Course recommendation method based on fine-grained interaction and knowledge graph

By integrating fine-grained interaction and knowledge graphs into the course recommendation system to supplement the interactive information between users and videos, the existing system cannot accurately distinguish user preferences and achieve more efficient course recommendations.

CN119941464AActive Publication Date: 2025-05-06ZHEJIANG TOPTHINKING INFORMATION TECH +1
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Patent Information

Application Number
CN202411978707.X
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2024-12-31
Publication Date
2025-05-06
Estimated Expiration
2044-12-31

AI Technical Summary

Technical Problem

The existing course recommendation system ignores the interaction between users and videos, which makes it impossible to accurately distinguish different users' preferences for different videos in the same course, thereby affecting the recommendation performance.

Method used

By collecting users' interaction behavior on the education platform, building a knowledge graph, acquiring a knowledge unit network of courses, and supplementing the interaction information between users and courses with the user, and using collaborative course collections to supplement collaborative information for courses to improve recommendation performance.

Benefits of technology

Through the combination of fine-grained interaction and knowledge graph, different users' preferences for different videos in the same course can be distinguished, and the accuracy and performance of course recommendations can be improved.

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Abstract

The invention discloses a course recommendation method based on fine-grained interaction and a knowledge graph, and the method comprises the steps: collecting interaction behaviors of a user on an education platform, and obtaining a learning record of the user; constructing a knowledge graph G; obtaining knowledge unit networks of the courses according to the knowledge graph, analyzing learning records of the user, and respectively obtaining collaborative knowledge unit network sets of the user and the courses; extracting rough preferences and detailed preferences of the user on courses, and generating a personalized course recommendation list based on the two preferences; according to the method, on the basis of fine-grained interaction and the knowledge graph, by means of rich semantic information of the knowledge graph, additional information is provided for a recommendation algorithm, a collaborative course set is used for supplementing collaborative information for courses, and interaction information of a user and a video is supplemented into interaction information of the user and the courses; the method can distinguish the difference between users who have interacted with different videos of the same course, and can also improve the recommendation performance.
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Description

Technical Field

[0001] The present invention relates to the technical field of course recommendation, and in particular to a course recommendation method based on fine-grained interaction and knowledge graph. Background Art

[0002] In recent years, with the rapid development of information technology, the field of education has also changed. Online education has developed. With the development of online education and the accumulation of course data, information overload is inevitable. Recommendation systems can effectively alleviate the problem of information overload, so course recommendation systems came into being. Knowledge graphs contain rich semantic information and can provide rich additional information for recommendation systems, thereby improving the performance of recommendation systems. Therefore, they are widely used in various recommendation systems.

[0003] Previous course recommendation systems mostly only considered the interaction between users and courses, ignoring the interaction between users and videos belonging to the courses, thus limiting the performance of the course recommendation system. Existing knowledge graph-based course recommendation systems mostly only consider the interaction between users and courses to recommend courses to users, ignoring the interaction between users and videos. For example, different users interact with different videos of the same course, which represents the different detailed content of the course that users pay attention to. If the interaction between users and videos is ignored, at the level of users and courses, the preferences of these users who interact with the same course will be regarded as similar, but in fact they are different, thus resulting in performance impairment. Summary of the invention

[0004] The purpose of the present invention is to provide a course recommendation method based on fine-grained interaction and knowledge graph, which provides additional information for the recommendation algorithm with the help of rich semantic information of knowledge graph, uses collaborative course collection to supplement collaborative information for courses, and supplements the interactive information between users and videos to the interactive information between users and courses, so as to improve the recommendation performance;

[0005] The specific technical solution is as follows: A course recommendation method based on fine-grained interaction and knowledge graph, including the following steps:

[0006] Step 1: Collect the user's interactive behavior on the education platform and obtain the user's learning record;

[0007] Preferably, the user's learning record is the user's course registration record and the user's video viewing record of the registered course; the courses on the education platform are counted to obtain a course set C = {c1, c2, ..., c i ,……,c M}, there are M courses in total; count the videos on the education platform, and get the video set V = {v1, v2, ..., v P}, there are a total of P videos; count the users on the education platform and get the user set U = {u1,u2,……,u i ,……,u N}, there are N users in total;

[0008] For user u, u∈U, its course registration set That is, user u has registered L u Courses, for registered courses c u,i , the video collection he has watched is It is expressed as: That is, user u has watched a total of belongs to course c u,i Video; get the interaction matrix between users and courses The elements of the interaction matrix Y are either 0 or 1. The value is 1, which means that user u has interacted with course c. i On the contrary, if it is 0, it means there is no interaction.

