040] Step 1: Outlier Removal Based on Box Plot. When analyzing the original assessment and prediction results data of the second-hand mobile phone market transactions, we noticed a significant range, from 1 to several thousand values, indicating a clear imbalance in the data. Directly applying this data to train a tensor model might affect the accuracy of the model's assessment and prediction due to this uneven distribution. Although normalization techniques may help with data preprocessing, they cannot completely overcome the challenges posed by the uneven sample distribution. We used a box plot outlier removal formula to remove outlier records from the original dataset. The box plot quickly identifies outliers—points far removed from the main data set. The box plot calculation results for the original dataset are shown below. Figure 2 As shown in Table 1 below, the calculated data results after box plot data feature calculation are as follows:

[0041] Table 1: Digitized features of box plots in the dataset

[0042] Min Max QL median QU IQR average value 1 4477 5.0 14.5 71.0 66.0 96.63

[0043] In this context, Min represents the minimum predicted evaluation result among the original 1996 sample data points for used mobile phones, while Max represents the maximum predicted evaluation result. QL and QU correspond to the lower and upper quartiles in the box plot's numerical characteristics, respectively. IQR represents the interquartile range, calculated as IQR = QU - QL, and is an important statistical indicator for measuring data dispersion, systematically extended by Tukey to box plot data analysis. Based on the numerical characteristics of the box plot, the lower limit is calculated as max{QL - 1.5IQR, Min} = 1, and the upper limit is calculated as min{QU + 1.5IQR, Max} = 170. Therefore, sample data with predicted evaluation results higher than 170 and lower than 1 are considered outliers and are removed. Ultimately, only 1729 out of the original 1996 sample data points remain.

[0044] Step Two: Feature Removal with High Missing Attribute Rates. Due to missing data in some sample data of the original used mobile phone dataset, and because modeling the relationship between feature attributes and evaluation prediction results is crucial for accurately predicting the evaluation results of used electronic products, it is necessary to remove samples with significant missing information to avoid affecting the accuracy of the evaluation prediction results. To ensure optimal experimental results, the integrity of the training samples, and the rationality of the experiment, samples with a missing attribute rate exceeding 50% in the original data were selectively removed. The missing rate of each feature attribute in the original used mobile phone sample data was calculated separately, and the missing rates for each feature attribute are shown in Table 2 below:

[0045] Table 2: Missing rates of feature attributes in the original dataset of second-hand mobile phones

[0046]

[0047] As can be seen from Table 2, the missing data rate of the feature attributes "standard", "model", "warranty period", "network mode" and "can be powered on" has exceeded 50%. At this time, if this information is put into the training set as training data, it will be difficult to reflect the relationship between the feature attributes and the evaluation prediction results due to the lack of effective training data samples, which will seriously affect the accuracy and effect of training. Therefore, in the subsequent processing and construction of Tucker tensor decomposition missing value completion prediction, the information of these 5 feature attributes will be removed.

[0048] Step 3: Identifying Feature Attributes Highly Correlated with Evaluation Prediction Values ​​Based on Mutual Information. When predicting missing values ​​in the evaluation results of used mobile phones, mutual information is used as a statistical tool to quantify the impact of each feature attribute on the evaluation prediction value. Given the limited size of our current used mobile phone dataset, we utilize the Tucker tensor decomposition model to construct a multi-dimensional feature attribute-evaluation prediction value relationship tensor, aiming for more accurate evaluation prediction of missing values ​​in the used mobile phone recycling evaluation. However, when the model contains too many attribute dimensions, the model complexity may increase dramatically, thus affecting the accuracy of the evaluation. Therefore, before formally constructing the Tucker tensor model, a mutual information feature attribute selection strategy is adopted. This strategy comprehensively considers the correlation between each feature attribute and the evaluation prediction value to select the feature attribute that contributes the most to the model.

[0049] Mutual information (MI) is a measure of statistical independence between two random variables. It is a more general method for measuring nonlinear relationships and can be used to identify relationships between datasets that are not detected by commonly used measures of linear correlation. For random variables... The formula for mutual information between T and T is described as follows:

[0050]

[0051] in It is the marginal distribution of the random variable Z. It is the marginal distribution of the random variable T, and It is the joint distribution of (Z, T). This represents the mutual information value between two random variables Z and T. In practical terms, random variable Z represents the characteristic attributes of used electronic products, and random variable T represents the predicted value of the used electronic product recycling assessment to be completed.

