CN112861006A - Recommendation method and system fusing meta-path semantics - Google Patents

Abstract

The invention provides a recommendation method fusing meta-path semantics, which comprises the following steps: calculating a relative importance bipartite graph of the user/commodity based on a heterogeneous information network and a user/commodity scoring data set; guiding information propagation between user/commodity nodes by using the relative importance bipartite graph to obtain embedded expression of the user/commodity nodes; and transforming the embedded expression of the user/commodity node to obtain a final embedded expression of the user/commodity node for recommendation, and inputting the final embedded expression of the user/commodity node into a recommendation model to obtain the prediction score of the user/commodity.

Description

Recommendation method and system for fusion of meta-path semantics

technical field

The invention relates to the technical field of computer networks, in particular to a method and system for recommending semantic fusion meta-paths.

Background technique

With the rapid development of the Internet, Internet users are faced with a large amount of information every day. In the absence of a clear purpose for the user, selecting products that may be of interest to the user for recommendation can greatly stimulate the potential interest of the user and help the user. Efficient decision-making by users in the case of information overload has become an important growth engine for the development of the Internet. Therefore, recommender systems play an increasingly important role in various Internet products.

Although many algorithms have been successfully applied to recommender systems, most of them focus on learning recommendation policies from single user-item interaction records (or plus simple content information). This will inevitably face the problems of data sparse, cold start, over-recommendation, etc. In recent years, some studies have pointed out that the use of other relationship data between users and users and between users and commodities can help recommender systems solve one or more of the above problems; some studies introduce social networks into recommender systems to solve The problem caused by the sparse data of some user transaction records; in order to further utilize more complex and heterogeneous user/commodity relationship data, some researches propose to introduce heterogeneous information network into the recommender system. The heterogeneous information network contains multiple types of nodes, and the edges between different types of nodes contain different types of semantics. Common examples are bibliographic information networks, social media networks, protein networks, and more. In recent years, the concept of meta-path has been proposed to represent the rich semantic information in heterogeneous information networks. A meta-path is a sequence composed of different types of nodes. Each meta-path records the path from the initial node to the end node. The edge type and node type of . Different meta-paths contain different semantic information, so introducing such rich semantic information into the recommender system is expected to solve the problems of data sparseness, cold start, and over-recommendation in the recommender system.

Introducing the meta-path into the propulsion system, since there are multiple meta-paths in the heterogeneous information network, how to effectively integrate the multiple meta-paths and extract consistent information from them for downstream tasks is the key to utilizing the meta-path. Existing methods employ assigning different weights to different paths, and then fuse meta-path semantics into regularization terms, prediction functions, or user/item latent representation modules, respectively. Although they have achieved good results, these methods only simply fuse different meta-paths through weight coefficients, and the model has limited ability to express the semantics of different meta-paths. In addition, there is little work on how to use meta-path semantics and node content/attribute information simultaneously for node embedding learning.

SUMMARY OF THE INVENTION

In view of the deficiencies of the prior art, the present invention proposes a recommendation method and system for integrating meta-path semantics, which can fuse semantic information contained in different meta-paths to obtain consistent semantic information, which can be applied to user/commodity embedding learning or recommended strategy.

In order to achieve the above object, the present invention proposes a recommendation method integrating meta-path semantics, including: calculating the relative importance bipartite graph of the user/product based on a heterogeneous information network and a user/product scoring data set; using the The relative importance bipartite graph guides the information dissemination between user/commodity nodes, and obtains the embedded expression of the user/commodity node; transforms the embedded expression of the user/commodity node to obtain the recommended The final embedded expression of the user/commodity node, and the final embedded expression of the user/commodity node is input into a recommendation model to obtain the prediction score of the user/commodity.

In order to achieve the above object, the present invention also proposes a recommendation system that integrates meta-path semantics, including: a meta-path early fusion module, which is used to calculate the relative user/commodity based on heterogeneous information networks and user/commodity scoring datasets. Importance bipartite graph; meta-path semantic embedding module for using the relative importance bipartite graph to guide information dissemination between user/commodity nodes to obtain the embedded expression of the user/commodity nodes; recommendation prediction module , which is used to transform the embedded expression of the user/commodity node to obtain the final embedded expression of the user/commodity node for recommendation, and input the final embedded expression of the user/commodity node into a recommendation model Get the predicted score for the user/item.