[0009] Step 2: Collect course data on the education platform and construct a knowledge graph G containing schools, teachers, course names, course videos, and course knowledge points;

[0010] Preferably, the knowledge graph G is expressed as:

[0011] G={(h,r,t)|h∈E,r∈R,t∈E}

[0012] Among them, h represents the head entity, r represents the relationship between the head entity and the tail entity, t represents the tail entity, E represents the entity set in the knowledge graph, R represents the relationship set in the knowledge graph, and (h, r, t) represents a triple; the head entity and the tail entity both include school, teacher, course name, course video, and course knowledge points.

[0013] Step 3: Use the knowledge graph G to obtain the knowledge unit network of the course, analyze the learning record of the user, and obtain the collaborative knowledge unit network set of the user and the course respectively;

[0014] Preferably, the method for respectively obtaining collaborative knowledge unit network sets of users and courses further comprises the following steps:

[0015] Step 3-1: Align the courses in the course collection with the course name entities in the knowledge graph, and align the videos in the video collection with the course video entities in the knowledge graph;

[0016] Step 3-2: Obtain the knowledge unit network of the course in the knowledge graph;

[0017] Step 3-3: Get course c i A collection of collaborative courses;

[0018] Step 3-4: Get course c i A collection of collaborative knowledge unit networks;

[0019] Step 3-5: Obtain the user's collaborative knowledge unit network collection.

[0020] Step 4: Extract the user's general and detailed preferences for courses, and generate a personalized course recommendation list based on the two preferences.

[0021] Preferably, the method for generating a personalized course recommendation list based on two preferences specifically comprises the following steps:

[0022] Step 4-1: Obtain the overall course core content of users and courses respectively;

[0023] Step 4-2: Obtain the user's overall video core content;

[0024] Step 4-3: Obtain the final representations of users and courses respectively;

[0025] Step 4-4: Based on the final representation o of user u u and candidate courses c i The final representation of ci Calculate the scores and recommend the top N courses to the user after sorting them in descending order.

[0026] Compared with the prior art, the beneficial effects achieved by the present invention are as follows: the course recommendation method of the present invention is based on fine-grained interaction and knowledge graph, and uses the rich semantic information of the knowledge graph to provide additional information for the recommendation algorithm, uses a collaborative course set to supplement the collaborative information for the course, and supplements the interaction information between the user and the video to the interaction information between the user and the course, so as to distinguish the differences between users who have interacted with different videos of the same course, thereby improving the recommendation performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] The accompanying drawings are used to provide further understanding of the present invention and constitute a part of the specification. They are used to explain the present invention together with the embodiments of the present invention and do not constitute a limitation of the present invention.

[0028] In the attached picture:

[0029] Figure 1 A flowchart of the steps of a course recommendation method based on fine-grained interaction and knowledge graph provided in an embodiment of the present invention.

[0030] Figure 2 A schematic diagram of a knowledge graph constructed according to an embodiment of the present invention. DETAILED DESCRIPTION

[0031] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0032] Combination Figure 1 to Figure 2 The present invention provides a technical solution: a course recommendation method based on fine-grained interaction and knowledge graph, specifically comprising:

[0033] Step 1: Obtain the user's learning record by collecting the user's interactive behavior on the education platform;

