Long-tail commodity recommendation method based on accurate symmetric positive definite manifold learning

Abstract

The invention belongs to the technical field of electronic commerce, and particularly relates to a long-tail commodity recommendation method based on precise symmetric positive definite manifold learning. Comprising the steps that user behavior data and commodity feature data of an e-commerce platform are acquired and preprocessed, and a commodity feature matrix and a user commodity interaction matrix are obtained; optimizing the user commodity interaction matrix by adopting a low-rank approximation technology, and updating the user commodity interaction matrix; mapping the commodity feature matrix into a symmetric positive definite matrix space, and performing manifold embedding processing to obtain an SPD feature vector of the commodity; carrying out aggregation processing on the SPD feature vectors of the commodities by adopting a popular full-connection layer to obtain aggregation features; mapping the aggregated features to Euclidean space by using a geometric preserving mapping function to obtain mapping features; processing the mapping features to obtain a long-tail commodity recommendation result; according to the method, the advertisement click rate of the e-commerce platform is improved, and the precision marketing and recommendation effects are remarkably enhanced.

Description

Technical Field

[0001] The present invention belongs to the technical field of e-commerce, and in particular relates to a long-tail commodity recommendation method based on precise symmetric positive definite manifold learning. Background Art

[0002] The phenomenon of long-tail distribution of product features is common in e-commerce recommendation systems. Traditional recommendation algorithms, such as neural network-based models, cannot effectively handle the recommendation of long-tail products. The reason is that these traditional neural network models fail to fully capture the complex nonlinear relationship between product features when processing long-tail distribution data; they encode items and users into Euclidean space, which is a flat geometric structure with polynomial expansion capacity. When faced with data in Euclidean space, people often tend to use the scale within the space for evaluation. However, this approach is not always appropriate. Such a way of thinking gives people a sense of limitation, as if we can only analyze within a specific space and scale, while ignoring the diversity and complexity of things themselves. In fact, long-tail distribution data can be traced back to a hierarchical structure, and the number of items grows exponentially. When encoding this data into Euclidean space, this imbalance makes it difficult to express the relationship between data through embedding vectors, thereby reducing the accuracy of the final prediction. Therefore, how to improve the recommendation effect of long-tail products in the recommendation system and meet personalized needs while maintaining recommendation diversity has become a hot topic in current research.

[0003] In recent years, the recommendation system method based on manifold learning has gradually attracted widespread attention in the academic community. Manifold learning can improve the model's ability to represent complex relationships by capturing the intrinsic geometric structure of data in high-dimensional space, especially showing significant advantages when dealing with nonlinear and sparse data. Among them, the symmetric positive definite (SPD) manifold, as a typical non-Euclidean space, can more effectively describe the geometric characteristics of high-dimensional data. Although manifold learning methods have made some progress in data representation, in the actual application of e-commerce recommendation systems, how to efficiently embed long-tail products into multi-manifold space still remains a challenge:

[0004] 1. Diversity of feature representation in manifold space: While improving the accuracy of recommendations, how to effectively enhance the intrinsic variability of recommendation results and the closeness of the relationship between different product feature spaces remains a major challenge. Traditional fully connected layers cannot fully utilize the structural characteristics of manifold geometry, which limits the performance of the recommendation system.

[0005] 2. Information loss in the reverse mapping process of manifold space: When mapping manifold data back to Euclidean space, how to avoid information loss and ensure the integrity of geometric properties and high-dimensional structure information is an important challenge. Summary of the invention

[0006] In view of the shortcomings of the prior art, the present invention proposes a long-tail product recommendation method based on precise symmetric positive definite manifold learning, which includes:

[0007] Obtain user behavior data and preprocess it, input the product feature vector obtained after preprocessing into the trained long-tail product recommendation model to obtain the long-tail product recommendation results;

[0008] The training process of the long-tail product recommendation model includes:

[0009] S1: Obtain user behavior data and product feature data from the e-commerce platform and pre-process them to obtain a product feature matrix and a user-product interaction matrix;

[0010] S2: Use low-rank approximation technology to optimize the user-item interaction matrix and update the user-item interaction matrix;