It can be seen from the above scheme that the advantages of the present invention are: the similarity based on different meta-paths is fused at an early stage to obtain consistent semantics, which is directly applied to the expression learning process of users/items; By taking meta-path-based semantics as the attention score between nodes, the learned embedding can preserve both the node content information and the topology information between nodes. In addition, the present invention proposes a broader recommendation model, and users can design the specific implementation of each module by themselves, such as calculating the similarity matrix based on meta-paths in different ways, using different structures to perform early fusion of multiple meta-paths, etc. , the model is simple and easy to build. Meanwhile, compared with the existing method, the actual recommendation effect of the present invention is better.

Description of drawings

FIG. 1 is a schematic diagram of the framework of the recommendation system integrating meta-path semantics according to the present invention.

FIG. 2 is a schematic diagram of the framework of the meta-path early fusion module of the present invention.

FIG. 3 is a schematic diagram of the framework of the meta-path semantic embedding module of the present invention.

Among them, reference numerals:

1: meta path

2: Bipartite graph of relative importance

3: Similarity matrix

4: Embedding expression from the previous round

5: Community Information

6: Embedding expression for the current round

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, but not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative work fall within the protection scope of the present invention.

It should be noted that the embodiments of the present invention and the features of the embodiments may be combined with each other under the condition of no conflict.

Referring to FIG. 1 , a recommendation method for integrating meta-path semantics provided by the present invention mainly includes three steps, namely, calculating the relative importance of users/commodities based on heterogeneous information networks and user/commodity rating datasets. Graph; use the relative importance bipartite graph to guide the information dissemination between user/commodity nodes to obtain the embedded expression of the user/commodity node; transform the embedded expression of the user/commodity node to obtain the recommended user/commodity node The final embedding expression of the node, and the final embedding expression of the user/item node is input into a recommendation model to obtain the prediction score of the user/item. A recommender system integrating meta-path semantics provided by the present invention mainly includes three modules, namely meta-path early fusion module, meta-path semantic embedding module and recommendation prediction module (Recommender), the three modules respectively correspond to the above three modules Steps, the following will introduce the role of each module in detail.

为结点类型A_i与结点类型A_j之间的邻接矩阵。根据不同应用场景，人工设计了L条元路径模板P₁,P₂,...,P_L。本发明的融合元路径语义的推荐系统主要包括以下三个模块：In the embodiment of the present invention, it is assumed that the user set is U, the commodity set is I, and the score value set is Y, where |U|=m and |I|=n. For simplicity, the scoring data is considered here, given a scoring record Π={u,v,y _uv |u∈U,v∈I,y _uv∈Y }. Given a heterogeneous information network G=(V, E), where the node type set is A, the edge type is R, and the meta-path is a path template defined on the node type and edge type, which can be determined by the node type passed through. Representation: P=(A ₁ A ₂ . . . A _l+1 ). definition

is the adjacency matrix between node type A _i and node type A _j . According to different application scenarios, L meta-path templates P ₁ , P ₂ ,...,P _L are manually designed. The recommendation system for integrating meta-path semantics of the present invention mainly includes the following three modules:

(1) Meta-path early fusion module

Please refer to Figure 2. The function of the early fusion module of the meta-path is mainly to calculate the relative importance of users/commodities based on the heterogeneous information network and the user/commodity scoring dataset; degree matrix for adaptive fusion.

连续相乘。该相似性正比于初始结点到末尾结点之间按照给定语义所经过的总路径数，设在第l条元路径下得到的用户/商品相似度矩阵为M^(l)。定义一个辅助二部图BG(U,I,E,R)，该辅助二部图的结点为用户和商品两种类型结点，只有用户结点与用户结点之间有连接。用户结点u与商品结点v之间连接的边具有特征向量r_uv,定义为：First, based on the given heterogeneous information network and L meta-paths, the data is preprocessed, and the similarity matrix of user/commodity nodes under different meta-path semantics is calculated. There are many specific similarity calculation methods. The present invention adopts PathCount similarity, and the specific calculation method is as follows: along the meta-path, the adjacency matrix between each adjacent two types of nodes is calculated.

