JP2010250377A - Link prediction system, method, and program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a scalable link prediction technology that can cope with the number of dozens to millions nodes. <P>SOLUTION: At first, similarity matrices W<SB>Z</SB>, W<SB>Y</SB>, W<SB>X</SB>, are low-rank approximated by a technology such as incomplete Cholesky decomposition. Then, the eigenvalue decomposition of low-rank approximate matrices of the similarity matrices W<SB>Z</SB>, W<SB>Y</SB>, W<SB>X</SB>is performed. Schematically, low-rank approximation is the approximation of one matrix by a product of two rectangular matrices. Here, low-rank approximation facilitates the calculation of eigenvalue decomposition. In the next step, eigenvalues of obtained low-rank approximate matrices of W<SB>Z</SB>, W<SB>Y</SB>, W<SB>X</SB>are used to constitute normalized Laplacian L. Since the normalized Laplacian L is obtained in this manner, V=<SB>Z</SB>, V=<SB>Y</SB>, V=<SB>X</SB>as matrices with respective eigenvectors of low-rank approximate matrices of W<SB>Z</SB>, W<SB>Y</SB>, W<SB>X</SB>arranged therein and L are used to favorably calculate the inverse matrix of the part of (σL+I). When the inverse matrix of (σL+I) is obtained, F can be calculated due to vec(F)=(σL+I)<SP>-1</SP>vec(F<SP>*</SP>). <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to a technique for predicting a link between a plurality of objects. A link here is the strength of the relationship between objects, for example.

  Conventionally, there is a desire to predict the strength of links or relationships between multiple objects or entities. For example, there is a technique called collaborative filtering that predicts a purchase relationship between a user and a product, which is described in Japanese Patent Application Laid-Open No. 2000-148864 related to the present applicant.

  Alternatively, a field called biological network prediction that predicts the presence or absence of an interaction between different proteins is also known.

  These are, for example, computer systems as so-called supervised learning in which correct answers are given for pairs between several objects, and based on these values the strength of links is predicted for the unknown unknown pairs. It was solved by.

  Since then, models have been proposed that consider multiple types of links, i.e. multi-type links, rather than a single link between pairs of objects. For example, in collaborative filtering, not only the purchase relationship between a pair of objects but also evaluation and browsing of product information are taken into consideration.

  Similarly, in biological network prediction, a plurality of types of links such as the type of action and the environment in which the action occurs can be considered. In order to realize such multi-type link prediction, the computer processing becomes complicated.

  By the way, a company that operates a social network service (SNS) or a company that operates a blog generally has hundreds of thousands to millions of users, and logs of activities between the users are as follows. Accumulated on the system server.

  The activities recorded in the log include reading a blog, writing a blog, adding a comment, bookmarking a blog, making friends with other users, and becoming a celebrity blog supporter.

  In the specification of Japanese Patent Application No. 2007-336919 related to the present applicant, in an online community such as SNS, a plurality of behavioral indicators such as writing a blog, reading a news, reading a message, etc. are handled uniformly. It describes the concept of action matrix that enables

  Therefore, based on the log in which the information is recorded, the user wants to recommend a celebrity blog from the commented data, the user wants to recommend a celebrity blog, or the celebrity blog. From activities related to blogging, requests such as measuring the degree of fanfare to celebrities have come out.

  This, in a way, results in the problem of multi-type link prediction. However, in the case of SNS and blog sites, the number of users is enormous, from hundreds of thousands to millions, and general algorithms used in collaborative filtering and biological network prediction are the scale of SNS and blog sites. There is a problem that it is not scalable enough.

  By the way, the approach to the link prediction problem based on machine learning has been known so far. This is (1) a pair-wise prediction model that predicts the presence or absence of a link between two arbitrary components or entities. And (2) a relational network model that is a model of the entire network structure.

  The pair-wise prediction model has the advantage that relatively large problems are easier to handle because it can be reduced to a normal supervised prediction framework. Therefore, a plurality of researchers including one of the inventors of the present application developed a method using pair-wise feature-based reasoning in a pair-wise prediction model.

  This method consists of (a) a link prediction method for semi-supervised prediction, (b) simultaneous prediction of multitype links, and (c) a novel method using a Kronecker product or Kronecker sum of node similarity matrices. It is characterized by using pairwise similarity and (d) being capable of high-speed calculation by the conjugate gradient method.

In this method, for the three sets of labels (node, node, link type), the first node set of the set is X, the second node set of the set is Y, and the set of link types of the set is Z. In this case, inference is performed based on the hypothesis that “similar nodes are likely to have the same label (property)”. If the inference based on the link propagation principle is expressed as an objective function of the optimization problem and is partially differentiated with an appropriate variable and set to zero, the equation to be solved is as follows.
(σL + I) vec (F) = vec (F ^* ) Equation 1

Here, σ is a small real number such as 0,01, for example, L is a Laplacian matrix, and L ≡ D − W
Where W is the Kronecker product or Kronecker sum of similarity matrices W _Z , W _Y , W _X between triplets X, Y, Z Z is the Kronecker product of D _Z , D _Y , D _X or The Kronecker sum, D _Z , D _Y , D _X is a diagonal matrix I having the sum of each row of W _Z , W _Y , W _{X as} a diagonal component I is a unit matrix F ^* of a predetermined size includes a known relationship to a portion thereof (node, node, link type) 3-order tensor F with three sets the components is not the same form as F ^*, the value of the calculation result is stored An unknown third-order tensor.

  In this approach, Equation 1 is solved using the conjugate gradient method.

  The above detailed discussion was presented on the topic of link propagation: a semi-supervised learning method for link prediction on March 13, 2009, the 73rd meeting of the Japanese Society for Artificial Intelligence. It is described in the distributed materials.

