Precision marketing data analysis method based on deep matrix decomposition
By combining deep matrix factorization and variational autoencoders, the problems of data sparsity and anonymous users in consumer profiling technology are solved, enabling more accurate product recommendations and marketing strategies, and improving the scalability and predictive performance of the recommendation system.
Patent Information
- Application Number
- CN202310392558.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-04-13
- Publication Date
- 2026-01-23
- Estimated Expiration
- 2043-04-13
AI Technical Summary
Existing consumer profiling technologies suffer from problems such as data sparsity and the high maintenance requirements of customer similarity matrices, which limit the accuracy and versatility of product recommendations and marketing, especially for anonymous users who cannot extract hidden attributes.
By using a preference prediction model based on deep matrix factorization and variational autoencoder, linear and nonlinear correlation information between customers and products is extracted. Collaborative filtering is used for matrix factorization, and variational autoencoder is combined to learn nonlinear correlation information to construct a precise marketing strategy.
It improves the accuracy and scalability of product recommendations, solves the problems of data sparsity and cold start, can more effectively predict customer preferences, and enhances the precision of marketing strategies.
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Figure CN116402536B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of data analysis, and more particularly to a method for precision marketing data analysis based on deep matrix factorization. Background Technology
[0002] AI-powered precision marketing can provide merchants with real-time records and feedback data of customer browsing, helping them understand consumers and push product information based on consumer needs and preferences to achieve the best marketing results.
[0003] Patent CN114723491A discloses a precision marketing data analysis method and system based on user profiling and data mining, and patent CN114579544A discloses a precision marketing data analysis method based on a data management platform. Both are based on consumer profiling technology.
[0004] However, consumer profiling technology suffers from problems such as data sparsity and the high maintenance requirements of customer similarity matrices. For a small number of anonymous users whose accurate data cannot be obtained, it is impossible to extract the hidden attributes of customers and products, which limits the accuracy and versatility of product recommendations and marketing. Summary of the Invention
[0005] This invention provides a precise marketing data analysis method based on deep matrix factorization, which extracts the linear and nonlinear hidden features between customers and products through data analysis, and accurately predicts customer preferences.
[0006] In a first aspect, embodiments of the present invention provide a method for precise marketing data analysis based on deep matrix factorization, including:
[0007] Obtain the customer-product interaction matrix, wherein each row of the interaction matrix corresponds to each customer, each column corresponds to each product, and the matrix elements contain customer-product interaction information;
[0008] The interaction matrix is decomposed based on the collaborative filtering method to obtain the latent vectors of each customer and each product. The latent vectors contain the linear association information between customers and products.
[0009] The vector pairs formed by the latent vectors of each customer and each product are input into the preference prediction model based on variational autoencoder for training. The model learns the nonlinear correlation information between customers and products. The trained model is used to output the preference of each customer for each product.
[0010] Using a trained preference prediction model, we can recommend precise marketing strategies.
[0011] Secondly, embodiments of the present invention also provide an electronic device, the electronic device comprising:
[0012] One or more processors;
[0013] Memory, used to store one or more programs.
[0014] When the one or more programs are executed by the one or more processors, the one or more processors implement the precision marketing recommendation method based on deep matrix factorization as described in any embodiment.
[0015] Thirdly, embodiments of the present invention also provide a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the precision marketing recommendation method based on deep matrix factorization as described in any embodiment.
[0016] This invention addresses three problems in recommendation systems: data sparsity, cold start, and scalability. It proposes a data processing method based on matrix factorization and deep learning. Matrix factorization reduces data dimensionality and extracts hidden information from customer-product interactions, learning the linear correlation between customers and products. A Variational Autoencoder (VAE) applies nonlinear factors through probability distributions to learn the nonlinear relationship between customers and products, further improving the accuracy of prediction results. In fuzzy contexts, this embodiment can extract more hidden information from nonlinear data, resulting in more effective predictions. Attached Figure Description
[0017] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.
[0018] Figure 1 This is a flowchart of a precision marketing recommendation method based on deep matrix factorization provided in an embodiment of the present invention.
[0019] Figure 2 This is a schematic diagram illustrating the specific structure and data processing of the VAE-based preference prediction model provided in this embodiment of the invention.
[0020] Figure 3 This is a schematic diagram of the structure of an electronic device provided in an embodiment of the present invention. Detailed Implementation
[0021] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention.
[0022] In the description of this invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.
[0023] In the description of this invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.