[0034] In this embodiment, the user's learning record is the user's course registration record and the user's video viewing record of the registered course; the courses on the education platform are counted to obtain a course set C = {c1, c2, ..., c i ,……,c M}, there are M courses in total; count the videos on the education platform, and get the video set V = {v1, v2, ..., v P}, there are a total of P videos; count the users on the education platform and get the user set U = {u1,u2,……,u i ,……,u N}, there are N users in total; for user u, u∈U, its course registration set That is, user u has registered L u Courses, for registered courses c u,i , the video collection he has watched is It is expressed as: That is, user u has watched a total of belongs to course c u,i Video; you can get the interaction matrix between users and courses The elements of the interaction matrix Y are either 0 or 1. uci The value is 1, which means that user u has interacted with course c. i , otherwise, if it is 0, it means there is no interaction;

[0035] Step 2: Collect course data on the education platform and build a knowledge graph G containing schools, teachers, course names, course videos, and course knowledge points;

[0036] In this embodiment, combined with Figure 2, the knowledge graph G is represented as:

[0037] G={(h,r,t)|h∈E,r∈R,t∈E}

[0038] Among them, h represents the head entity, r represents the relationship between the head entity and the tail entity, t represents the tail entity, E represents the entity set in the knowledge graph, R represents the relationship set in the knowledge graph, and (h, r, t) represents a triple; the head entity and the tail entity both include school, teacher, course name, course video, and course knowledge points; combined Figure 2 As shown, for example, (course name 1, contains, course video 1), where course name 1 can be linear algebra, relation is contains, and course video 1 can be an introduction to determinants, then the meaning is linear algebra (course name) contains (relation) introduction to determinants (course video);

[0039] For example, in actual operation, the opposite relationships are represented separately, that is, the included (relationship) in (course video 1, included, course name 1) represents a different relationship from the included (relationship) mentioned above, and directed edges are used to distinguish them in the constructed knowledge graph.

[0040] Step 3: Obtain the knowledge unit network of the course based on the knowledge graph, and analyze the user's learning records to obtain the collaborative knowledge unit network sets of the user and the course respectively;

[0041] In this embodiment, the method of respectively obtaining collaborative knowledge unit network sets of users and courses further includes:

[0042] Step 3-1: Add the courses {c1, c2, ..., c i ,……,c M} is aligned with the course name entity in the knowledge graph to obtain in, Represents the course name entity; the videos in the video collection {v1, v2, ..., v P} is aligned with the course video entity in the knowledge graph to obtain in, Represents a course video entity;

[0043] Step 3-2: Obtain the knowledge unit network of the course in the knowledge graph;

[0044] For example, in the knowledge graph, all course name entities are initial entities; i , whose entity in the knowledge graph is get The initial entity set It is expressed as:

[0045]

[0046] And The directly connected entities are first-order entities, and we get The first-order entity set It is expressed as:

[0047]

[0048] Furthermore, the course name entity The first-order triple set in the knowledge graph is defined as:

[0049]

[0050] Finally, it is expanded to the L-order entity set and the set of L-order triples Respectively expressed as:

[0051]

[0052] Preferably, the set of triples of each order of a course is called the knowledge unit network of the course; i The knowledge unit network is Combination Figure 2 As shown, taking the expansion to the second-order triple set as an example, course name 1 is the entity after course 1 is aligned, and the first-order triple set T of course name 1 课程名称1,1 is {(course name 1, taught by, teacher 1), (course name 1, taught by, teacher 2), (course name 1, including, course video 1), (course name 1, including, course video 2)}; the second-order triple set T of course name 1 课程名称1,2 The knowledge unit network KUN of course 1 is {(teacher 1, employed, school 1), (teacher 2, employed, school 1), (course video 1, covers, course knowledge point 1), (course video 1, covers, course knowledge point 2), (course video 2, covers, course knowledge point 2)}; 课程1 For {T 课程名称1,1 , T 课程名称1,2}.

[0053] Step 3-3: Get the co-course set of the course;

[0054] For example, for course c i , its collaborative course set consists of the courses that the users who have interacted with the course have interacted with. i A collection of collaborative courses Defined as:

[0055]

[0056] Where c is the course, is the course registration set of user u”, and u” is the interactive course c i In the user set, u' is a user who has interacted with course c i Users, is user u' and course c i If there is interaction, it is 1, if there is no interaction, it is 0.