Japanese Patent Application No. 2007-336919

The 73rd Annual Meeting of the Japanese Society for Artificial Intelligence The Society for Artificial Intelligence Basic Research Meeting March 13, 2009, Hisashi Kashima, Kaoru Kato, Yoshihiro Yamanishi, Masaru Sugiyama, Koji Tsuda Announcement titled `` Learning Method '' and disseminated proceedings

The method proposed in the above paper has solved the multi-type link prediction problem faster than before, but it has been found that it is not sufficient in terms of scalability. That is, according to a trial calculation, in the method proposed in the above paper, when there is one link type, when the size of one node set X is M and the size of the other node set Y is N, the amount of calculation is
The order of O (M ³ N ² + M ² N ³ ), and the amount of computer memory that must be reserved for calculation is the order of O (MN). Then, when the number of users and the number of documents ranges from hundreds of thousands to millions, such as SNS and blog site, it becomes difficult to perform link prediction.

  Accordingly, an object of the present invention is to provide a technique that enables a computer to perform link prediction in a scalable manner with respect to a set having an enormous number of objects such as an SNS and a blog site. is there.

そして、クロネッカー和の場合、以下の形式で

で表されることを仮定する。ここで、
D_Z ^-1/2Ｗ_ZD_Z ^-1/2、D_Y ^-1/2Ｗ_YD_Y ^-1/2、D_X ^-1/2Ｗ_XD_X ^-1/2は、それぞれ一般性を失うことなく表記の簡略化のため新しい行列Ｗ_Z、Ｗ_Y、W_Xとみなすことができることから、本願発明では以降ラプラシアン行列Ｌを以下の一般形式で表すことにする。

ただし、演算子は下記の数６の通りで、クロネッカー積とクロネッカー和は統一的に記述される。また、クロネッカー積の場合、c=1.0で、クロネッカー和の場合、c=3.0となるが、cの値は演算子によらず、データから学習して、最適な値を効率的に求めることも可能である。 In order to achieve the above object, the present invention provides the above formula 1, that is,
Starting from (σL + I) vec (F) = vec (F ^* ), the normalized Laplacian matrix L is, in the case of Kronecker product, of the form

And in the case of Kronecker sum,

It is assumed that here,
D _Z ^-1/2 W _Z D _Z ^-1/2 , D _Y ^-1/2 W _Y D _Y ^-1/2 , D _X ^-1/2 W _X D _X ^-1/2 Since it can be regarded as new matrices W _Z , W _Y , W _{X in} order to simplify the notation without losing them, the Laplacian matrix L will be represented in the following general form in the present invention.

However, the operator is as in Equation 6 below, and the Kronecker product and the Kronecker sum are described uniformly. In the case of the Kronecker product, c = 1.0, and in the case of the Kronecker sum, c = 3.0. However, the value of c is learned from the data regardless of the operator, and the optimum value can be obtained efficiently. Is possible.

  According to one aspect of the present invention, there is provided a technique implemented by computer processing to obtain an exact solution that enables Equation 1 to be solved in a scalable manner even for a set having a large number of objects. Is done.

In this method, first, the similarity matrices W _Z , W _Y , and W _X are decomposed into eigenvalues, respectively.

In the next stage, the matrix L is constructed using the obtained eigenvalues of W _Z , W _Y , and W _X. The expressions constituting the matrix L differ depending on whether W _Z , W _Y , and W _X are Kronecker products or Kronecker sums.

When the matrix L is obtained in this way, the inverse matrix of the (σL + I) portion is obtained by using V _Z , V _Y , V _X and L, which are matrices in which the eigenvectors of W _Z , W _Y , and W _X are arranged. Is advantageously calculated. When the inverse matrix of (σL + I) is obtained,
Since vec (F) = (σL + I) ⁻¹ vec (F ^* ), F is calculated. In this exact solution, the order of complexity is O (M ³ + N ³ ), which is substantially less than O (M ³ N ² + M ² N ³ ) in the above method. .

  According to the second aspect of the present invention, there is provided a method for obtaining an approximate solution by computer processing, which requires only a small amount of calculation and a memory amount of a computer without sacrificing the accuracy of the exact solution. .

In this method, the similarity matrices W _Z , W _Y , and W _X are first approximated to a low rank by a technique such as incomplete Cholesky decomposition. Also, with this method, the solution that is one step before the final solution is stored in the computer memory in a compact manner, so that a portion requiring a large solution can be quickly retrieved on demand. .

Next, eigenvalue decomposition of the low rank approximation matrix of the similarity matrix W _Z , W _Y , W _X is performed. The low-rank approximation is to approximate one matrix by the product of two rectangular matrices. Here, the low-rank approximation facilitates calculation of eigenvalue decomposition.

In the next stage, the matrix L˜ is constructed using the eigenvalues of the obtained low rank approximation matrix of W _Z , W _Y , W _X. The expressions constituting the matrix L˜ differ depending on whether W _Z , W _Y , and W _X are Kronecker products or Kronecker sums.

Thus the matrix L ~ is obtained, W _Z, W _Y, using W _X of each eigenvector is a matrix obtained by arranging V ~ _Z a low rank approximation matrix, V ~ _Y, a ~ V ~ _X and L, The inverse matrix of the part (σL ~ + I) is advantageously calculated. When the inverse matrix of (σL ~ + I) is obtained,
Since vec (F) = (σL˜ + I) ⁻¹ vec (F ^* ), F is calculated.

In the case of this approximate solution, if the link type is one, the shorter size of the W _X rectangular matrix decomposition is d _M , and the shorter size of the W _Y rectangular matrix decomposition is d _N.
The computational complexity is O (Md _M ² + Nd _N ² + d _M ³ + d _N ³ )
Memory is O (Nd _M + Md _N + d _M ² + d _N ² )
As shown in FIG.

Basically, d _M and d _N are selected so that d _M << M and d _N << N within a range in which the prediction accuracy of the link is maintained as desired.