Multiply continuously. The similarity is proportional to the total number of paths traversed between the initial node and the end node according to the given semantics, and the user/commodity similarity matrix obtained under the lth meta-path is M ^(l) . Define an auxiliary bipartite graph BG(U,I,E,R), the nodes of this auxiliary bipartite graph are user and commodity nodes, and only the user node and the user node are connected. The edge connecting the user node u and the commodity node v has a feature vector r _uv , which is defined as:

是第l条元路径语义下第u个用户和第v个商品之间的相似度。本发明将r_uv看作为用户u和商品v之间的复杂关系的多维表示。该辅助二部图的边集E定义为：

在该辅助二部图上，一个顶点i的邻居集合定义为：Ν(i)＝{j|(i,j)∈E}，i∈U∪I。Among them, in formula (1),

is the similarity between the u-th user and the v-th item under the semantics of the l-th meta-path. The present invention views r _uv as a multi-dimensional representation of the complex relationship between user u and item v. The edge set E of this auxiliary bipartite graph is defined as:

On this auxiliary bipartite graph, the neighbor set of a vertex i is defined as: N(i)={j|(i,j)∈E}, i∈U∪I.

The meta-path early fusion module maps the feature vector r _uv on the auxiliary bipartite graph to the relative importance between the user node u and the commodity node v, so as to convert the complex and inconsistent similarity under multiple meta-paths Transform into similarity under consistent semantics. Further, the present invention specifically adopts a multi-layer perceptron (Multi-Layer Perceptron, MLP) to map the feature vector r _uv of the edge to a scalar e _uv , and normalizes it through the Softmax function:

e _uv =MLP(r _uv ) (2)

If only the value calculated according to formula (2) is used as the relative importance between the user node u and the commodity node v, the importance of the user node u to the commodity node v is the same as that of the commodity node v to the user node. The relative importance of u is equal. In fact, the user node u and the commodity node v have different neighbor nodes and their numbers, so the importance of the user node u to the commodity node v is the same as the commodity node v to the user node u. The _{relative importance} of the The relative importance of point u to commodity node v, and α _vu is defined as the relative importance of commodity node v to user node u.

其中α_U→I由所有用户到商品的相对重要性组成，α_I→U由所有商品到用户的相对重要性组成。该矩阵α可看作为一个辅助二部图的加权邻接矩阵，本发明将其称为相对重要性二部图。该相对重要性二部图α将在下一元路径语义嵌入模块中用于指导用户/商品的嵌入更新。put all alpha _uv with alpha _vu and construct the matrix

where α _U→I consists of the relative importance of all users to items, and α _I→U consists of the relative importance of all items to users. The matrix α can be regarded as a weighted adjacency matrix of an auxiliary bipartite graph, which is called a relative importance bipartite graph in the present invention. This relative importance bipartite graph α will be used in the next meta-path semantic embedding module to guide the user/item embedding update.

(2) Meta-Path Semantic Embedding Module

Please refer to Figure 3. The function of the meta-path semantic embedding module is mainly to perform the embedding learning of user/commodity nodes based on the relative importance bipartite graph α output by the meta-path early fusion module, and the learned user/commodity nodes At the same time, the content features of nodes and the topological structure between nodes (ie, the topological relationship of the graph corresponding to the relative importance bipartite graph α) can be maintained.

其中每一行代表一个用户的p维原始特征，商品结点的原始特征为

其中每一行代表一个商品的q维原始特征。首先，经过一个特征变换，将用户结点的原始特征

商品结点的原始特征

转换到同一个空间内，并将其进行拼接，得到用户/商品结点的初始特征矩阵

Let the original feature of the user node be

Each row represents the p-dimensional original feature of a user, and the original feature of the commodity node is

where each row represents the q-dimensional raw features of an item. First, after a feature transformation, the original features of the user node are transformed into

Raw features of commodity nodes

Convert to the same space and splicing it to get the initial feature matrix of the user/commodity node