[0052] To better select feature attributes based on mutual information values ​​and avoid the influence of different units, the value range of mutual information (MI) is mapped to between 0 and 1, resulting in more representative standardized mutual information (NMI). The final standardized mutual information value is then used to determine the NMI for random variables. The formula for mutual information between T and T is as follows:

[0053]

[0054] in, Represents random variables entropy, Represents random variables entropy, This represents the mutual information value between two random variables Z and T. This represents the standardized mutual information value between two random variables Z and T. In practical terms, random variable Z represents the characteristic attributes of used electronic products, and random variable T represents the predicted value of the used electronic product recycling assessment to be completed.

[0055] Finally, for the remaining 11 feature attributes, note that since only 11 attributes remain, only the standardized mutual information of these 11 features was calculated. The uncalculated feature attributes are represented by N / A in the table below. Their standardized mutual information values ​​are shown in Table 3:

[0056] Table 3: Standardized mutual information values ​​between feature attributes and evaluation prediction results

[0057]

[0058] Finally, the standardized mutual information values ​​of the remaining feature attributes and the predicted values ​​of the recycling assessment are calculated. As can be seen from Table 3 above, the mutual information values ​​of ID lock, function options, color, storage capacity and memory are the highest. Finally, these 5 feature attributes are retained to construct the Tucker decomposition tensor completion prediction model, which is used to compare with other prediction methods in the experiment of missing value completion prediction based on Tucker tensor decomposition.

[0059] Step Four: Before proceeding to Step Four, let's introduce the definition of Tucker decomposition completion. The Tucker tensor can be decomposed into the product of a core tensor and factor matrices along each dimension. Using an n-order tensor... For example, its Tucker decomposition is as follows:

[0060]

[0061] in, Known as the core tensor, This is called the factor matrix. This represents the pattern product of the tensor in the nth dimension.

[0062] After the first three steps, the five feature attributes with the highest standardized mutual information values ​​are retained: ID lock, function options, color, storage capacity, and memory. We define the entire tensor constructed based on the training dataset as a sparse tensor. Define the core tensor. Representing the core interaction of 5 characteristic attributes, The symbol representing the set of real numbers indicates that... It is ( , , , , A set of real tensors of size ). This represents the dimensional length of the interaction with the ID lock's feature attributes. Based on continuous experimentation, it has been found that when... The effect is best when the value is 3, so choose 3. =3; This represents the length of the dimension interacting with the feature attributes of the function options. Based on continuous experimentation, it has been found that when... The effect is best when the value is 3, so choose 3. =3; This represents the dimensional length of the interaction with color feature attributes. Based on continuous experimentation, it has been found that when... The effect is best when the value is 3, so choose 3. =3; This represents the length of the dimension that interacts with the storage capacity characteristic attribute. Based on continuous experiments, it has been found that when... The effect is best when the value is 3, so choose 3. =3; This represents the dimensional length of the interaction with memory feature attributes. Based on continuous experiments, it has been found that when... The effect is best when the value is 3, so choose 3. =3. Define the factor matrix. Represents the ID lock characteristic attribute, The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of ID lock feature attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents the feature attributes of the function options. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of feature attribute values ​​for the function option. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents color characteristic attributes. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of color feature attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents storage capacity characteristics. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of color feature attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents memory characteristics and attributes. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of memory characteristic attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3. Define error disturbance. Representing the difference before and after the loss function, based on continuous experiments, it has been found that when... At this value, the best results are achieved; exceeding or falling below this value will lead to poorer prediction results. Definition The learning rate, representing gradient descent, has been found through continuous experimentation to be... At this value, the best results are achieved; exceeding or falling below this value will lead to poorer prediction results. Definition The contribution of the regularization penalty term is represented by the value of the regularization penalty term. Based on continuous experiments, it has been found that when... At this value, the best results are achieved; exceeding or falling below this value will lead to poorer prediction results. Definition Represents the complete tensor after Tucker tensor decomposition to complete the predicted missing evaluation values. Definition This represents the specific position of a non-zero element in the tensor, therefore it is defined as follows: Representative is sparse tensor A non-zero element at a given position. (Definition) Represents the complete tensor after Tucker decomposition and completion. The element that completes the position. Definition This represents the pattern product of the tensor in the nth dimension, therefore The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in a given dimension. Definition The first element representing the ID lock feature attribute matrix Row vectors of a row; definition The first of the feature attribute matrix representing the functional options Row vectors of a row; definition The first element representing the color feature attribute matrix Row vectors of a row; definition The first of the storage capacity feature attribute matrix Row vectors of a row; definition The first element representing the memory feature attribute matrix The row vector of a row. This represents the Kronecker product of matrices.