  According to the present invention, there is provided a scalable link prediction technique capable of handling hundreds of thousands to millions of nodes by greatly reducing the amount of computation and the required memory size of a computer. It opens the way to the application of link prediction techniques to online community systems such as blog sites.

It is a figure which shows that a client computer is connected to an online community server via the internet. It is a figure which shows the hardware constitutions of a client computer. It is a figure which shows the hardware constitutions of an online community server. It is a functional logic block diagram of the Example of this invention. It is a figure which shows the example of the link between two sets. It is a figure which shows the example of user information. It is a figure which shows the outline | summary flowchart of the whole process of the process of this invention. It is a figure which shows the flowchart of a parameter setting process. It is a figure which shows the example of a similarity matrix. It is a figure which shows the example of a similarity matrix. It is a figure which shows the example of a similarity matrix. It is a flowchart of the process which solves the equation of link prediction. It is a flowchart of the process which calculates | requires the exact solution of link prediction. It is a flowchart of the process which calculates | requires the approximate solution of link prediction.

  Embodiments of the present invention will be described below with reference to the drawings. Unless otherwise noted, the same reference numerals refer to the same objects throughout the drawings. Also, what is described below is an example in which the present invention is used on an online community system, but this is an embodiment of the present invention, and the present invention will be described in the contents described in this example. It should be understood that the present invention can be used for applications that perform link prediction between arbitrary objects, such as collaborative filtering and biological network prediction, without intending to be limited.

  1, a plurality of client computers 106a, 106b,... 106z are connected to the online community server 102 via the Internet 104. In the system of FIG. 1, a user of a client computer logs in to the online community server 102 through a line of the Internet 104 through a Web browser. Specifically, a predetermined URL is typed into a Web browser and a predetermined page is displayed. The login may be performed using a predetermined dedicated client application program instead of the Web browser.

  When logging in, the user of the client computer uses a given user ID and a password associated therewith. Once a client computer user logs in, he or she writes a diary in the online community, browses the diary of another person who is allowed access, writes a comment, sees the news, and is a good friend Activities such as creating groups with each other, chatting, and searching for hobby communities.

  Next, referring to FIG. 2, a hardware block diagram of the client computer indicated by reference numerals 106a, 106b... 106z in FIG. In FIG. 2, the client computer has a main memory 206, a CPU 204, and an IDE controller 208, which are connected to the bus 202. In addition, a display controller 214, a communication interface 218, a USB interface 220, an audio interface 222, and a keyboard / mouse controller 228 are connected to the bus 202. A hard disk drive (HDD) 210 and a DVD drive 212 are connected to the IDE controller 208. The DVD drive 212 is used for introducing a program from a CD-ROM or DVD as necessary. A display device 216 having an LCD screen is preferably connected to the display controller 214. The display device 216 displays an online community screen through a Web browser.

  Devices such as a dedicated controller and an acceleration sensor / device can be connected to the USB interface 220 as necessary. These devices can be used to improve operability within the online community.

  A keyboard 230 and a mouse 232 are connected to the keyboard / mouse controller 228. The keyboard 230 is typically used in an online community to write a chat message or describe community content to be searched. The mouse 232 is used in the online community to click on a link to read news, select and execute an action from a menu, or select a diary to read.

  The CPU 204 may be, for example, any one based on a 32-bit architecture or a 64-bit architecture, such as Intel Pentium (trademark of Intel Corporation) 4, Core (trademark) 2 Duo, AMD Athlon (trademark), or the like. Can be used.

  The hard disk drive 210 stores at least an operating system and a web browser (not shown) that runs on the operating system, and the operating system is loaded into the main memory 206 when the system starts up. . As the operating system, Windows XP (a trademark of Microsoft Corporation), Windows Vista (a trademark of Microsoft Corporation), Linux (a trademark of Linus Torvalds), or the like can be used.

  The communication interface 218 uses the TCP / IP communication function provided by the operating system to communicate with the online community server 102 using the Ethernet (trademark) protocol or the like.

  FIG. 3 is a schematic block diagram of a hardware configuration on the online community provider side. As shown in FIG. 3, client computers 106 a, 106 b... 106 z are connected to the communication interface 302 of the online community server 102 via the Internet 104. The communication interface 302 is further connected to a bus 304, and a CPU 306, a main memory (RAM) 308, and a hard disk drive (HDD) 310 are connected to the bus 304.

  Although not shown, a keyboard, a mouse, and a display may be further connected to the online community server 102, and the entire online community server 102 may be managed and maintained by these.

  The hard disk drive 310 of the online community server 102 stores a correspondence table of user IDs and passwords for login management of operating systems, client computers, client computers 106a, 106b,... 106z. . The hard disk drive 310 further stores software such as Apache for causing the online community server 102 to function as a Web server, and is loaded into the main memory 308 when the online community server 102 is started up. To do. As a result, the client computers 106a, 106b,... 106z can access the online community server 102 using the TCP / IP protocol.

  In addition, the hard disk drive 310 of the online community server 102 further includes information on messages, diaries or blogs, bulletin boards, etc. of each user of the online community service, and information on the online community service. An HTML file and a multimedia format such as a graphic image, a moving image file, and a music file are stored.

  The user can write in the diary or the blog and the bulletin board, and other users can read the blog and the bulletin board and make comments according to the authorized authority.

As will be described in detail later, the hard disk drive 310 includes a module for calculating a multitype link prediction according to the present invention, that is, a module for calculating a third-order tensor F ^* , a module for calculating a similarity matrix, A module for calculating eigenvalues and eigenvectors of a matrix, a module for calculating a Kronecker product or Kronecker sum of a matrix, and a module for performing other necessary matrix calculations are stored.

  The configuration of blogs, bulletin boards, etc. and user access control for them can be realized with tools of well-known programming languages such as Perl, Ruby, PHP, Servlet, JSP. Alternatively, C, C ++, C #, Java (trademark of Sun Microsystems), etc. can be used.