分别代表待学习的用户/商品结点特征的变换矩阵。元路径语义嵌入模块通过信息传播机制进行用户/商品结点嵌入的更新，为了将元路径语义与结点特征进行融合，本发明设计利用元路径早期融合模块输出的相对重要性二部图α指导信息传播的进行。即，相对重要性二部图α作为一种注意力机制，指导着用户/商品结点之间的信息传播。设信息传播一共进行Γ轮迭代，每一轮迭代可分为两个过程进行，并且在此需要说明的是，可将得到的初始特征矩阵作为第一轮的用户/商品结点的嵌入表达。例如对于Γ轮迭代中的第t轮来说，首先，本发明采用一个聚合器(Aggregator)将每个结点i(用户/商品)的邻居结点的当下特征

j∈Ν(i)进行聚合，得到聚合后的每个结点i的社区信息

其中：Among them, in formula (5),

respectively represent the transformation matrices of the user/commodity node features to be learned. The meta-path semantic embedding module updates the user/commodity node embedding through the information dissemination mechanism. In order to fuse the meta-path semantics and node features, the present invention uses the relative importance bipartite graph α output by the meta-path early fusion module to guide information dissemination. That is, the relative importance bipartite graph α acts as an attention mechanism to guide the information dissemination between user/commodity nodes. It is assumed that information propagation is carried out in a total of Γ rounds of iteration, and each round of iteration can be divided into two processes, and it should be noted here that the obtained initial feature matrix can be used as the embedding expression of the user/commodity nodes in the first round. For example, for the t-th round in the Γ round iteration, firstly, the present invention uses an aggregator (Aggregator) to combine the current features of the neighbor nodes of each node i (user/commodity)

j∈N(i) is aggregated to obtain the aggregated community information of each node i

in:

之后，再将每个结点i的前一轮的嵌入表达

及其对应的社区信息

一并通过一个过滤器(Filter)，进行一次信息的筛选，并得到更新后的当前轮的每个结点i的嵌入表达

Get the community information of each node i

After that, the embedding expression of the previous round of each node i is expressed

and its corresponding community information

Through a filter (Filter) together, filter the information once, and get the updated embedded expression of each node i of the current round

The function of formula (6) is to fuse the meta-path-based semantics with the content information of nodes through an aggregator, learn from the attention mechanism, and skillfully use meta-path semantic information to guide the information dissemination of nodes, so that the learned At the same time, the content information of the node and the topological relationship between the nodes are preserved.

(3) Recommendation prediction module

)，用于评价用户与商品之间的相对重要性，并相应设计了两个推荐预测模型。The role of the recommendation prediction module is mainly to perform the downstream prediction task of the model on the basis of the updated node (user/product) embedding expression H ^Γ (after Γ round iteration) obtained by the meta-path semantic embedding module. Based on the embedding of user/commodity nodes, the framework proposed in the present invention can easily perform prediction tasks such as classification and regression, and the present invention focuses on scoring data (prediction score

), used to evaluate the relative importance between users and products, and designed two recommendation prediction models accordingly.

i∈U∪I。基于此，本发明提出了两种推荐预测模型：线性模型与双线性模型。线性模型将用户的最终嵌入表达与商品的最终嵌入表达进行拼接后的向量进行加权得到预测分数

加权系数为

双线性模型利用一个对称正定矩阵，将用户的最终嵌入表达与商品的最终嵌入表达通过一个双线性函数映射得到预测分数

双线性模型与线性模型相比，引入了二次交互，具有更多的灵活性。其中，First, the embedding expression H ^Γ of the node (user/item) is transformed through a fully connected layer to obtain the final embedding expression for the recommended user/item:

i∈U∪I. Based on this, the present invention proposes two recommendation prediction models: a linear model and a bilinear model. The linear model weights the vector obtained by splicing the final embedding expression of the user and the final embedding expression of the item to obtain the prediction score

The weighting factor is

The bilinear model uses a symmetric positive definite matrix to map the final embedding expression of the user and the final embedding expression of the item through a bilinear function to obtain the prediction score

Compared with the linear model, the bilinear model introduces quadratic interaction and has more flexibility. in,

Linear model:

Bilinear model:

)，本发明构建了回归损失

并通过随机梯度下降法对模型进行优化。In addition, for scoring data (predicted score

), the present invention constructs a regression loss

The model is optimized by stochastic gradient descent.

The present invention is experimented on two real datasets MovieLens-1M and Yelp. The evaluation indicators used are the common RMSE (root mean square error) and MAE (mean absolute error). For any user, the calculation formulas of its RMSE and MAE are:

分别为该用户对第i个测试样本的真实评价得分与预测的该用户对第i个测试样本的评价得分。where m is the number of test samples, y _i and

are the user's real evaluation score for the i-th test sample and the predicted user's evaluation score for the i-th test sample.