[0024] Figure 1 This is a flowchart illustrating a precision marketing recommendation method based on deep matrix factorization, provided in an embodiment of the present invention. This method is suitable for situations where product marketing strategies are determined through data analysis and is executed by an electronic device. Figure 1 As shown, the method specifically includes:
[0025] S110. Obtain the interaction matrix between customers and products, wherein each row of the interaction matrix corresponds to each customer, each column corresponds to each product, and the matrix elements contain the interaction information between customers and products.
[0026] The interaction matrix is derived from the customer database, product data and database, as well as the customer-product interaction database, and clearly reflects the customer-product interaction information in matrix form.
[0027] Optionally, customer-product interaction data can be extracted from the customer database and marketing database, and the interaction data can be segmented according to at least one of gender, age, average monthly spending, and marital status.
[0028] Next, remove temporary information, images, and videos from the segmented data. For example, missing values can be added or removed to clean the original data, removing traces of temporary data and enabling error logging and request interface backup. When a customer accesses certain images or videos, the data occupies a significant amount of storage space, and this information is automatically downloaded the next time the webpage is viewed. This data has no reference value and affects the efficiency and quality of recommendations, so it is also removed.
[0029] Finally, a customer-product interaction matrix is constructed from the cleaned data. Each row of the matrix corresponds to a customer, and each column corresponds to a product. The matrix elements represent the interaction information between the customer and the product. At this point, the interaction matrix is relatively sparse. When the system only records explicit user feedback, it will form a highly sparse matrix.
[0030] S120. The interaction matrix is decomposed based on the collaborative filtering method to obtain the latent vectors of each customer and each product. The latent vectors contain linear association information between customers and products.
[0031] Considering the sparsity of the interaction matrix, this step reduces its dimensionality using matrix factorization in a collaborative filtering algorithm. Matrix factorization decomposes the interaction matrix into smaller matrices, transforming the original high-dimensional and sparse features into low-dimensional latent vectors.
[0032] In the smaller matrix obtained after decomposition, each row represents a customer's latent vector. Each column represents the latent vector of a product. C| represents the total number of customers, and |D| represents the total number of products. The latent vectors represent the hidden features of customers and products, and are their respective representations in the latent space. Since matrix factorization is a linear operation, the latent vectors contain information about the linear relationships between customers and products.
[0033] S130. Input the vector pairs formed by the latent vectors of each customer and each product into the preference prediction model based on variational autoencoder for training, learn the nonlinear correlation information between customers and products, and use the trained model to output the preference of each customer for each product.
[0034] Since the relationship between customers and products is usually non-linear and has deeper and more complex relationships, this step uses a deep learning model based on variational autoencoders to learn the non-linear association information between customers and products in order to improve the accuracy and success rate of recommendations.
[0035] Variational autoencoders (VAEs) can not only improve latent factor-based models but also manage nonlinear probabilistic latent variable models. They are widely used in image generation but have not yet been applied in precision marketing. This embodiment notes the similarity between the probability-based variable generation method of VAEs and the dependency modeling between goods in a customer sequence. A preference prediction model is constructed based on VAEs. The model input is a vector pair consisting of latent vectors of customers and goods, and the model output is the customer's preference for the goods.
[0036] Optionally, the specific structure of the model is as follows: Figure 2 As shown, it includes an optimized VAE and a preference prediction module. The optimized VAE is used to reconstruct the input vector pairs so that the reconstructed vectors include both linear and non-linear correlation information between products and customers. The preference prediction module is used to calculate the customer's preference for the products based on the reconstructed vector pairs.
[0037] A conventional Visual Encoding Engine (VAE) consists of an encoding part and a decoding part. The encoding part maps the input data X to the latent space Z, and the encoding process can be represented as f: X × α → Z. The decoding part recovers the input data X based on the distribution of the latent space, and the decoding process can be represented as Z × β → X. However, conventional VAEs cannot guarantee the regularity of the latent space and cannot generate new content from random encoding. Without explicit regularization, some combinations in the latent space become meaningless once decoded. To overcome this deficiency, this embodiment optimizes the conventional VAE by introducing a random component in the generation process β of the decoding part. This enriches the latent space and variability of the generative model, and the mined features are input into the decoder to reconstruct the data representation X. The reconstructed X will then contain richer information, which is more conducive to improving prediction accuracy.
[0038] Based on the above principles, in one specific implementation, the model training process includes the following steps:
[0039] Step 1: Input the latent vectors of any customer and any item into the encoding part of the variational autoencoder to obtain a latent vector. Optionally, one-hot encoding is used to encode the latent vectors of the customer and item into canonical basis vectors; the canonical basis vectors of the customer and item are then input into the encoding part of the variational autoencoder to obtain a latent vector. For example, the c-th customer is encoded as the c-th canonical basis vector e. c The d-th product code is the d-th canonical basis vector e. d .