[0011] S3: Map the product feature matrix into the symmetric positive definite matrix space and perform manifold embedding to obtain the SPD feature vector of the product;

[0012] S4: Use the popular fully connected layer to aggregate the feature vectors of SPD products to obtain aggregate features;

[0013] S5: Use the geometry-preserving mapping function to map the aggregated features to the Euclidean space to obtain the mapped features;

[0014] S6: Process the mapped features to obtain long-tail product recommendation results;

[0015] S7: Calculate the total loss function based on the long-tail product recommendation results and the user-product interaction matrix, adjust the model parameters based on the total loss function, and obtain a trained long-tail product recommendation model.

[0016] Preferably, the user-product interaction matrix is ​​optimized as follows:

[0017]

[0018] Among them, R represents the user-item interaction matrix, λ lr represents the low-rank weight, R k Represents a low-rank approximation of the user-item interaction matrix.

[0019] Preferably, the SPD matrix is ​​aggregated using a popular fully connected layer and expressed as:

[0020]

[0021] Among them, F agg represents the aggregated feature, k represents the number of product features, and w irepresents the weight of the i-th product, F i ' Represents the SPD feature vector of the i-th product.

[0022] Preferably, the total loss function is the sum of the diversity constraint loss and the mapping loss; wherein the diversity constraint loss is calculated by the error balance loss and the overall diversity metric.

[0023] Furthermore, the error balance loss is expressed as:

[0024]

[0025] in, Represents the error balance loss value, R u,i represents the actual rating of product i by user u, represents the predicted rating of user u for product i, λ3 represents the distance weight, d(F′ i ,F′ j ) represents the commodity SPD feature vector F′ i and F′ j Ω represents the set of user-item pairs, and I represents the set of item pairs.

[0026] Furthermore, the overall diversity metric is expressed as:

[0027]

[0028] Where D represents the overall diversity measure, d(F′ i ,F′ j ) represents the commodity SPD feature vector F′ i and F′ j The distance between.

[0029] Furthermore, the diversity constraint loss is expressed as:

[0030]

[0031] Among them, L1 represents the diversity constraint loss value, represents the error balance loss value, γ represents the diversity weight, and D represents the overall diversity measure.

[0032] Furthermore, the mapping loss is expressed as:

[0033]

[0034] Among them, L2 represents the mapping loss, F agg represents the aggregation feature, G(F agg ) represents the commodity aggregation feature F agg The mapping feature obtained after mapping, G-1 is a function that maps the mapped features back to the manifold space, and μ represents the regularization coefficient.

[0035] The beneficial effects of the present invention are as follows: the present invention introduces SPD manifold learning and utilizes its ability to capture the geometric structure of data in non-Euclidean space to accurately describe the intrinsic relationship of long-tail product features, effectively improving the prediction accuracy of the recommendation model for long-tail products; the present invention optimizes the interaction matrix through low-rank approximation technology, while improving the accuracy of recommendations, it enhances the intrinsic variability of recommendation results, and effectively takes into account the accuracy and diversity of recommendations. The present invention adopts a geometry-preserving mapping function to preserve the geometric structure and relationship information of product features during the mapping process, ensuring the integrity of the embedded characteristics of the manifold space in the Euclidean space, thereby improving the robustness and generalization ability of the recommendation model. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 This is a flow chart of the training of the long-tail product recommendation model in the present invention. DETAILED DESCRIPTION

[0037] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0038] The present invention proposes a long-tail product recommendation method based on precise symmetric positive definite manifold learning, and the method comprises the following contents:

[0039] The user behavior data is obtained and preprocessed, and the product feature vector obtained after preprocessing is input into the trained long-tail product recommendation model to obtain the long-tail product recommendation result.

[0040] like Figure 1 As shown in the figure, the training process of the long-tail product recommendation model includes:

[0041] S1: Obtain user behavior data and product feature data from the e-commerce platform and preprocess them to obtain a product feature matrix and a user-product interaction matrix.

[0042] User behavior data includes user u i For product i j The actual score or interaction level can be calculated using the following data:

[0043] Explicit feedback: for example, a user's rating of a product (e.g., 1-5 stars);

[0044] Implicit feedback: such as user clicks on products, browsing time, adding to shopping carts, favorites, number of purchases, etc.