[0063] Based on the above definitions, a Tucker tensor decomposition completion prediction model is constructed. The accuracy of the model is verified using a test set, ultimately achieving the completion prediction of missing values ​​in the evaluation results of used electronic product recycling. The specific prediction construction process is simplified into four steps, and the textual explanation can be found below.

[0064] 1: Initialize a 5-dimensional Tucker sparse tensor based on the existing training dataset. Based on the initialized 5-dimensional Tucker sparse tensor Initialize a 5-dimensional core tensor and the corresponding 5-dimensional factor matrix ;

[0065] 2: Constructing a 5-dimensional Tucker tensor after missing value imputation prediction based on the gradient descent iterative algorithm of Tucker tensor decomposition and completion. ;

[0066] 3: Calculate and compare the predicted tensors With the real tensor The difference between them is lost;

[0067] 4: Update and adjust the core tensor based on the difference value using the gradient descent update algorithm. Repeat steps 2, 3, and 4 for the factor matrices I, F, C, S, and M until convergence. At this point, it can be considered that the missing values ​​of the evaluation results have been recovered to complete the predicted Tucker tensor. With the actual tensor Since they are the same tensor and approximately equal, the missing value information in the recovery evaluation result can be considered to have been completed and predicted.

[0068] A more detailed description of Algorithm 1 is as follows:

[0069]

[0070]

[0071] Algorithm 1 is explained as follows: First, input a sparse Tucker tensor constructed based on the training set. The values ​​of the lattice elements inside this tensor represent the evaluation results of the used electronic products, and then a minimum error value is input. It is used to determine the condition for the final gradient descent iteration to stop, that is, to represent the tensor after completing the missing values ​​of the evaluation result in the output. With the real sparse Tucker tensor The difference loss reached the convergence condition. Step 1, based on the definition of the Tucker tensor decomposition formula, defines a randomly initialized 5th-order core tensor. This represents the core interaction relationships across five characteristic attribute dimensions of secondhand electronic products, and can be randomly initialized. Then, five factor matrices are initialized separately. , , , , Their internal values ​​are derived from the basic attribute values ​​representing the five feature attribute dimensions. Step 2: Hyperparameters The degree of suppression of the regularization term in the loss function, representing the final error calculated in the gradient descent iterations, is a hyperparameter. The greater the penalty from the regularization term on the overall loss function, the better it can prevent overfitting. The step size in step 3... This represents the learning rate and step size of the Tucker decomposition completion learning formula during gradient descent. The larger the value, the greater the fluctuation. Gradient descent might reach the lower convergence minimum more quickly, but it could also skip the minimum, leading to convergence failure. Steps 4-16 can be considered a complete epoch iteration. Each complete iteration calculates the previous... With this The degree of decrease, when the degree of decrease is less than the minimum error value At this point, gradient descent has reached a stable state, and convergence is considered achieved. Steps 5-13 represent traversing the training set to construct the sparse Tucker tensor. Each element in the dataset (i.e., the training set sample data) is then backpropagated based on the gradient form of the Tucker decomposition formula to update its core tensor parameters and factor matrix parameters. This represents the pattern product of the tensor in the nth dimension. This represents the Kronecker product of matrices. Step 6 formula. This represents the core tensor based on the decomposition definition of the Tucker tensor. In respectively The product of the dimension and the corresponding factor matrix yields the completed new element. Step 7 uses the Tucker decomposition formula to analyze the ID lock feature attribute factor matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the ID lock feature attribute matrix. The The row vector of the row, where the ← symbol represents the update and replacement operation; Step 8 formula is based on the Tucker decomposition formula for the feature attribute factor matrix of the function options. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated by backpropagation to update the feature attribute matrix of the function options. The The row vector of the row, where the ← symbol indicates an update / replacement operation; step 9's formula is based on the Tucker decomposition formula for the color feature attribute factor matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated by backpropagation to the color feature attribute matrix. The The row vector of the row, where the ← symbol represents the update and replacement operation; Step 10 formula is based on the Tucker decomposition formula for the storage capacity feature attribute factor matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the storage capacity feature attribute matrix. The The row vector of rows, where the ← symbol represents the update and replacement operation; step 11 formula is based on the Tucker decomposition formula for the memory feature attribute factor matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the memory feature attribute matrix. The The row vector of the row, where the ← symbol represents the update / replacement operation; the formula in step 12 is based on the Tucker decomposition formula for the core tensor. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the entire core tensor. Here, the left arrow (←) sign indicates the update / replacement operation. The completion of step 13 indicates that the tensor is sparse. Every non-zero element in the array has been traversed. Step 14 then represents using... Update and replace The formula for calculating the loss function in step 15 is as follows:

[0072]

[0073] in, Represents a sparse Tucker tensor constructed based on the training set. . This represents the latest kernel tensor parameters after training. Represents the tensor The least squares result is calculated by combining all non-zero elements in the data with the elements after Tucker decomposition and completion. , , , , This represents the feature attribute value parameter matrix after training and updating. This represents the L2 regularization term of the core tensor. Represents the ID lock feature attribute factor matrix L2 regularization term, Represents the feature attribute factor matrix of functional options L2 regularization term, Represents the color feature attribute factor matrix L2 regularization term, Represents the storage capacity characteristic attribute factor matrix L2 regularization term, Represents the memory feature attribute factor matrix The L2 regularization term. The process up to step 16 can be considered a complete epoch iteration, meaning one complete iteration through the training data has been performed. Step 17 represents obtaining a complete, padded tensor. .

[0074] To verify the actual predictive performance of this invention, a validation experiment was conducted on a personal laptop computer with an Intel Core i5 8th generation CPU and 16GB of RAM. As mentioned earlier, the dataset used in the experiment came from a second-hand mobile phone trading platform. After data preprocessing in the first three steps, five attributes ("ID lock", "function", "color", "storage", and "memory") were selected to predict the recycling evaluation results. 596 data samples were used as the training dataset, and 148 data samples were used as the test dataset. The general setup of the experiment is shown in Table 4.

[0075] Table 4: Basic Experimental Setup

[0076] The missing value completion prediction method based on Tucker tensor decomposition proposed in this invention is experimentally validated using three error evaluation metrics (MAE, MRE, and RMSE) to assess the experimental results. The standard formulas are as follows:

[0077]

[0078]

[0079]

[0080] In the standard formula for evaluation indicators, This represents the actual recycling assessment result value of the i-th used electronic product. This represents the supplementary predicted value for the recycling assessment result of the i-th used electronic product. MAE is the mean absolute error, which is the average of the absolute values ​​of the errors. MRE is the relative error, which refers to the ratio of the error to the true value. RMSE is the root mean square error, which is sensitive to outliers, but they can reflect the distribution of prediction errors.

[0081] Five prediction methods were used as baselines: resampling-based logistic regression (LRR), multi-view radial basis function neural network (MV-RBF), stochastic multilayer perceptron with trainable perturbation factor (S-MLPT), Firefly algorithm-backpropagation neural network (FA-BPNN), and prediction method based on CP decomposition (CP). The experimental verification results of this invention are as follows:

[0082] Table 5: MAE Evaluation Indicators Results for Different Prediction Methods

[0083]

[0084] As shown in Table 5, under the prediction evaluation index MAE (Absolute Error), the missing value completion prediction method based on Tucker tensor decomposition proposed in this invention has the lowest error among all the baseline prediction methods. Taking a 5% training set ratio as an example, our method's prediction error value is 19.99, which is about 17.3% lower than the highest value of 24.19. This indicates that the missing value completion prediction method based on Tucker tensor decomposition has higher prediction accuracy and better performance.