[0040] Step 2: Construct Gaussian distributions for the customer and product dimensions, respectively, centered on each element of the latent vector. To ensure the regularity of the latent space, this step starts from the two basic information dimensions of customers and products, representing the latent vector as a joint normal distribution of the customer and product dimensions. Specifically, a Gaussian prior distribution with a non-zero mean and a precision matrix are placed on the latent vectors of customers and products, forming a multivariate normal distribution. The latent space is the space formed by this multivariate normal distribution. For example, the latent space of customers and products can be represented as (u c ,v d )∈R C ×R D To avoid overfitting, the Kullback-Leibler divergence can be used to balance the latent space. The balancing process can be expressed as: KL(p β (Z|X)||q α (Z))=E[q α (Z)]-E[p β (Z|X)].
[0041] Step 3: Randomly sample the Gaussian distributions for both the customer and product dimensions, obtaining two random vectors. This step uses the aforementioned multivariate normal distribution to randomly sample the latent space Z to obtain other latent information between customers and products. Specifically, the Gaussian distributions for the customer and product dimensions form two separate dense layers. These two separate dense layers return the mean and variance of the two Gaussian multivariate distributions, thus the latent space can be represented as... Where K represents the dimension of the latent space (i.e., the number of elements in the latent vector), μ1(u c ), μ2(v d Let ) represent the means of two Gaussian distributions for the customer dimension and the product dimension, respectively. Let represent the variances of two Gaussian distributions for customers and goods, respectively. Based on these Gaussian distributions, a space of two random vectors can be formed: Where N(μ, ∑) represents the K-dimensional multivariate normal distribution μ of the mean vector and the diagonal covariance matrix ∑, and its probability density function is: This pair of random vectors represents the association information between random customers and random goods. Sampling is performed on the spaces P and Q of this pair of random vectors to form another pair of random vectors. The sampling process can be represented as Variational(uc,vd)=(p,q)∈R K ×R K ,sample sample
[0042] In one specific implementation, the process of generating two random vectors includes:
[0043] Using each element of the potential vector as a unit, perform the following operations: S1-1, determine the mean and variance of the Gaussian distribution of the customer dimension centered on the current element; S1-2, obtain the probability density function of the Gaussian distribution based on the mean and variance; S1-3, perform random sampling on the Gaussian distribution based on the probability density function. The random sampled values corresponding to each element constitute a random vector of the customer dimension.
[0044] Simultaneously, taking each element of the potential vector as a unit, the following operations are performed: S2-1, determine the mean and variance of the Gaussian distribution of the product dimension centered on the current element; S2-2, obtain the probability density function of the Gaussian distribution based on the mean and variance; S2-3, perform random sampling on the Gaussian distribution based on the probability density function. The random sampled values corresponding to each element constitute a random vector of the product dimension.
[0045] The random vectors for the customer dimension and the random vectors for the product dimension are sample p and sample q, respectively.
[0046] Step 4: Input the two random vectors into the decoding part of the variational autoencoder to reconstruct the latent vectors of the customer and the product. The reconstructed latent vectors contain linear and nonlinear correlation information between the customer and the product.
[0047] Step 5: Based on the reconstructed latent vectors of the customer and the product, predict the customer's preference for the product. Optionally, calculate the dot product of the customer's reconstructed latent vector and the product's reconstructed latent vector as the customer's preference for the product.
[0048] Step Six: Adjust model parameters based on feedback by minimizing the difference between the predicted and labeled preferences. The training function aims to minimize the error. The goal is to minimize the error between the actual and predicted values, where h(e) u ,e i ) represents the latent vector predicted by the model as e. u The client has a latent vector of e i The preference level of the product, r represents the marked preference level of the customer for the product; It can be adopted Alternatively, it can be used x and y represent arbitrary variables, corresponding to h(e) respectively. u ,e i ) and r.
[0049] Repeat steps one through six for each vector pair to gradually complete model training. The trained model can predict the preference of any customer for any product. It should be noted that steps two and three are the core of optimizing VAE compared to conventional VAE. They ensure the regularity of the latent space from the two dimensions of customers and products, and enrich the latent feature information through random sampling of the latent space, making the reconstructed data information more comprehensive and further improving the prediction accuracy.
[0050] After model training is complete, offline evaluation metrics are used to assess the results. These metrics include Precision@R (accuracy), Recall@R (recall rate), and HitRate@R (whether the customer's target selection is in the top-R recommendation list).