The experiments compared the following three types of comparison methods: (1) other data than user/item rating data in heterogeneous graphs are used as features of users or items, such as FM [11] and GCMC [12]; (2) based on Meta-path methods: such as FMG [13] and HAN [14]; (3) methods based on knowledge graph embedding, such as CKE [15] and KGAT [16].

The MovieLens-1M dataset used (https://grouplens.org/datasets/movielens/) contains the following different node types: user (user, abbreviated as U), movie (movie, abbreviated as M), gender ( gender, abbreviated as Gd), occupation (occupation, abbreviated as Ocp), movie type (type, abbreviated as T). The included relationships are: UM (the user watches a movie), UOcp (the user belongs to a certain occupation), UGd (the user belongs to a certain gender), and MT (the movie belongs to a certain genre). The artificially designed meta-paths are: UM, UMUM, UMTM, UMTMUM, UOcpUM, UGdUM.

The Yelp dataset used (https://www.yelp.com/dataset challenge) contains the following different node types: user (user, abbreviated U), business (business, abbreviated B), compliment (compliment, abbreviated Comp) ), city (city, abbreviated Ci), type (category, abbreviated Cat). It includes the following relationships: user reviews merchants (UB), users are friends with users (UU), users praise (UComp), merchants belong to a certain city (BCi), and merchants belong to a certain type (BCat). The following meta-paths were manually designed: UB, UBUB, UUB, UCompUB, UBCiB, UBCatB, UBCatBUB.

The one-hot encoding of user ID and product ID is used as the original input feature, the feature dimension k is 128, and the final feature dimension d is 24. The total number of iteration rounds Γ is taken as 1. The Adam optimizer is used, and the learning rate is set to 0.001. Experiments were conducted under three different train/test settings with the number of training samples/test samples: 40%/60%, 60%/40%, 80%/20%. The experimental results on the two datasets are shown in Table 1 and Table 2, respectively.

Table 1

Table 1 shows the experimental results on the MovieLens-1M dataset. Among them, ↓ indicates that the smaller the value, the better the model performance. Bold indicates the optimal experimental result, and underline indicates the sub-optimal experimental result among existing methods.

Table 2

Table 2 shows the experimental results on the Yelp dataset. Among them, ↓ indicates that the smaller the value, the better the model performance. Bold indicates the optimal experimental result of the present invention, and underline indicates the sub-optimal experimental result in the existing method. Since the Yelp dataset is very sparse when the training set proportions are 40% and 60%, the two types of knowledge graph-based embedding methods are not applicable, so they are not compared, and are marked as '-' in the table.

It can be seen from the experimental results in Table 1 and Table 2 that compared with the existing methods, the present invention has achieved leading performance on both real data sets.

To sum up, the present invention will perform early fusion based on the similarity of different meta-paths, obtain consistent semantics, and directly apply it to the expression learning process of users/items; drawing on the information dissemination mechanism and attention model, innovatively based on meta-paths As the attention score between nodes, the learned embedding can retain both the node content information and the topology information between nodes. In addition, the present invention proposes a broader recommendation model, and users can design the specific implementation of each module by themselves, such as calculating the similarity matrix based on meta-paths in different ways, using different structures to perform early fusion of multiple meta-paths, etc. , the model is simple and easy to build. Meanwhile, compared with the existing method, the actual recommendation effect of the present invention is better.

Claims (10)

1. a recommendation method of fusion meta-path semantics, is characterized in that, comprises: Calculate the relative importance bipartite graph of the user/commodity nodes based on the heterogeneous information network and the user/commodity scoring data set; Use the relative importance bipartite graph to guide the information dissemination between user/commodity nodes, and obtain the embedded expression of the user/commodity nodes; Transform the embedded expression of the user/commodity node to obtain the final embedded expression of the user/commodity node for recommendation, and input the final embedded expression of the user/commodity node into a recommendation model to obtain the The predicted score for the user/item. 2. The method for recommending fusion meta-path semantics as claimed in claim 1, wherein, based on a heterogeneous information network and a user/commodity scoring data set, the relative importance of the user/commodity node is calculated in two parts Figures include: Calculate the similarity matrix M ^(l) of the user/commodity nodes in different meta-paths; Calculate the feature vector r _uv of the edge between the user/commodity nodes; A multi-layer perceptron is used to map the feature vector r _uv of the edge into a scalar e _uv , and normalized by the Softmax function to construct a bipartite graph α of the relative importance of the user/commodity; in,