[0085] Table 6: Results of MRE Evaluation Indicators for Different Prediction Methods

[0086] Under the Mean Relative Error (MRE) evaluation metric, as shown in Table 6, the missing value completion prediction method based on Tucker tensor decomposition of this invention has the lowest deviation among all the baseline prediction methods and is closer to the actual recovery evaluation result. Taking a 5% training set ratio as an example, our method's prediction error value of 0.72 is about 17.2% lower than the highest value of 0.87. This fully demonstrates the stability and higher accuracy of the missing value completion prediction method based on Tucker tensor decomposition.

[0087] Table 7: RMSE Evaluation Indicators Results for Different Prediction Methods

[0088] The root mean square error (RMSE) has a quadratic term, which significantly amplifies extreme errors, thus substantially increasing the metric. This fully demonstrates the predictive stability and accuracy of the model method. Table 7 shows that the missing value completion prediction method based on Tucker tensor decomposition of this invention has a lower RMSE among all baseline prediction methods. For example, with a 10% training set ratio, our method's prediction error is 31.86, a decrease of approximately 24.8% compared to the highest value of 42.36. Therefore, the missing value completion prediction method based on Tucker tensor decomposition can produce fewer extreme error predictions, thereby achieving higher accuracy.

[0089] Based on the above analysis, this invention can achieve higher accuracy in predicting recycling assessment results in the recycling of second-hand electronic products, thereby helping to promote the development of the second-hand electronic product recycling industry. After the model training is completed, this invention can be directly provided to relevant practitioners for use, thereby achieving efficient recycling.

[0090] The present invention has been described in detail above through embodiments. Those skilled in the art should understand that certain modifications and variations can be made to the present invention without departing from the spirit and essence of the present invention. For example, the initialization values ​​of the core tensors of the Tucker model can still achieve the purpose of the present invention. Within the technical scope disclosed in the present invention, equivalent substitutions or changes made according to the technical solution and inventive concept of the present invention should be covered within the protection scope of the present invention.

Claims

1. A missing value completion prediction method based on Tucker tensor decomposition, characterized in that, Includes the following steps: Step 1: Remove outlier data based on box plots; Step 2: Remove features with high missing rates. Step 3: Identify feature attributes that are highly correlated with the recovery evaluation results based on mutual information; Step 4: Construct a Tucker tensor decomposition completion prediction model based on the training data, verify the accuracy of the model using the test set, and finally achieve the completion prediction of missing values ​​in the evaluation results of used electronic product recycling.

2. The missing value completion prediction method based on Tucker tensor decomposition as described in claim 1, characterized in that: Step one employs the box plot data removal method systematically promoted by Tukey to calculate the maximum value (Max), minimum value (Min), lower quartile (QL), and upper quartile (QU) of the box plot, thereby obtaining the interquartile range (IQR). The calculation formula is IQR = QU - QL. Finally, the lower limit value can be calculated as max{QL - 1.5IQR, Min}, and the upper limit value can be calculated as min{QU + 1.5IQR, Max}, thus removing relevant abnormal data that are greater than the upper limit and less than the lower limit.

3. The missing value completion prediction method based on Tucker tensor decomposition as described in claim 1, characterized in that: Step 2: Selectively remove samples from the original sample whose data feature attribute information missing rate exceeds 50%.

4. The missing value completion prediction method based on Tucker tensor decomposition as described in claim 1, characterized in that: In step three, before formally constructing the Tucker tensor model, a feature attribute selection strategy based on mutual information is employed. This strategy comprehensively considers the correlation between each feature attribute and the recovery evaluation result value to select the feature attribute that contributes the most to the model. Mutual information (MI) is a measure of the statistical independence between two random variables. The formula for mutual information between T and T is described as follows: in It is the marginal distribution of the random variable Z. It is the marginal distribution of the random variable T, and It is the joint distribution of (Z, T). This represents the mutual information value between two random variables Z and T. In practical terms, random variable Z represents the characteristic attributes of used electronic products, and random variable T represents the predicted value of the used electronic product recycling assessment to be completed. Mapping the value range of mutual information (MI) to between 0 and 1 yields the more representative standardized mutual information (NMI). The judgment (NMI) is made based on the final standardized mutual information value. The formula for mutual information between T and T is as follows: in, Represents random variables entropy, Represents random variables entropy, This represents the mutual information value between two random variables Z and T. Then, it represents the standardized mutual information value between two random variables Z and T. In practical terms, random variable Z represents the characteristic attributes of second-hand electronic products, and random variable T represents the predicted value of the second-hand electronic product recycling assessment to be completed. Finally, the five characteristic attributes with the highest mutual information values ​​are retained to construct the Tucker decomposition and completion tensor model.

