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

Abstract

The application belongs to the technical field of electronic commerce and specifically relates to a long-tail commodity recommendation method based on accurate symmetric positive definite manifold learning; comprising: obtaining user behavior data and commodity feature data of an e-commerce platform and preprocessing the same to obtain a commodity feature matrix and a user commodity interaction matrix; using a low-rank approximation technology to optimize the user commodity interaction matrix 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; using a popular full connection layer to perform aggregation processing on the SPD feature vector of the commodity to obtain an aggregated feature; using a geometric preserving mapping function to map the aggregated feature to a Euclidean space to obtain a mapped feature; and processing the mapped feature to obtain a long-tail commodity recommendation result; the application realizes the improvement of the advertisement click rate of the e-commerce platform and significantly enhances the effect of accurate marketing and recommendation.

Description

Technical Field

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

[0002] E-commerce recommendation systems are prone to the phenomenon of long-tail distributions of product features. Traditional recommendation algorithms, such as neural network-based models, are unable to effectively handle long-tail product recommendations. This is because these traditional neural network models fail to fully capture the complex nonlinear relationships between product features when dealing with long-tail data. They encode items and users in Euclidean space, a flat geometric structure with polynomial expansion capacity. When faced with data in Euclidean space, people often tend to use the scale within this space for evaluation. However, this approach is not always appropriate. Such thinking creates a sense of limitation, as if analysis is restricted to a specific space and scale, while ignoring the diversity and complexity of objects. In fact, long-tail data can be traced back to hierarchical structures with an exponentially growing number of items. When encoding this data in Euclidean space, this imbalance makes it difficult to express relationships between the data through embedding vectors, thereby reducing the accuracy of the final prediction. Therefore, how to improve the recommendation performance of long-tail products in recommendation systems while maintaining recommendation diversity and meeting personalized needs has become a hot topic of research.

[0003] In recent years, recommendation system methods based on manifold learning have 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, symmetric positive definite (SPD) manifolds, as typical non-Euclidean spaces, can more effectively describe the geometric characteristics of high-dimensional data. Although manifold learning methods have made certain 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 faces challenges:

[0004] 1. Diversity in Feature Representation in Manifold Spaces: While improving recommendation accuracy, effectively enhancing the inherent variability of recommendation results and the close relationship between different product feature spaces remains a major challenge. Traditional fully connected layers cannot fully utilize the structural characteristics of manifold geometry, limiting the performance of recommendation systems.

[0005] 2. Information loss during 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 characteristics and high-dimensional structural information is an important challenge. Summary of the Invention

[0006] In response to the shortcomings of the existing technology, 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, then input the obtained product feature vector 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 preprocess 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 a symmetric positive definite matrix space and perform manifold embedding to obtain the product’s SPD feature vector;

[0012] S4: Use the popular fully connected layer to aggregate the feature vectors of SPD products to obtain aggregated 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 features, k represents the number of product features, 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 user u on product i, 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 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 metric, d(F′ i ,F′ j ) represents the product 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 aggregate feature, G(F agg ) represents the product 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: by introducing SPD manifold learning and utilizing its ability to capture the geometric structure of data in non-Euclidean space, the present invention accurately describes the intrinsic relationships 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, thereby improving the accuracy of recommendations while enhancing the intrinsic variability of recommendation results, effectively balancing 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 Euclidean space, thereby improving the robustness and generalization ability of the recommendation model. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0037] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. 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 making creative efforts are within the scope of protection of the present invention.

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

[0039] Obtain user behavior data and preprocess it, then input the product feature vector obtained after preprocessing into the trained long-tail product recommendation model to obtain the long-tail product recommendation results.

[0040] like Figure 1 As shown in Figure 2, 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 the product feature matrix and user-product interaction matrix.

[0042] User behavior data includes user u i For product i j The actual rating 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, collections, purchase times, and other behaviors.

[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 ontology 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 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, 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. Its basic idea is to represent high-dimensional data as a combination of low-rank matrices through matrix decomposition technology. This not only reduces data storage requirements but also improves data interpretability.

[0053] Singular value decomposition (SVD) can be used to achieve low-rank approximation. Given a matrix R, it can be decomposed into:

[0054] R≈UΣV T

[0055] Where 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 not only helps us reduce the amount of computation, but also removes 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; λ lr Represents the low-rank weight, controlling the influence of low-rank approximation; R k Represents a 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 means matrix F′ i Is a positive definite matrix. To ensure that the feature matrix satisfies the SPD property, the original product feature matrix is ​​processed as follows, that is, the product feature matrix is ​​mapped into the space of symmetric positive definite matrices (SPD matrices):

[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 characteristic 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. This property ensures the stability of the matrix in various applications.