[0045] Product feature data describes the attribute characteristics of a product and can usually be extracted through the product’s own descriptive data or contextual information.

[0046] Sources of product feature data may include:

[0047] Product characteristics: classification (such as category, subcategory); description (such as title, keywords, tags); attributes (such as price, brand, rating), etc.

[0048] Contextual information: time characteristics (such as listing and delisting time, promotion time period); geographical characteristics (such as regional preferences), etc.

[0049] The product feature data is encoded to obtain the product feature matrix F, and the actual user ratings of the products are normalized to obtain the user-product interaction matrix R.

[0050] S2: Use low-rank approximation technology to optimize the user-item interaction matrix and update the user-item interaction matrix.

[0051] In order to effectively improve the processing efficiency of the model and enhance the recommendation ability for long-tail products, the low-rank approximation technology is used to optimize the processing efficiency of high-dimensional features and solve the problem of sparsity of long-tail product features.

[0052] Low-rank approximation technology provides an effective method for processing the user-item interaction matrix R. The basic idea is to represent high-dimensional data as a combination of low-rank matrices through matrix decomposition technology, which not only reduces the storage requirements of the data, but also improves the interpretability of the data.

[0053] Low-rank approximation can be achieved using singular value decomposition (SVD). Given a matrix R, it can be decomposed into:

[0054] R≈UΣV T

[0055] Among them, U and V are the left and right singular vector matrices respectively, and Σ is a diagonal matrix containing singular values. By retaining the largest k singular values, a low-rank approximation can be obtained:

[0056]

[0057] This low-rank approximation can not only help us reduce the amount of computation, but also remove noise to a certain extent, making the recommendation system more robust when dealing with long-tail products.

[0058] The optimization problem is expressed as:

[0059]

[0060] Where R represents the user-item interaction matrix; lr represents the low-rank weight, controlling the influence of low-rank approximation; R k Represents the low-rank approximation of the user-item interaction matrix, which is used to constrain the low-rank property of R.

[0061] The optimized R k As the new user-item interaction matrix R.

[0062] S3: Map the product feature matrix into the symmetric positive definite matrix space and perform manifold embedding processing to obtain the SPD feature vector of the product.

[0063] The SPD matrix set is defined as:

[0064]

[0065] where F′ i >0 indicates matrix F′ i Is a positive definite matrix. To ensure that the feature matrix satisfies the properties of SPD, the original product feature matrix is ​​processed as follows, that is, the product feature matrix is ​​mapped to the space of symmetric positive definite matrix (SPD matrix):

[0066] 1. Symmetry: Ensure matrix symmetry;

[0067] 2. Positive definiteness: F i (2) =Fi i (1) +∈I (where ∈>0), ensuring the matrix is ​​positive definite.

[0068] The SPD feature matrix has the following important properties:

[0069] 1. Symmetry: Each SPD matrix is ​​a symmetric matrix, that is, F′ i =F′ i T .

[0070] 2. Reversibility: All SPD matrices are reversible, meaning their determinant is greater than zero.

[0071] 3. Positive Eigenvalues: All eigenvalues ​​are positive, which ensures the stability of the matrix in various applications.

[0072] In order to effectively capture the nonlinear relationship between product features, the present invention designs a manifold embedding mapping function, which is defined as follows:

[0073]

[0074] Here, the matrix square root is calculated by eigenvalue decomposition (EVD):

[0075]

[0076] F′ i is the SPD feature vector after manifold embedding mapping, and F i (2) Rewrite it as F i ; F i 1 / 2 Denotes the matrix F i The square root of . Through the mapping here, the following points are ensured:

[0077] 1. Maintain positive definiteness: SPD feature vector F′ after mapping i It is still a positive definite matrix, so that the subsequent manifold learning process can continue to apply the geometric properties of the SPD manifold.

[0078] 2. Preservation of manifold structure: Using square root mapping can preserve the geometric relationship between features, which is of great significance for subsequent manifold analysis and learning.

[0079] S4: Use the popular fully connected layer to aggregate the SPD feature vectors of the products to obtain aggregate features.