[0057] Step 3-4: Obtain the network set of collaborative knowledge units of the course;

[0058] For example, for course c i , and its collaborative knowledge unit network set is defined as follows:

[0059]

[0060] Among them, KUN c Represents the knowledge unit network of course c.

[0061] Step 3-5: Obtain the user's collaborative knowledge unit network set;

[0062] For example, for user u, through its course registration set Get its collaborative knowledge unit network set CS u,KUN , C.S. u,KUN Further expressed as:

[0063]

[0064] in, Represents the knowledge unit network formed by user u based on his course registration set.

[0065] Step 4: Extract the user's general and detailed preferences for courses, and generate a personalized course recommendation list based on the two preferences;

[0066] In this embodiment, generating a personalized course recommendation list based on two preferences further includes:

[0067] Step 4-1: Obtain the overall course core content of users and courses respectively;

[0068] For example, for user u, the core content of the course that he is interested in is obtained through the videos he has watched; the core content of the course consists of two parts: the course body and the video interaction. u,i Video Collection Course c that user u follows u,i Core content of the course The definition is as follows:

[0069]

[0070] Among them, the course video entity It's video v j After the entities are aligned in the knowledge graph, Represents the course video entity e vj The embedding representation of , d is the embedding dimension, represents the trainable parameters, Represents the number of courses that user u has watched. u,i The video interaction part of the core content of the course extracted from the video, the course name entity Representative course c u,i After the entities are aligned in the knowledge graph, Represents the course name entity The embedding representation of It is the core part of the course. represents the trainable parameters, represents matrix multiplication, Represents the course c that user u follows u,i The core content of the course;

[0071] Specifically, for user u, the overall course core content CCC u The definition is as follows:

[0072]

[0073] in, σ is the sigmoid activation function, For collection The number of elements contained in; the overall course core content CCC of user u u It can be regarded as their general preference;

[0074] Furthermore, for course c i , the core content of its overall course The definition is as follows:

[0075]

[0076] in, Course Title Entity and Course c i and c j Entities aligned in the knowledge graph, Represents the course name entity c i and c jThe embedding representation of , σ is the sigmoid activation function, Representative course c i A collection of collaborative courses The number of elements in , represents the trainable parameters, stands for matrix multiplication; course c i The overall core content of the course Able to provide course c i Supplementary collaborative information.

[0077] Step 4-2: Obtain the user's overall video core content;

[0078] In this embodiment, for user u, the overall video core content VCC u The definition is as follows:

[0079]

[0080] in, User u overall video core content VCC u It can be regarded as their detailed preferences under the general preferences.

[0081] Step 4-3: Obtain the final representations of users and courses respectively;

[0082] For example, for user u, the course c he has interacted with u,i The knowledge unit network is Course c u,i The set of l-order triples of Course c u,i The first-order response The definition is as follows:

[0083]

[0084]

[0085] in, Represents a set of triples The number of elements in , and are all scalar weights, “·” represents the dot product operation of the vector, “· T " represents the transposition operation, σ represents the sigmoid activation function, and "·||·" represents the concatenation operation of the vector. represents the unit vector of a vector, represents matrix multiplication, is a trainable parameter, “||·||2” represents the 2-norm of the vector;

[0086] Specifically, the course cu,i The initial entity embedding As its 0th order response

[0087] Aggregate Courses u,i The various order responses are obtained It is expressed as:

[0088]

[0089] in,

[0090] Furthermore, the collaborative knowledge unit network set CS of aggregated user u is u,KUN The information in the , get the final representation o of user u u , expressed as:

[0091]

[0092] in, |CS u,KUN | represents the knowledge unit network set CS u,KUN The number of elements in ;

[0093] Preferably, for course c i , get course c i The process of obtaining the final representation of the user is roughly similar to that of obtaining the final representation of the user; i The knowledge unit network is Course c i The set of l-order triples of Course c i The first-order response The definition is as follows:

[0094]

[0095] in, Course Title Entity For course c i Entities aligned in the knowledge graph, Represents a set of triples The elements in is a scalar weight, “·” represents the dot product operation of the vector, “· T " represents the transposition operation, σ represents the sigmoid activation function, and "·||·" represents the concatenation operation of the vector. represents the unit vector of a vector, represents matrix multiplication, is a trainable parameter, ||·||2 represents the 2-norm of the vector;