[0072] In order to effectively capture the nonlinear relationship between product features, this paper 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 Represents the matrix F i The square root of . Through the mapping here, the following points are ensured:

[0077] 1. Maintain positive definiteness: SPD eigenvector 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 aggregated 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, allowing the recommendation system to more accurately capture user preferences. The feature aggregation formula is expressed as:

[0081]

[0082] Among them, F agg represents the aggregated features, k represents the number of product features, 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 product, and the similarity between product features.

[0084] In the manifold fully connected layer, not only the effectiveness of aggregation must be considered, but also the geometric structure of the manifold must be maintained. 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 the weights should be 1 to ensure the rationality of aggregation:

[0089]

[0090] To address 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 optimization algorithms.

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

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

[0093]

[0094] Where: M represents the manifold space set. F agg is the aggregated feature of the product. p is the mapping order, which controls the degree of nonlinearity in the mapping. By choosing an appropriate p, we can adjust the complexity of the mapping, 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. Preserving geometric properties: The mapping function G can map points on the manifold to Euclidean space while preserving the geometric structure of the manifold. For example, for points on the SPD manifold, their positive definiteness and symmetry can still be preserved after mapping.

[0097] 2. Capturing nonlinear relationships: By selecting an appropriate p, we can effectively capture nonlinear relationships between product features. When the p value is small, the mapping function becomes more sensitive to feature changes, thus 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) networks, neural collaborative filtering (NCF) networks, and self-attention mechanisms (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 users' 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′ are calculated by 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 more complex loss functions to optimize the performance of the recommendation system. Incorporating 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 user u on product i, 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 and F′ j Ω represents the set of user-item pairs, and I represents the set of item pairs.

[0113] When browsing products, users typically prefer to see different types of products rather than duplicate recommendations for similar items. An effective diversity strategy can guide users to discover products they might not have considered, thereby increasing their likelihood of purchase. Therefore, when optimizing recommendation systems, incorporating diversity constraints ensures that recommendations are not only accurate but also offer a diverse selection, 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 using the generalized Riemannian distance to ensure 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 a higher diversity of the recommendation results.

[0118] When optimizing recommendation results, it is necessary to incorporate diversity constraints into the loss function to balance the accuracy and diversity of 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] During the mapping process from manifold space to Euclidean space, loss of feature information can lead to degraded model performance. Therefore, ensuring information integrity is crucial. Information-preserving constraints quantify the difference between features before and after mapping to evaluate the effectiveness of the 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. G(F 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 impact 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 using this total loss function. When the maximum number of iterations is reached or the total loss function converges, training is stopped, the model parameters are saved, and the trained long-tail product recommendation model is obtained. User behavior data and product feature data are obtained and preprocessed. This preprocessed data is then fed into the trained long-tail product recommendation model to obtain 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, 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-time series 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 plans 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 scope of protection of the present invention.

Claims

1. A long-tail product recommendation method based on exact symmetric positive manifold learning, characterized by: include: Obtain user behavior data and product feature data and preprocess them, then 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 preprocess 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 use the manifold embedding mapping function to perform manifold embedding processing to obtain the product's SPD feature vector; the manifold embedding mapping function is expressed as: Among them, F′ i represents the SPD feature vector after manifold embedding mapping, Represents the feature matrix F of the i-th product i The square root of ; Φ() represents the mapping function; S4: Use the popular fully connected layer to aggregate the SPD feature vectors of the products to obtain aggregated 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 exact 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 exact symmetric positive manifold learning according to claim 1 is characterized in that: The popular fully connected layer is used to aggregate the SPD matrix and express it as: Among them, F agg represents the aggregated features, k represents the number of product features, 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 exact symmetric positive manifold learning according to claim 1, 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 exact 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 user u on product i, 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 and 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 exact symmetric positive manifold learning according to claim 4, characterized in that: The overall diversity measure is expressed as: Where D represents the overall diversity metric, d(F′ i ,F′ j ) represents the product SPD feature vector F′ i and 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 exact 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 aggregate feature, G(F agg ) represents the product 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.

Images