  Furthermore, it is possible to embed JavaScript (trademark) in the HTML file as appropriate and configure the system to cooperate with Perl, Ruby, PHP, etc.

  Content such as blogs, bulletin boards, news, etc. can be stored in a content management database (CMDB) and managed centrally.

  As the online community server 102, models such as IBM (trademark of International Business Machines Corporation) System X, System i, System p, which can be purchased from International Business Machines Corporation. You can use any server. In this case, usable operating systems include AIX (trademark of International Business Machines Corporation), UNIX (trademark of The Open Group), Linux (trademark), Windows (trademark) 2003 Server, and the like.

FIG. 4 shows a block diagram of functions according to the present invention. In FIG. 4, user information 402 is user profile information stored in the hard disk drive 310 of the online community server 102. Specifically, as shown in FIG. 6, users x ₁ and x ₂ , X ₃ ... includes personal information such as gender, age, etc. Further, as shown in FIG. 6, the user information 402 preferably also includes information on links related to units such as individual articles of content such as blogs.

  The content 404 is a collection of blog articles, comments, and other information stored in the hard disk drive 310. For active sites, content is written and augmented daily by users.

  The log 406 is a file that records all activities such as user login, logout, blog writing, reading, and commenting, and is also stored in the hard disk drive 310. An example of the log 406 is as follows.

Date Time Activity User ID
-------------------------------------------------- ------------------------
2009/03/01 09: 51: 001 JST Login 0146230
2009/03/01 10: 00: 050 JST Logout 0099321
2009/03/01 10: 11: 130 JST PostBlog 0146230 PostID: 004524082
2009/03/01 10: 12: 020 JST OpenMsg 2965124 MsgID: 019348003

  As described above, the log 406 includes at least the date and time of activity, the content of the activity, and the user ID of the user who performed the activity. Further, when the blog is written (PostBlog), the ID of the written article is also stored. When the message is opened (OpenMsg), the ID of the opened message is stored. That is, it is possible to acquire the relationship between the user and the content element based on the entry of the log 406.

The calculation module 408 is a program module that performs processing according to the present invention, and is a module for calculating the multi-type link prediction according to the present invention described above, that is, a module that calculates the third-order tensor F ^* , A module for calculating a similarity matrix, a module for obtaining eigenvalues and eigenvectors of a matrix, a module for calculating a Kronecker product or Kronecker sum of a matrix, and a module for performing other necessary matrix calculations are included. These modules are preferably written in a well-known programming language such as JAVA ™, C, C ++, C #, etc., stored in hard disk drive 310 in an executable binary format, or operating system It is loaded into memory by the function and is executable.

The calculation module 408 generates the teacher tensor data F ^* 410, the similarity matrix 412, and the eigenvalue / eigenvector 414 of the similarity matrix using the information of the user information 402, the content 404, and the log 406. The teacher tensor data 414, the similarity matrix 412 and the eigenvalue / eigenvector 414 of the similarity matrix generated in this way are preferably temporarily placed in the RAM 308. For this reason, it is desirable that the RAM capacity of the online community server 102 be as large as possible.

  The calculation module 408 further performs calculation using information of the teacher tensor data 410, the similarity matrix 412 and the eigenvalue / eigenvector 414 of the similarity matrix, and generates a link prediction result F416. The actual calculation process will be described later with reference to flowcharts in FIG.

  In this case, the link prediction result F416 is a third-order tensor that includes, as elements, link prediction values between all users and all content elements. Thus, this result can be used to predict elements of content that are highly relevant to a particular user or group of users, and thus to identify content that should be recommended.

FIG. 5 is a diagram schematically showing a link relationship between the user set X and the content element set Y in the online community in this embodiment. In FIG. 5, here is an example in which user x ₁ reads content element y ₂ , user x ₂ comments on content element y ₁ , and user x ₃ writes content element y _3. It is shown. Such an existing link relationship is reflected as teacher tensor data F ^* 410. Such link-related information can be extracted from the log 406 in practice.

  In the example of FIG. 5, X ≠ Y, but there is also a model in which links between users are examined in the user set X. In this case, X = Y.

  Next, the link prediction processing algorithm of the present invention will be described with reference to FIG. In particular, FIG. 7 is a schematic flowchart of the entire process.

  Step 702 is a step for setting parameters, that is, a step for preparing data such as a teacher data tensor and a similarity matrix in advance, which will be described in detail later in connection with the flowchart of FIG.

  Step 704 is the step of solving the link prediction equation, which is the basis of the present invention, and will be described later in detail with reference to the flowchart of FIG.

  Step 706 is a step of outputting a prediction value obtained by solving a link prediction equation. In this embodiment, the obtained prediction value is a prediction value of each link between each user and each content element, and this value is used to predict a high link prediction with a specific content element. Since it can be presumed which user has the value, the content can be recommended based on this value.

  Next, steps for setting parameters will be described with reference to the flowchart of FIG. The log 406 records the user ID, the content element ID, and the type of activity. Therefore, in step 802 of FIG. 8, the calculation module 408 scans the entries of the log 406, for example, to count the links between the user and the content elements for each type.

So, let N be the number of user-content element pairs whose link is known, N ⁺ the number of pairs known to have a link, and N the number of pairs known to have no link. ^- and when the component [F ^*] _{i tensor, j,} the _k, i if you know that there is a link k between the ^{j ε + = N / N +} , between i and j If k links is known not ε ^- = N / N ^- so as to store. If the existence of a link is not known, 0 is simply entered. Here, the user who knows whether or not there is a link is a case where the presence or absence of the relationship of i, j, and k is obtained from the log 406, otherwise the presence or absence of the link is unknown, that is, the target of prediction It is defined as a thing. It should be understood that such a method of giving the parameter of F ^* is an example, and the present invention is not limited to such a method of giving the parameter.