[0096] Aggregate Courses i The various order responses are obtained It is expressed as:

[0097]

[0098] For example, the course c i A collection of collaborative knowledge unit networks The information obtained after aggregation is used as the course c i Additional information for course c i Final characterization It is expressed as:

[0099]

[0100] Step 4-4: Based on the final representation o of user u u and candidate courses c i The final representation of ci Calculate the scores and recommend the top N courses sorted in descending order to the user;

[0101] Exemplarily, the score calculation formula is as follows:

[0102]

[0103] Among them, σ represents the sigmoid activation function, “·” represents the dot product operation of the vector, and “· T ” represents the transpose operation;

[0104] Specifically, the recommended loss function is:

[0105]

[0106] Among them, BCE is the binary cross entropy loss function, is the set of courses that user u has not interacted with;

[0107] The knowledge graph loss function is

[0108]

[0109] Among them, “·” represents the dot product operation of the vector, and ||·||2 represents the 2-norm of the vector;

[0110] The overall loss function is

[0111] Loss=RecLoss+KGLoss.

[0112] That is, the overall loss function composed of the recommendation loss function and the knowledge graph loss function is obtained.

[0113] It should be noted that, in this article, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device.

[0114] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art can still modify the technical solutions described in the aforementioned embodiments or replace some of the technical features therein by equivalents. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A course recommendation method based on fine-grained interaction and knowledge graph, characterized by: The method comprises the following steps: Step 1: Collect users’ interactive behaviors on the education platform and obtain their learning records; Step 2: Collect course data on the education platform and construct a knowledge graph G containing schools, teachers, course names, course videos, and course knowledge points; Step 3: Use the knowledge graph G to obtain the knowledge unit network of the course, analyze the learning record of the user, and obtain the collaborative knowledge unit network set of the user and the course respectively; Step 4: Extract the user's general preference and detailed preference for courses, and generate a personalized course recommendation list based on the two preferences. The user's general preference for courses is the user's overall course core content, and the detailed preference is the user's overall video core content.

2. A course recommendation method based on fine-grained interaction and knowledge graph according to claim 1, characterized in that: In step 1, the user's learning record is the user's course registration record and the user's video viewing record of the registered course; the courses on the education platform are counted to obtain a course set C = {c1, c2, ..., c i ,……,c M }, there are M courses in total; The video on the statistical education platform is obtained as the video set V = {v1, v2, ..., v P }, there are P videos in total; Count the users on the education platform and get the user set U = {u1,u2,……,u i ,……,u N }, there are N users in total; For user u, u∈U, its course registration set That is, user u has registered L u Courses, for registered courses c u,i , the video collection he has watched is It is expressed as: That is, user u has watched a total of belongs to course c u,i Video; get the interaction matrix between users and courses The elements of the interaction matrix Y are either 0 or 1. The value is 1, which means that user u has interacted with course c. i On the contrary, if it is 0, it means there is no interaction.

3. A course recommendation method based on fine-grained interaction and knowledge graph according to claim 2, characterized in that: The knowledge graph G is expressed as: G={(h,r,t)|h∈E,r∈R,t∈E} Among them, h represents the head entity, r represents the relationship between the head entity and the tail entity, t represents the tail entity, E represents the entity set in the knowledge graph, R represents the relationship set in the knowledge graph, and (h, r, t) represents a triple; the head entity and the tail entity both include school, teacher, course name, course video, and course knowledge points.

4. According to claim 3, a course recommendation method based on fine-grained interaction and knowledge graph is characterized by: The method of obtaining collaborative knowledge unit network sets of users and courses respectively further includes: Step 3-1: Add the courses {c1, c2, ..., c i ,……,c M } is aligned with the course name entity in the knowledge graph to obtain The videos in the video collection {v1, v2, ..., v P } is aligned with the course video entity in the knowledge graph to obtain Said Represents the course name entity, Represents a course video entity; Step 3-2: Obtain the knowledge unit network of the course in the knowledge graph; Step 3-3: Get course c i A collection of collaborative courses; Step 3-4: Get course c i A collection of collaborative knowledge unit networks; Step 3-5: Obtain the user's collaborative knowledge unit network collection.