[0080] The core idea of ​​the manifold fully connected layer is to use weighted summation to aggregate representations from different product features. By assigning appropriate weights to each feature, the aggregation effect can be optimized so that the recommendation system can capture user preferences more accurately; the feature aggregation formula is expressed as:

[0081]

[0082] Among them, F agg represents the aggregated feature, k represents the number of product features, and w i represents the weight of the i-th product, F′ i Represents the SPD feature vector of the i-th product.

[0083] In order to better apply this aggregation strategy, we need to reasonably define the weight w i These weights can be calculated based on a variety of factors, such as the user's historical interactions, the popularity of the item, and the similarity between item features.

[0084] In the manifold fully connected layer, we must not only consider the effectiveness of aggregation, but also maintain the geometric structure of the manifold. This requires us to introduce constraints in the weight calculation and feature aggregation process so that the aggregation result still lies in the predefined manifold.

[0085] To achieve this, the following constraints can be used:

[0086] Constraint 1: Aggregated features must remain on the SPD manifold:

[0087]

[0088] Constraint 2: The sum of weights should be 1 to ensure the rationality of aggregation:

[0089]

[0090] In order to solve the above constraints, a projection method can be used to ensure that the aggregation result satisfies the properties of the manifold. This can be achieved through an optimization algorithm.

[0091] S5: Use the geometry-preserving mapping function to map the aggregated features into the Euclidean space to obtain the mapped features.

[0092] Construct a high-order geometric mapping function to ensure that the manifold space features retain geometric properties and nonlinear relationships when mapped back to Euclidean space, thereby reducing information loss. Traditional linear mapping methods often fail to capture the inherent structure of data when dealing with complex features, especially when there are nonlinear relationships between features. To this end, the present invention adopts a high-order mapping function and utilizes the geometric properties of the manifold to ensure that the nonlinear properties of the features can be effectively retained during the mapping process. The mapping function is defined as:

[0093]

[0094] Where: M represents the manifold space set. agg is the aggregate feature of the product. p is the mapping order, which controls the nonlinearity of the mapping. By choosing a suitable p, the complexity of the mapping can be adjusted, thereby affecting the model's ability to learn data features.

[0095] During the mapping process, the present invention focuses on the following aspects:

[0096] 1. Maintaining geometric properties: The mapping function G can map points on the manifold to Euclidean space while maintaining the geometric structure of the manifold. For example, for points on the SPD manifold, their positive definiteness and symmetry can still be maintained after mapping.

[0097] 2. Capturing nonlinear relationships: By selecting an appropriate p, the nonlinear relationship between product features can be effectively captured. When the p value is small, the mapping function is more sensitive to feature changes, thereby better capturing subtle nonlinear interactions.

[0098] After mapping, the feature expression is G(F agg). In order to ensure the validity of the mapping, we further process the mapped features. Consider the norm of the mapped features:

[0099]

[0100] S6: Process the mapped features to obtain long-tail product recommendation results.

[0101] Preferably, in order to effectively process the mapped features, especially the recommendation of long-tail products, the present invention introduces deep learning structures such as multi-layer perceptron (MLP) network, neural collaborative filtering (NCF) network and self-attention mechanism (such as Transformer) to process the mapped features. Specifically, the features input into these networks are the mapping feature norms obtained after geometric mapping. Through the multi-layer network layer, the model can automatically learn the complex nonlinear relationship between users and products, and predict the user's interest in long-tail products based on the mapped features.

[0102] S7: Calculate the total loss function based on the long-tail product recommendation results and the user-product interaction matrix, adjust the model parameters based on the total loss function, and obtain a trained long-tail product recommendation model.

[0103] The total loss function is the sum of the diversity constraint loss and the mapping loss; among them, the diversity constraint loss is calculated by the error balance loss and the overall diversity metric.

[0104] Diversity Constrained Loss:

[0105] The generalized Riemannian distance is the length of the shortest path between two points on the manifold. (i.e., SPD matrix), the generalized Riemannian distance can be calculated using the following formula:

[0106] d(F′ i ,F′ j )=||log(F′ i 1 / 2 F′ j F′ i 1 / 2 )|| F

[0107] Among them, ∥·∥ F represents the Frobenius norm, which is defined as:

[0108]

[0109] Here, A is an arbitrary matrix, a pq are the elements of the matrix A.