  In step 804, the calculation module 408 performs the set X similarity matrix, the set Y similarity matrix, and the set Z at the user set X, the content set Y, and the link between X and Y in FIG. Compute the similarity matrix. Here, let Z be the set of links between X and Y for later convenience.

The similarity matrix of X, Y, and Z is basically calculated based on the property between the elements of each set. FIG. 9 shows an example of the similarity matrix of the set X of users. Such a similarity matrix can be generated, for example, by calculating a normalized distance between the feature vectors of each user as shown in FIG. Therefore, the similarity matrix is an M × M matrix where M is the number of users. At this time, the value of each element is non-negative and normalized so that the diagonal component is 1 or other value. This similarity matrix is denoted as W _X.

FIG. 10 shows an example of the similarity matrix of the content set Y. Such a similarity matrix, for example, parses each element of the content, extracts keywords, constructs feature vectors as a sequence of those keywords, and sets normalized distances between the feature vectors. It can be generated by calculating. Therefore, the similarity matrix is an N × N matrix where N is the number of content elements. At this time, the value of each element is non-negative and normalized so that the diagonal component is 1 or other value. This similarity matrix is denoted as W _Y.

FIG. 11 shows an example of the similarity matrix of the link set Z. Such a similarity matrix can be generated, for example, by listing user and content elements associated with individual links to construct feature vectors and calculating normalized distances between those feature vectors. can do. Therefore, the similarity matrix is a T × T matrix, where T is the number of types of links. At this time, the value of each element is non-negative and normalized so that the diagonal component is 1 or other value. This similarity matrix is denoted as W _Z.

  These notations are followed in the description of the algorithm below.

  In step 806, the calculation module 408 obtains a variable value for solving the link prediction equation. The variable value here is, for example, the third-order tensor F for storing the result.

  Next, processing for solving a link prediction equation will be described with reference to FIG. In step 1202 of FIG. 12, the calculation module 408 determines whether the dimension of the similarity matrix is smaller than a predetermined threshold. This is for determining whether to apply the exact solution or the approximate solution according to the present invention. That is, the exact solution gives more accurate link prediction, but requires a large amount of computer memory and requires more computation time, so it is determined which one to apply based on the dimension of the similarity matrix.

  If it is determined that the dimension of the similarity matrix is smaller than the predetermined threshold value, in step 1204, the calculation module 408 executes a step of obtaining an exact solution of the link prediction equation. The step of obtaining the exact solution of the link prediction equation will be described in more detail later with reference to FIG.

  If it is determined that the dimension of the similarity matrix is larger than the predetermined threshold, the calculation module 408 executes a step of obtaining an approximate solution of the link prediction equation in step 1206. The step of obtaining an approximate solution of the link prediction equation will be described in more detail later with reference to FIG.

Next, an algorithm for solving the link prediction equation according to the present invention will be described.
First, Formula 1 quoted from the said nonpatent literature is redisplayed.
(σL + I) vec (F ) = vec (F *) ··· Formula 1

  In the description of this embodiment, it is stated that the diagonal components of the similarity matrix are all 1, but in reality the elements of the similarity matrix represent the similarity between the elements of the set. Is possible. However, in the present invention, it is an important condition that the sum of each row of such a similarity matrix can be efficiently calculated.

When obtaining an exact solution, the sum of each row of the similarity matrix can be efficiently obtained by a simple method, but when obtaining an approximate solution, the sum of each row of the matrix that approximates the similarity matrix can also be obtained efficiently. In this way, the similarity matrix can be normalized as shown in the above formulas 1 and 2 using the sum of each row or the maximum value of the sum of rows. For example, in the step of obtaining an approximate solution, the amount of calculation for calculating the sum of each row of an N × N similarity matrix takes O (N ² ) steps, but from the N × d low-rank approximation matrix of the similarity matrix, Only O (Nd) is required for calculating the sum of sums.

よって、解きたい式は、下記の式２のようになる。なお、係数σは、例えば0.01と設定される。

ここで、類似度行列の間に作用される演算子は下記のとおりであり、

すなわち、クロネッカー積

または、クロネッカー和

のどちらかである。クロネッカー積またはクロネッカー和のどちらを使うかは、用途による。vec(Ｆ)という記法は、テンソルＦの成分を並べてベクトルにしたものである。 Then, L is a normalized Laplacian matrix, and when Equation 3 is reprinted, it is expressed as follows.

Therefore, the equation to be solved is as shown in Equation 2 below. The coefficient σ is set to 0.01, for example.

Here, the operators operated during the similarity matrix are as follows:

That is, the Kronecker product

Or Kronecker sum

Either. Whether to use Kronecker product or Kronecker sum depends on the application. The notation vec (F) is a vector in which the components of the tensor F are arranged.

  In this case, there is no clear standard whether to use the Kronecker product or the Kronecker sum, but since the method of the present invention is fast, it is possible to efficiently select the optimal one by learning from the data. Depending on the problem you want, you can use the one with good accuracy.

The definition is as follows:
Kronecker product: If all three elements of similarity matrix are similar, the triplet is similar. Kronecker sum: Two elements in three elements of similarity matrix are the same, and the remaining 1 If the two are similar, the triplet is similar.

この逆行列をいかに高速に、且つ少ないメモリでコンパクトに解くかが、この技法のキーである。 Thus, the solution of Equation 2 is written as Equation 3 below.

The key to this technique is how to solve this inverse matrix quickly and compactly with a small amount of memory.

  For Vec () notation, Kronecker product, Kronecker sum, AJ Laub, Matrix for Scientists and Engineers, Society for Industrial and Applied Mathematics 2005; or David A. Harville, Matrix Algebra from a Statistician's Perspective, Springer Verlag 1997, etc. Please refer to.