5. The missing value completion prediction method based on Tucker tensor decomposition as described in claim 1, characterized in that: Step four involves constructing a Tucker decomposition and completion tensor model based on the training data, and validating the model's accuracy using a test set. Before proceeding with step four, the definition of Tucker decomposition and completion is introduced: a Tucker tensor is decomposed into the product of a core tensor and factor matrices along each dimension, resulting in an n-order tensor. The Tucker decomposition is as follows: in, Known as the core tensor, This is called the factor matrix. Let represent the pattern product of the tensor in the nth dimension. After the first three steps, the five feature attributes with the highest standardized mutual information values ​​are retained: ID lock, function options, color, storage capacity, and memory. We define the entire tensor constructed based on the training dataset as a sparse tensor. Define the core tensor Representing the core interaction of 5 characteristic attributes, The symbol representing the set of real numbers indicates that... It is ( , , , , A set of real tensors of size ). This represents the dimensional length of the interaction with the ID lock's feature attributes. This represents the length of the dimension interacting with the feature attributes of the function options. This represents the length of the dimension of interaction with color feature attributes. This represents the length of the dimension that interacts with the storage capacity characteristic attribute; This represents the dimensionality of the interaction with memory feature attributes, defining the factor matrix. Represents the ID lock characteristic attribute, The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of ID lock feature attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents the feature attributes of the function options. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of feature attribute values ​​for the function option. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents color characteristic attributes. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of color feature attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents storage capacity characteristic attribute, The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of color feature attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3; Define the factor matrix Represents memory characteristics and attributes. The symbol representing the set of real numbers indicates that... It is ( , A set of real matrixes of size 1000. Indicates the number of memory characteristic attribute values. Representation and core tensor The length of the interaction dimension remains consistent with the core tensor. of The values ​​are the same, therefore the values ​​are taken as follows: =3, define error disturbance The difference before and after the loss function is defined as follows: The learning rate, representing gradient descent, is defined as follows: The contribution of the regularization penalty term is defined as follows: Represents the complete tensor after Tucker tensor decomposition to complete the predicted missing evaluation values, defined as follows: This represents the specific position of a non-zero element in the tensor, therefore it is defined as follows: Representative is the sparse tensor Non-zero elements at position, defined Represents the complete tensor after Tucker decomposition and completion. The element for positional completion is defined. This represents the pattern product of the tensor in the nth dimension, therefore The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in dimension; The representative tensor in the th Pattern product in dimension, defined The first element representing the ID lock feature attribute matrix Row vectors of a row; definition The first of the feature attribute matrix representing the functional options Row vectors of a row; definition The first element representing the color feature attribute matrix Row vectors of a row; definition The first of the storage capacity feature attribute matrix Row vectors of a row; definition The first element representing the memory feature attribute matrix The row vector of a row. Let Kronecker product represent the matrix. Based on the above definition, a Tucker tensor decomposition completion prediction model is constructed. The accuracy of the model is verified using a test set. Finally, the model achieves the completion prediction of missing values ​​in the evaluation results of used electronic product recycling. The specific prediction process is simplified into four steps, which can be explained in the following text: Step 1: Initialize a 5-dimensional Tucker sparse tensor based on the existing training dataset. Based on the initialized 5-dimensional Tucker sparse tensor Initialize a 5-dimensional core tensor and the corresponding 5-dimensional factor matrix ; Step 2: Construct a 5-dimensional Tucker tensor after missing value imputation prediction based on the gradient descent iterative algorithm of Tucker tensor decomposition and completion. ; Step 3: Calculate the comparison prediction tensor With the real tensor The difference between them is lost; Step 4: Update and adjust the core tensor based on the difference value using the gradient descent update algorithm. Factor matrix Repeat steps 2, 3, and 4 until convergence.