[0051]
[0052]
[0053]
[0054] Wherein, RecSet represents the sequence of products recalled by the strategy, BehSet represents the sequence of products that the customer has interacted with in the next N days, RecSet∩BecSet represents the intersection of RecSet and BehSet, count(RecSet∩BecSet) represents the total number of elements in RecSet∩BehSet, count(RecSet) represents the total number of elements in RecSet, and count(BehSet) represents the total number of elements in BehSet; {predictTopR} represents the set of products in the top-R recommendation list, {click} represents the set of products that the customer has clicked, {predictTopR}∩{click} represents the intersection of {predictTopR} and {click}, and I{predictTopR}∩{click} represents the number of times {predictTopR} and {click} have an intersection.
[0055] S140. Utilize the trained preference prediction model to recommend precise marketing strategies.
[0056] The trained model is scalable and can be extended to any customer and any product, solving the cold start problem in precision marketing. Optionally, the latent vectors of the target customer and the target product are input into the trained preference prediction model to predict the target customer's preference for the target product; based on the preference, a precision marketing strategy is determined.
[0057] This embodiment addresses three issues in recommender systems: data sparsity, cold start, and scalability. It proposes a data processing method based on matrix factorization and deep learning. Matrix factorization reduces data dimensionality and extracts hidden information from customer-product interactions, learning the linear correlation between customers and products. A Virtual Object (VAE) applies a nonlinear factor through probability distribution to learn the nonlinear relationship between customers and products, further improving the accuracy of prediction results. In fuzzy contexts, this embodiment can extract more hidden information from nonlinear data, resulting in more effective predictions. Specifically, during the training phase, the VAE forces the distributions of both customers and products to approximate a centered and reduced standard normal distribution, regularizing the latent vector distributions of customers and products and ensuring good properties in their latent space. Random sampling of the regularized distribution is then used as input to the decoder, introducing richer random information into the decoding process, improving the accuracy and generalization of data reconstruction, and enabling more convenient generalization of precise marketing predictions.
[0058] Figure 3 This is a schematic diagram of the structure of an electronic device provided in an embodiment of the present invention, such as... Figure 3 As shown, the device includes a processor 60, a memory 61, an input device 62, and an output device 63; the number of processors 60 in the device can be one or more. Figure 3 Taking a processor 60 as an example; the processor 60, memory 61, input device 62, and output device 63 in the device can be connected via a bus or other means. Figure 3 Taking the example of a connection between China and Israel via a bus.
[0059] The memory 61, as a computer-readable storage medium, can be used to store software programs, computer-executable programs, and modules, such as the program instructions / modules corresponding to the precision marketing recommendation method based on deep matrix factorization in this embodiment of the invention. The processor 60 executes various functional applications and data processing of the device by running the software programs, instructions, and modules stored in the memory 61, thereby implementing the aforementioned precision marketing recommendation method based on deep matrix factorization.
[0060] The memory 61 may primarily include a program storage area and a data storage area. The program storage area may store the operating system and at least one application program required for a given function; the data storage area may store data created based on terminal usage. Furthermore, the memory 61 may include high-speed random access memory and non-volatile memory, such as at least one disk storage device, flash memory, or other non-volatile solid-state storage device. In some instances, the memory 61 may further include memory remotely located relative to the processor 60, which can be connected to the device via a network. Examples of such networks include, but are not limited to, the Internet, intranets, local area networks, mobile communication networks, and combinations thereof.
[0061] Input device 62 can be used to receive input digital or character information, and to generate key signal inputs related to user settings and function control of the device. Output device 63 may include display devices such as a display screen.
[0062] This invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the precision marketing recommendation method based on deep matrix factorization of any embodiment.
[0063] The computer storage medium of this invention can be any combination of one or more computer-readable media. A computer-readable medium can be a computer-readable signal medium or a computer-readable storage medium. A computer-readable storage medium can be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof. More specific examples of computer-readable storage media (a non-exhaustive list) include: an electrical connection having one or more wires, a portable computer disk, a hard disk, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage device, magnetic storage device, or any suitable combination thereof. In this document, a computer-readable storage medium can be any tangible medium that contains or stores a program that can be used by or in conjunction with an instruction execution system, apparatus, or device.
[0064] Computer-readable signal media may include data signals propagated in baseband or as part of a carrier wave, carrying computer-readable program code. Such propagated data signals may take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination thereof. Computer-readable signal media may also be any computer-readable medium other than computer-readable storage media, capable of sending, propagating, or transmitting programs for use by or in connection with an instruction execution system, apparatus, or device.