Returning to FIG. 13, in step 1302, the calculation module 408 calculates eigenvalue decomposition of W _X , W _Y , and W _Z. Here, AJ Laub, Matrix for Scientists and Engineers, Society for Industrial and Applied Write down useful theorems described in Mathematics 2005.

または、クロネッカー和

の固有ベクトルは、ともに

で与えられる。
また、固有値は、クロネッカー積の場合

で与えられ、クロネッカー和の場合

で与えられる。 <Theorem 1>
{λ _X ⁽¹⁾ , λ _X ⁽²⁾ , ..., λ _X ^(M) },
{λ _Y ⁽¹⁾ , λ _Y ⁽²⁾ , ..., λ _Y ^(N) },
{λ _Z ⁽¹⁾ , λ _Z ⁽²⁾ , ..., λ _Z ^(T) }
_Are the eigenvalues of W _X , W _Y , and W _Z , respectively.
Also, _let V _X , V _Y , and V _{Z be} a matrix in which eigenvectors (vertical vectors) of W _X , W _Y , and W _Z are arranged, respectively.
Then Kronecker product

Or Kronecker sum

The eigenvectors of are both

Given in.
Also, the eigenvalue is the case of Kronecker product

In case of Kronecker sum

Given in.

Therefore, in step 1302, eigenvalue decomposition of W _X , W _Y , and W _Z is performed.
_{W X = V X diag (λ} X (1), λ X (2), ..., λ X (M)) V X T
W _Y = V _Y diag (λ _Y ⁽¹⁾ , λ _Y ⁽²⁾ , ..., λ _Y ^(N) ) V _Y ^T
W _Z = V _Z diag (λ _Z ⁽¹⁾ , λ _Z ⁽²⁾ , ..., λ _Z ^(T) ) V _Z ^T
And Here, diag () is a diagonal matrix.

Therefore, in the case of the Kronecker product, the third-order tensor L that holds the eigenvalue is
[L] _{i, j, k} ≡ λ _X ⁽ⁱ⁾ λ _Y ^(j) λ _Z ^(k)
In the case of Kronecker sum, the third-order tensor L holding the eigenvalue is
[L] _i, defined ^{_{j, k ≡ λ X (i}} ) + λ Y and ^{_{(j) + λ Z (k}} ).
In the present invention, it is possible to predict the link by combining both the Kronecker product and the Kronecker sum, and the third-order tensor L that holds the eigenvalue at that time is calculated.
[L] _{i, j, k} ≡α (λ _X ⁽ⁱ⁾ λ _Y ^(j) λ _Z ^(k) ) + β (λ _X ⁽ⁱ⁾ + λ _Y ^(j) + λ _Z ^(k) ) .
However, α and β are arbitrary real numbers and can be set by learning optimum values from the data. Since the value of the third-order tensor L in the present invention can be efficiently calculated from any function of the eigenvalue of the similarity matrix or the eigenvalue of the approximate matrix, it is limited to only those obtained from the Kronecker sum and Kronecker product is not. Further, which function is used for the value of the third-order tensor L can be efficiently selected by learning from the data.

と書き直せる。ここで

であることを利用すると、

と書き直せる。 Then, according to Theorem 1, the inverse matrix of Equation 3 is

Can be rewritten. here

If you use that,

Can be rewritten.

のように定義すると、解は次のように書ける。

Since ((cσ + 1) I−σdiag (vec (L))) is a diagonal matrix, the inverse matrix can be easily obtained. Therefore, in step 1304, the third floor tensor D is

, The solution can be written as

This equation can be further simplified as follows.

ここで、×₁などの演算は、テンソルのモード積で、T. G. Kolder & B. W. Bader, "Tensor decomposition and applications" Tech. Rep. SAND2007-6702, Sandia National Laboratories などにその詳細な定義が記載されている。 As a result, the following formula 5 is obtained. In step 1306, using this equation 5, the resulting F is obtained.

Here, operations such as × ₁ are tensor mode products, and detailed definitions are described in TG Kolder & BW Bader, "Tensor decomposition and applications" Tech. Rep. SAND2007-6702, Sandia National Laboratories, etc. .

Next, the approximate solution will be described with reference to the flowchart of FIG. The approximate solution is used when the similarity matrix or the solution F is too large to fit in the computer memory. In step 1402, in order to deal with the problem that the similarity matrix is too large, the calculation module 408 uses a method such as incomplete Cholesky decomposition to calculate the similarity matrix W _X , W _Y , W as follows. Approximate _Z with low rank.

Here, G _X is an M × M matrix, G _Y is an N × N matrix, and G _Z is a T × T matrix,
M> M ~, N> N ~, T> T ~. By using the incomplete Cholesky decomposition, this decomposition can be executed with the amount of calculation O (MM ~ ² + NN ~ ² + TT ~ ² ) without explicitly configuring the similarity matrix in the memory.

In step 1404, the calculation module 408:
^{_{G X T G X, G Y}} T G Y, determine the eigenvalue decomposition of the G _Z ^T G _{Z.
^{G X T G X = U ~}} X diag (λ ~ X (1), λ ~ X (2), ..., λ ~ X (M ~)) U ~ X T
G _Y ^T G _Y = U ~ _Y diag (λ ~ _Y ⁽¹⁾ , λ ~ _Y ⁽²⁾ , ..., λ ~ _Y ^{(N ~)} ) U ~ _X ^T
G _Z ^T G _Z = U ~ _Z diag (λ ~ _Z ⁽¹⁾ , λ ~ _Z ⁽²⁾ , ..., λ ~ _Z ^{(N ~)} ) U ~ _Z ^T
That is, here, U to _X , U to _Y , and U to _Z are respectively
G _X ^T G _X, a G _Y ^T G _Y, matrix obtained by arranging eigenvectors of G _Z ^T G _Z (column vector).