6. The missing value completion prediction method based on Tucker tensor decomposition as described in claim 5, characterized in that, The gradient descent-based update algorithm is as follows: First, input a sparse Tucker tensor constructed based on the training set. The values ​​of the lattice elements inside this tensor represent the evaluation results of the used electronic products, and then a minimum error value is input. It is used to determine the condition for the final gradient descent iteration to stop, that is, to represent the tensor after completing the missing values ​​of the evaluation result in the output. With the real sparse Tucker tensor The difference loss reached the convergence condition. In step 1, according to the definition of the Tucker tensor decomposition formula, a 5th-order core tensor was randomly initialized. The core interaction relationships representing the five characteristic attribute dimensions of second-hand electronic products can be randomly initialized, and then the five factor matrices can be initialized separately. , , , , Their internal values ​​are derived from the basic attribute values ​​representing the five feature attribute dimensions. (Step 2: Hyperparameters) The degree of suppression of the regularization term in the loss function representing the final error calculated in the gradient descent iterations; the step size in step 3. This represents the learning rate and step size of the Tucker decomposition completion learning formula during gradient descent. The larger the value, the greater the fluctuation. Gradient descent may reach the lower convergence minimum more quickly, but it may also skip the convergence minimum, leading to convergence failure. Steps 4-16 are considered a complete epoch. Each complete iteration calculates the previous... With this The degree of decrease, when the degree of decrease is less than the minimum error value At this point, gradient descent has reached a stable state, and convergence is considered achieved. Steps 5-13 represent traversing the training set to construct the sparse Tucker tensor. Each element in the dataset represents a sample data point in the training set. Then, based on the gradient form of the Tucker decomposition formula, backpropagation is used to update its core tensor parameters and factor matrix parameters. This represents the pattern product of the tensor in the nth dimension. The formula for the Kronecker product of matrices is shown in step 6. This represents the core tensor based on the decomposition definition of the Tucker tensor. In respectively The product of the dimension and the corresponding factor matrix yields the completed new element. Step 7, the formula is based on the Tucker decomposition formula for the ID lock feature attribute matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the ID lock feature attribute factor matrix. The The row vector of the row, where the ← symbol represents the update and replacement operation; Step 8 formula is based on the Tucker decomposition formula for the feature attribute factor matrix of the function options. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated by backpropagation to update the feature attribute matrix of the function options. The The row vector of the row, where the ← symbol indicates an update / replacement operation; step 9's formula is based on the Tucker decomposition formula for the color feature attribute factor matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated by backpropagation to the color feature attribute matrix. The The row vector of the row, where the ← symbol represents the update and replacement operation; Step 10 formula is based on the Tucker decomposition formula for the storage capacity feature attribute factor matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the storage capacity feature attribute matrix. The The row vector of rows, where the ← symbol represents the update and replacement operation; step 11 formula is based on the Tucker decomposition formula for the memory feature attribute factor matrix. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the memory feature attribute matrix. The The row vector of the row, where the ← symbol represents the update / replacement operation; the formula in step 12 is based on the Tucker decomposition formula for the core tensor. The gradient update formula is obtained by taking the derivative. The error between the predicted and completed elements and the true element values ​​is calculated and backpropagated to update the entire core tensor. Here, the left arrow sign indicates the update and replacement operation. The completion of step 13 indicates that the tensor is sparse. Every non-zero element in the array has been traversed; step 14 represents using... Update and replace The formula for calculating the loss function in step 15 is as follows: in, Represents a sparse Tucker tensor constructed based on the training set. , This represents the latest kernel tensor parameters after training. Represents the tensor The least squares result is calculated by combining all non-zero elements in the data with the elements after Tucker decomposition and completion. , , , , This represents the feature attribute value parameter matrix after training and updating. This represents the L2 regularization term of the core tensor; Represents the ID lock feature attribute factor matrix L2 regularization term; Represents the feature attribute factor matrix of functional options L2 regularization term; Represents the color feature attribute factor matrix L2 regularization term; Represents the storage capacity characteristic attribute factor matrix L2 regularization term; Represents the memory feature attribute factor matrix The L2 regularization term, up to step 16, is considered a complete epoch iteration, meaning a complete iteration of the training data has been completed. Step 17 indicates that a complete, padded tensor has been obtained. .

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