[0110] Considering F′ i and F′ j The symmetry and positive definiteness of F′ i 1 / 2 F′ j F′ i 1 / 2 , we can obtain the relative positions of the two on the SPD manifold. This method is particularly suitable for capturing the interaction between product features in complex environments. By introducing the generalized Riemannian distance, we can construct a more complex loss function to optimize the performance of the recommendation system. Combining the nonlinear relationship between features, we can define a new loss function as follows:

[0111]

[0112] in, Represents the error balance loss value, R u,i represents the actual rating of product i by user u, represents the predicted rating of user u for product i, λ3 represents the distance weight, d(F′ i ,F′ j ) represents the commodity SPD feature vector F′ i and F′ j Ω represents the set of user-item pairs, and I represents the set of item pairs.

[0113] When users browse products, they usually want to see different types of products rather than repeated recommendations of similar products. An effective diversity strategy can guide users to discover products they have not considered before, thereby increasing the possibility of purchase. Therefore, when optimizing the recommendation system, diversity constraints are introduced to ensure that the recommendation results not only meet the accuracy requirements, but also provide a variety of choices, thereby enhancing the user experience.

[0114] Diversity constraints:

[0115]

[0116] Where: D represents the overall diversity measure; F i and F j Represent the SPD feature vectors of product i and j respectively. Here, the distance d(F′ i ,F′ j ) can be calculated by the generalized Riemannian distance, ensuring that the nonlinear relationship between features is captured.

[0117] Through this design, the smaller the distance between features, the larger the corresponding diversity metric D, indicating that the diversity of the recommendation results is lower. Conversely, the larger the distance between features, the smaller the diversity metric D, reflecting that the diversity of the recommendation results is higher.

[0118] When optimizing the recommendation results, it is necessary to incorporate the diversity constraint into the loss function to balance the accuracy and diversity of the recommendations. The diversity constraint loss can be expressed as:

[0119]

[0120] Among them, L1 represents the diversity constraint loss value, represents the error balance loss value, γ represents the diversity weight, which is used to balance the relationship between accuracy and diversity; D represents the overall diversity measure.

[0121] Mapping loss:

[0122] In the process of mapping from manifold space to Euclidean space, the loss of feature information may lead to a decline in model performance. Therefore, it is crucial to ensure the integrity of information. Information preservation constraints evaluate the effectiveness of mapping by quantifying the difference between features before and after mapping, enhance the consistency of feature representation, and ensure information integrity during the mapping process.

[0123] The goal of the information preservation constraint is to minimize the difference between the original features and the inverse mapping features during the mapping process. The information preservation constraint, i.e., the mapping loss, is expressed as:

[0124]

[0125] Among them: F agg Represents the aggregate characteristics of the product. agg ) is the feature after high-order geometric mapping. G -1 is the function that maps the mapped features back to the manifold space. ∥·∥ F represents the Frobenius norm, which is used to measure the square root of the sum of the squares of the elements of the matrix. μ is the regularization coefficient, which is used to control the influence of the regularization term on the loss function.

[0126] The total loss function is the sum of the diversity constraint loss and the mapping loss. The model parameters are adjusted through the total loss function. When the maximum preset number of iterations is reached or the total loss function converges, the training is stopped, the model parameters are saved, and the trained long-tail product recommendation model is obtained. The user behavior data and product feature data are obtained and preprocessed, and the preprocessed data is input into the trained long-tail product recommendation model to obtain the long-tail product recommendation results.