G _X ^T G _X , G _Y ^T G _Y , G _Z ^T G _Z are less because orders can be much smaller than G _X G _X ^T , G _Y G _Y ^T , G _Z G _Z ^T Eigenvalue decomposition can be performed in the memory.

In the next step 1406, eigenvectors of G _X G _X ^T , G _Y G _Y ^T , and G _Z G _Z ^T are obtained as follows.
V ~ _X = G _X U ~ _X diag (λ ~ _X ⁽¹⁾ , λ ~ _X ⁽²⁾ , ..., λ ~ _X ^{(M ~)} ) ^-1/2
V ~ _Y = G _Y U ~ _Y diag (λ ~ _Y ⁽¹⁾ , λ ~ _Y ⁽²⁾ , ..., λ ~ _Y ^{(N ~)} ) ^-1/2
V ~ _Z = G _Z U ~ _Z diag (λ ~ _Z ⁽¹⁾ , λ ~ _Z ⁽²⁾ , ..., λ ~ _Z ^{(T ~)} ) ^-1/2

Using the above results, the inverse matrix of Equation 2 is as follows.

Where L ~ is Kronecker product
[L ~] _{i, j, k} ≡ λ ~ _X ⁽ⁱ⁾ λ ~ _Y ^(j) λ ~ _Z ^(k)
For Kronecker sum
_{[L ~] i, j,} k ≡ λ ~ X (i) + λ ~ Y (j) + λ ~ Z (k)
It is defined as

なので、このままでは厳密解の場合のように、式２の逆行列を変形することはできない。 However, unlike the exact solution,

Therefore, the inverse matrix of Equation 2 cannot be transformed as it is in the exact solution.

のようになる。 So, using Woodbury's formula described in CM Bishop, Pattern Recognition and Machine Learning, 2006, Springer Verlag,

become that way.

Therefore, the third-order tensor D is defined as shown in Equation 6 below.

Then, the solution is expressed as follows.

From this, F is calculated | required as follows.

Ｆ自体のサイズが大きいため、式８の結果を予め計算して保持しておくことにより、ステップ１４１２で、必要な予測だけ、オンデマンドで計算することが可能となる。 In particular, in step 1410, keep the following as the central part of the equation:

Since the size of F itself is large, by calculating and holding the result of Equation 8 in advance, it becomes possible to calculate only the necessary prediction on demand in Step 1412.

  Thus, in the resulting tensor F, the value of the component (i, j, k) stores the value of the link k between the element i of the set X and the element j of the set Y, and the value is large. It is interpreted that the likelihood of link k is higher.

  As described above, the multi-type link prediction technique of the present invention has been described by taking the online community server as an example. However, the present invention is not limited to this, and any arbitrary method such as collaborative filtering, biological network prediction, etc. It can be applied to an application example of multi-type link prediction.

  In the above embodiment, the multi-type link prediction program is executed on the Web server. However, those skilled in the art will understand that it can be similarly executed on a stand-alone computer. Will.

402 ... user information 404 ... content 406 ... log 408 ... calculation module 410 ... teacher tensor data F ^*
412 ... Similarity matrix data 414 ... Eigenvector, eigenvalue data 416 ... Link prediction result

Claims (18)