e _uv =MLP(r _uv ),

is the similarity between the user node u and the commodity node v along the l-th meta-path. 3. The method for recommending fusion meta-path semantics as claimed in claim 2, wherein the scalar e _uv is normalized through a Softmax function, comprising:

Among them, _αuv is defined as the relative importance of user node u to commodity node v, and _αvu is defined as the relative importance of commodity node v to user node u. 4. The method for recommending fusion meta-path semantics according to claim 1, wherein the relative importance bipartite graph is used to guide the information dissemination between user/commodity nodes to obtain the user/commodity nodes Embedding expressions include: The original features of the user node

The original feature of the commodity node is

After conversion to the same space, the initial feature matrix of the user/commodity node is obtained by splicing

The current features of the neighbor nodes of the user/item node

Aggregate through an aggregator to obtain the community information of the user/commodity node

Express the embedding of the user/commodity node in the previous round

and said community information

Through a filter, the embedded expression of the user/commodity node in the current round is obtained

Among them, multiple rounds of information dissemination are carried out,

As the embedded expression of the user/commodity node in the first round,

respectively represent the transformation matrices of the user/commodity node features to be learned,

5 . The method for recommending fusion meta-path semantics according to claim 1 , wherein the recommendation model comprises: a linear model or a bilinear model. 6 . 6. A recommendation system integrating meta-path semantics, characterized in that it comprises: The meta-path early fusion module is used to calculate the relative importance bipartite graph of the user/commodity nodes based on the heterogeneous information network and the user/commodity scoring data set; a meta-path semantic embedding module, used for using the relative importance bipartite graph to guide the information dissemination between user/commodity nodes, to obtain the embedded expression of the user/commodity nodes; A recommendation prediction module, configured to transform the embedded expression of the user/commodity node to obtain the final embedded expression of the user/commodity node for recommendation, and input the final embedded expression of the user/commodity node A recommendation model obtains the predicted score for the user/item. 7. The recommendation system for fusion meta-path semantics according to claim 6, wherein the meta-path early fusion module comprises: Similarity matrix calculation submodule, for calculating the similarity matrix M ⁽¹⁾ of the user/commodity node in different meta-paths; a feature vector calculation submodule, used for calculating the feature vector r _uv of the edge between the user/commodity nodes; The relative importance bipartite graph construction sub-module is used to map the feature vector r _uv of the edge into a scalar e _uv using a multi-layer perceptron, and normalize it with the Softmax function to construct the user/commodity Relative importance bipartite graph α; in,

e _uv =MLP(r _uv ),

is the similarity between the user node u and the commodity node v of the l-th meta-path. 8. The recommendation system of fusion meta-path semantics according to claim 7, wherein the scalar e _uv is normalized through Softmax function, comprising:

Among them, _αuv is defined as the relative importance of user node u to commodity node v, and _αvu is defined as the relative importance of commodity node v to user node u. 9. The recommender system integrating meta-path semantics according to claim 6, wherein the meta-path semantic embedding module comprises: The initial feature matrix splicing sub-module is used to combine the original features of the user node

The original feature of the commodity node is

After conversion to the same space, the initial feature matrix of the user/commodity node is obtained by splicing

Aggregation sub-module, used to combine the current features of the neighbor nodes of the user/commodity node

Aggregate through an aggregator to obtain the community information of the user/commodity node

Filter sub-module, express the embedded expression of the user/commodity node in the previous round

and said community information

Through a filter, the embedded expression h _i t of the user/commodity node in the current round is obtained; Among them, multiple rounds of information dissemination are carried out,

As the embedded expression of the user/commodity node in the first round,

respectively represent the transformation matrices of the user/commodity node features to be learned,

10 . The recommendation system integrating meta-path semantics according to claim 6 , wherein the recommendation model comprises: a linear model or a bilinear model. 11 .

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