5. According to claim 4, a course recommendation method based on fine-grained interaction and knowledge graph is characterized by: The method for obtaining the knowledge unit network of a course in the knowledge graph is specifically as follows: in the knowledge graph, all course name entities are initial entities; i , whose entity in the knowledge graph is get The initial entity set It is expressed as: and The directly connected entities are first-order entities, and we get The first-order entity set It is expressed as: Course Title Entity The set of first-order triples in the knowledge graph Defined as: Expand to L-order entity collection and the set of L-order triples Respectively expressed as: The set of triples of each order of a course is called the knowledge unit network of the course; i The knowledge unit network is 6. A course recommendation method based on fine-grained interaction and knowledge graph according to claim 5, characterized in that: The sets in step 3-3, step 3-4 and step 3-5 are respectively expressed as: The course c i The collaborative course set of c is composed of the courses that the users who have interacted with this course have interacted with. i The set of co-courses is defined as It is expressed as: Where c is the course, is the course registration set of user u”, and u” is the interactive course c i In the user set, u' is a user who has interacted with course c i Users, is user u' and course c i The interaction record is 1 if there is interaction, and 0 if there is no interaction; The course c i The set of collaborative knowledge unit networks is defined as It is expressed as: Among them, KUN c Indicates course c i A network of knowledge units; The collaborative knowledge unit network set of user u: For user u, through its course registration set Get its collaborative knowledge unit network set CS u,KUN , C.S. u,KUN It is expressed as: in, Represents the knowledge unit network formed by user u based on his course registration set.

7. A course recommendation method based on fine-grained interaction and knowledge graph according to claim 6, characterized in that: The method for generating a personalized course recommendation list based on two preferences specifically includes the following steps: Step 4-1: Obtain the overall course core content of users and courses respectively; Step 4-2: Obtain the user's overall video core content; Step 4-3: Obtain the final representations of users and courses respectively; Step 4-4: Based on the final representation o of user u u and candidate courses c i The final representation Calculate the scores and recommend the top N courses to the user after sorting them in descending order.

8. The course recommendation method based on fine-grained interaction and knowledge graph according to claim 7 is characterized by: In step 4-3, the final representations of users and courses are obtained as follows: for user u, the courses c that he has interacted with u,i The knowledge unit network is Course c u,i The set of l-order triples of Course c u,i The first-order response The definition is as follows: in, Represents a set of triples The number of elements in , and are all scalar weights, "·" represents the dot product operation of vectors, "· T " represents the transposition operation, σ represents the sigmoid activation function, "·||·" represents the concatenation operation of the vector, represents the unit vector of a vector, represents matrix multiplication, is a trainable parameter, "||·||2" represents the 2-norm of the vector; The course c u,i The initial entity embedding As its 0th order response Aggregate Courses u,i The various order responses are obtained It is expressed as: in, Aggregate user u's collaborative knowledge unit network set CS u,KUN The information in the , get the final representation o of user u u , expressed as: in, |CS u,KUN | represents the knowledge unit network set CS u,KUN The number of elements in ; For course c i , get course c i The process of obtaining the final representation of the user is roughly similar to that of obtaining the final representation of the user; i The knowledge unit network is Course c i The set of l-order triples of Course c i The first-order response The definition is as follows: in, Course Title Entity For course c i Entities aligned in the knowledge graph, Represents a set of triples The elements in is a scalar weight, "·" represents the dot product operation of vectors, "· T " represents the transposition operation, σ represents the sigmoid activation function, "·||·" represents the concatenation operation of the vector, represents the unit vector of a vector, represents matrix multiplication, is a trainable parameter, ||·||2 represents the 2-norm of the vector; Aggregate Courses i The various order responses are obtained It is expressed as: The course c i A collection of collaborative knowledge unit networks The information obtained after aggregation is used as the course c i Additional information for course c i Final characterization It is expressed as:

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