[0127] In summary, the present invention converts the input high-dimensional sparse feature vector into a low-dimensional dense vector through the precise symmetric positive definite (SPD) manifold embedding method, retaining the geometric structure of the data; measures the association between users and products through a model based on generalized Riemannian distance, compresses data and extracts key information; combines low-rank approximation technology to improve the efficiency of data processing; introduces a manifold fully connected layer to simulate the complex interactive relationship between user and product features through a deep neural network; improves the richness of recommendation results through diversity constraints, and uses high-order geometric mapping methods to extract complex feature relationships; and improves the generalization ability of the model through information preservation and regularization constraints. Finally, the user interest features and product features are input into the prediction model to achieve personalized recommendations and advertising click-through rate prediction. The present invention achieves an increase in the click-through rate of e-commerce platform advertisements by optimizing the modeling of complex interest features behind user behavior and combining the invisible associations between non-temporal features, significantly enhancing the precision marketing and recommendation effects.

[0128] The above embodiments further illustrate the purpose, technical solutions and advantages of the present invention in detail. It should be understood that the above embodiments are only preferred implementation modes of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made to the present invention within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A long-tail product recommendation method based on precise symmetric positive manifold learning, characterized in that: include: Obtain user behavior data and product feature data and preprocess them, input the preprocessed data into the trained long-tail product recommendation model to obtain long-tail product recommendation results; The training process of the long-tail product recommendation model includes: S1: Obtain user behavior data and product feature data from the e-commerce platform and pre-process them to obtain a product feature matrix and a user-product interaction matrix; S2: Use low-rank approximation technology to optimize the user-item interaction matrix and update the user-item interaction matrix; S3: Map the product feature matrix into the symmetric positive definite matrix space and perform manifold embedding to obtain the SPD feature vector of the product; S4: Use the popular fully connected layer to aggregate the SPD feature vectors of the products to obtain aggregate features; S5: Use the geometry-preserving mapping function to map the aggregated features to the Euclidean space to obtain the mapped features; S6: Process the mapped features to obtain long-tail product recommendation results; S7: Calculate the total loss function based on the long-tail product recommendation results and the user-product interaction matrix, adjust the model parameters based on the total loss function, and obtain a trained long-tail product recommendation model.

2. The long-tail product recommendation method based on precise symmetric positive manifold learning according to claim 1 is characterized in that: The user-product interaction matrix is ​​optimized as follows: Among them, R represents the user-item interaction matrix, λ lr represents the low-rank weight, R k Represents a low-rank approximation of the user-item interaction matrix.

3. The long-tail product recommendation method based on precise symmetric positive manifold learning according to claim 1 is characterized in that: The SPD matrix is ​​aggregated using a popular fully connected layer and expressed as: Among them, F agg represents the aggregated feature, k represents the number of product features, and w i represents the weight of the i-th product, F i ′ Represents the SPD feature vector of the i-th product.

4. The long-tail product recommendation method based on precise symmetric positive manifold learning according to claim 1 is characterized in that: The total loss function is the sum of the diversity constraint loss and the mapping loss; among them, the diversity constraint loss is calculated by the error balance loss and the overall diversity metric.

5. The long-tail product recommendation method based on precise symmetric positive manifold learning according to claim 4 is characterized in that: The error balance loss is expressed as: in, Represents the error balance loss value, R u,i represents the actual rating of product i by user u, represents the predicted rating of user u for product i, λ3 represents the distance weight, d(F i ′ ,F j ′ ) represents the product SPD feature vector F i ′ With F j ′ Ω represents the set of user-item pairs, and I represents the set of item pairs.

6. The long-tail product recommendation method based on precise symmetric positive manifold learning according to claim 4 is characterized in that: The overall diversity measure is expressed as: Where D represents the overall diversity measure, d(F i μ ,F j ′ ) represents the commodity SPD feature vector F i ′ With F j ′ The distance between.

7. The long-tail product recommendation method based on exact symmetric positive manifold learning according to claim 4 is characterized in that: The diversity constraint loss is expressed as: Among them, L1 represents the diversity constraint loss value, represents the error balance loss value, β represents the diversity weight, and D represents the overall diversity measure.

8. The long-tail product recommendation method based on precise symmetric positive manifold learning according to claim 4 is characterized in that: The mapping loss is expressed as: Among them, L2 represents the mapping loss, F agg represents the aggregation feature, G(F agg ) represents the commodity aggregation feature F agg The mapping feature obtained after mapping, G -1 is a function that maps the mapped features back to the manifold space, and μ represents the regularization coefficient.

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