[0065] Program code contained on a computer-readable medium may be transmitted using any suitable medium, including but not limited to wireless, wire, optical fiber, RF, etc., or any suitable combination thereof.
[0066] Computer program code for performing the operations of this invention can be written in one or more programming languages or a combination thereof. Programming languages include object-oriented programming languages—such as Java, Smalltalk, and C++—as well as conventional procedural programming languages—such as C or similar programming languages. The program code can be executed entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In cases involving remote computers, the remote computer can be connected to the user's computer via any type of network, including a local area network (LAN) or a wide area network (WAN), or it can be connected to an external computer (e.g., via the Internet using an Internet service provider).
[0067] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the technical solutions of the embodiments of the present invention.
Claims
1. A precision marketing data analysis method based on deep matrix factorization, characterized in that, include: Obtain the customer-product interaction matrix, wherein each row of the interaction matrix corresponds to each customer, each column corresponds to each product, and the matrix elements contain customer-product interaction information; The interaction matrix is decomposed based on the collaborative filtering method to obtain the latent vectors of each customer and each product. The latent vectors contain the linear association information between customers and products. The vector pairs formed by the latent vectors of each customer and each product are input into the preference prediction model based on variational autoencoder for training. The model learns the nonlinear correlation information between customers and products. The trained model is used to output the preference of each customer for each product. Using a trained preference prediction model, we can recommend precise marketing strategies. The training process includes: inputting the latent vectors of any customer and any product into the encoding part of a variational autoencoder to obtain a latent vector; constructing Gaussian distributions for the customer dimension and product dimension, respectively, with each element of the latent vector as the center; randomly sampling each Gaussian distribution for the customer dimension and product dimension to obtain two random vectors; inputting the two random vectors into the decoding part of the variational autoencoder to reconstruct the latent vectors of the customer and the product, the reconstructed latent vectors containing linear and non-linear association information between the customer and the product; predicting the customer's preference for the product based on the customer's and the reconstructed latent vectors, and the product's and the reconstructed latent vectors; and adjusting the model parameters by minimizing the difference between the predicted preference and the labeled preference.
2. The method according to claim 1, wherein inputting the latent vectors of any customer and any product into the encoding part of the variational autoencoder to obtain a latent vector comprises: One-hot encoding is used to encode the latent vectors of the customers and products into canonical basis vectors; The canonical basis vectors of the customer and the product are input into the encoding part of the variational autoencoder to obtain a latent vector.
3. The method according to claim 1, wherein randomly sampling each Gaussian distribution of the customer dimension and the product dimension to obtain two random vectors includes: Perform the following operations on an element-by-element basis: S1-1. Determine the mean and variance of the Gaussian distribution of the customer dimension centered on the current element; S1-2. Based on the mean and variance, obtain the probability density function of the Gaussian distribution; S1-3. Randomly sample the Gaussian distribution according to the probability density function; A random vector for the customer dimension is formed by the random sampled values corresponding to each element.
4. The method according to claim 1, before randomly sampling each Gaussian distribution of the customer dimension and the product dimension to obtain two random vectors, further includes: The Kullback-Leibler divergence is used to balance the latent vector space formed by each Gaussian distribution to obtain the final latent vector space.
5. The method according to claim 1, characterized in that, The step of predicting the customer's preference for the product based on the customer's reconstructed latent vector and the product's reconstructed latent vector includes: Calculate the dot product of the reconstructed latent vector of the customer and the reconstructed latent vector of the product, and use it as the customer's preference for the product.
6. The method according to claim 1, characterized in that, The method of recommending precise marketing strategies using a trained preference prediction model includes: The latent vectors of the target customer and the target product are input into the trained preference prediction model to predict the target customer's preference for the target product. Based on the stated preferences, a targeted marketing strategy will be recommended.
7. The method according to claim 1, characterized in that, The acquisition of the customer-product interaction matrix includes: Extract customer-product interaction data from customer and marketing databases; Data is segmented according to at least one of the following: gender, age, average monthly spending, and marital status; Remove temporary information, images, and videos from the segmented data; A customer-product interaction matrix is constructed from the cleaned data.
8. An electronic device, characterized in that, include: One or more processors; Memory, used to store one or more programs. When the one or more programs are executed by the one or more processors, the one or more processors implement the precision marketing data analysis method based on deep matrix factorization as described in any one of claims 1-7.
9. A computer-readable storage medium, characterized in that, It stores a computer program that, when executed by a processor, implements the precision marketing data analysis method based on deep matrix factorization as described in any one of claims 1-7.
Citation Information
Patent Citations
Personalized recommendation method based on linear variational auto-encoder
CN114238779A