A system for predicting a multitype link between data of a first set of nodes and data of a second set of nodes by computer processing, comprising:
A memory capable of reading and writing data by the computer;
Data of the first set of nodes stored in the memory;
Data of the second set of nodes stored in the memory;
Means for storing in the memory a set of multi-type link information between the first node and the second node;
Means for calculating, from the multi-type link information, the first node, the second node, and third-order tensor teacher data therebetween;
Means for calculating data of a first similarity matrix of the first set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Means for calculating data of a second similarity matrix of the second set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Means for calculating data of a third similarity matrix of the set of multitype link information so that its diagonal component is normalized to 1 or a predetermined value;
Means for calculating eigenvalue decomposition of each of the first similarity matrix, the second similarity matrix, and the third similarity matrix and obtaining eigenvector matrix data in which the eigenvalues and eigenvectors are arranged. When,
Means for calculating a Kronecker product or Kronecker sum of the eigenvector matrix;
Means for calculating a third-order tensor parameter from the value of the eigenvalue;
Means for calculating multi-type link prediction based on the third-order tensor teacher data, the Kronecker product or Kronecker sum of the eigenvector matrix, and the third-order tensor parameters;
Multi-type link prediction system. When the means for calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker product, each of the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ λ _Y ^(j) λ _Z The multi-type link prediction system of claim 1, determined based on a value of ^(k) . When the means for calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker sum, each of the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, and k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ + λ _Y ^(j) + The multi-type link prediction system of claim 1, determined based on a value of λ _Z ^(k) . A method for predicting a multitype link between data of a first set of nodes and data of a second set of nodes by computer processing, comprising:
Loading data of the first set of nodes into memory of the computer;
Loading data of the second set of nodes into the memory;
Loading a set of multi-type link information between the first node and the second node into the memory;
Calculating, from the multitype link information, the first node, the second node, and third-order tensor teacher data therebetween;
Calculating the first similarity matrix data of the first set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Calculating data of a second similarity matrix of the second set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Calculating data of a third similarity matrix of the set of multitype link information so that its diagonal component is normalized to 1 or a predetermined value;
Calculating eigenvalue decomposition of each of the first similarity matrix, the second similarity matrix, and the third similarity matrix and obtaining eigenvector matrix data in which the eigenvalues and eigenvectors are arranged. When,
Calculating a Kronecker product or Kronecker sum of the eigenvector matrix;
Calculating a third-order tensor parameter from the value of the eigenvalue;
Calculating multi-type link prediction based on the third-order tensor teacher data, the Kronecker product or Kronecker sum of the eigenvector matrix, and the third-order tensor parameters;
Multi-type link prediction method. When the step of calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker product, each of the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ λ _Y ^(j) λ _Z The multi-type link prediction method according to claim 4, wherein the multi-type link prediction method is determined based on a value of ^(k) . If the step of calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker sum, the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, and k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ + λ _Y ^(j) + λ _Z ^(k) is the basis of the value determination, multi-type link prediction method of claim 4. A program for predicting a multi-type link between data of a set of first nodes and data of a set of second nodes by computer processing,
In the computer,
Loading data of the first set of nodes into memory of the computer;
Loading data of the second set of nodes into the memory;
Loading a set of multi-type link information between the first node and the second node into the memory;
Calculating, from the multitype link information, the first node, the second node, and third-order tensor teacher data therebetween;
Calculating the first similarity matrix data of the first set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Calculating data of a second similarity matrix of the second set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Calculating data of a third similarity matrix of the set of multitype link information so that its diagonal component is normalized to 1 or a predetermined value;
Calculating eigenvalue decomposition of each of the first similarity matrix, the second similarity matrix, and the third similarity matrix and obtaining eigenvector matrix data in which the eigenvalues and eigenvectors are arranged. When,
Calculating a Kronecker product or Kronecker sum of the eigenvector matrix;
Calculating a third-order tensor parameter from the value of the eigenvalue;
Performing the third-order tensor teacher data, the Kronecker product or Kronecker sum of the eigenvector matrix, and calculating a multi-type link prediction based on the third-order tensor parameters.
Multi-type link prediction program. When the step of calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker product, each of the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ λ _Y ^(j) λ _Z The multi-type link prediction program according to claim 7, which is determined based on the value of ^(k) . If the step of calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker sum, the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, and k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ + λ _Y ^(j) + The multi-type link prediction program according to claim 7, which is determined based on a value of λ _Z ^(k) . A system for predicting a multitype link between data of a first set of nodes and data of a second set of nodes by computer processing, comprising:
A memory capable of reading and writing data by the computer;
Data of the first set of nodes stored in the memory;
Data of the second set of nodes stored in the memory;
Means for storing in the memory a set of multi-type link information between the first node and the second node;
Means for calculating, from the multi-type link information, the first node, the second node, and third-order tensor teacher data therebetween;
Means for calculating data of a first similarity matrix of the first set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Means for calculating data of a second similarity matrix of the second set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Means for calculating data of a third similarity matrix of the set of multitype link information so that its diagonal component is normalized to 1 or a predetermined value;
Means for determining a low rank approximate decomposition of the first similarity matrix, the second similarity matrix, and the third similarity matrix;
Means for obtaining eigenvalue decomposition of the transposed matrix of the low rank approximate decomposition;
Means for obtaining data of an eigenvector matrix in which the eigenvalues of the low rank approximate decomposition and eigenvectors are arranged using the eigenvalue decomposition;
Means for calculating a Kronecker product or Kronecker sum of the eigenvector matrix;
Means for calculating a third-order tensor parameter from the value of the eigenvalue;
Means for calculating multi-type link prediction based on the third-order tensor teacher data, the Kronecker product or Kronecker sum of the eigenvector matrix, and the third-order tensor parameters;
Multi-type link prediction system. When the means for calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker product, each of the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ λ _Y ^(j) λ _Z The multi-type link prediction system of claim 10, wherein the multi-type link prediction system is determined based on a value of ^(k) . When the means for calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker sum, each of the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, and k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ + λ _Y ^(j) + The multitype link prediction system of claim 10, wherein the multitype link prediction system is determined based on a value of λ _Z ^(k) . A method for predicting a multitype link between data of a first set of nodes and data of a second set of nodes by computer processing, comprising:
Loading data of the first set of nodes into memory of the computer;
Loading data of the second set of nodes into the memory;
Loading a set of multi-type link information between the first node and the second node into the memory;
Calculating, from the multitype link information, the first node, the second node, and third-order tensor teacher data therebetween;
Calculating the first similarity matrix data of the first set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Calculating data of a second similarity matrix of the second set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Calculating data of a third similarity matrix of the set of multitype link information so that its diagonal component is normalized to 1 or a predetermined value;
Obtaining a low rank approximate decomposition of the first similarity matrix, the second similarity matrix, and the third similarity matrix;
Obtaining eigenvalue decomposition of the transposed matrix of the low rank approximate decomposition;
Using the eigenvalue decomposition to determine the eigenvalue of the low rank approximate decomposition and eigenvector matrix data in which eigenvectors are arranged;
Calculating a Kronecker product or Kronecker sum of the eigenvector matrix;
Calculating a third-order tensor parameter from the value of the eigenvalue;
Calculating multi-type link prediction based on the third-order tensor teacher data, the Kronecker product or Kronecker sum of the eigenvector matrix, and the third-order tensor parameters;
Multi-type link prediction method. When the step of calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker product, each of the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ λ _Y ^(j) λ _Z ^(k) is the basis of the value determination, multi-type link prediction method of claim 13. If the step of calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker sum, the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, and k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ + λ _Y ^(j) + The multi-type link prediction method according to claim 13, wherein the multi-type link prediction method is determined based on a value of λ _Z ^(k) . A program for predicting a multi-type link between data of a set of first nodes and data of a set of second nodes by computer processing,
In the computer,
Loading data of the first set of nodes into memory of the computer;
Loading data of the second set of nodes into the memory;
Loading a set of multi-type link information between the first node and the second node into the memory;
Calculating, from the multitype link information, the first node, the second node, and third-order tensor teacher data therebetween;
Calculating the first similarity matrix data of the first set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Calculating data of a second similarity matrix of the second set of nodes so that its diagonal component is normalized to 1 or a predetermined value;
Calculating data of a third similarity matrix of the set of multitype link information so that its diagonal component is normalized to 1 or a predetermined value;
Obtaining a low rank approximate decomposition of the first similarity matrix, the second similarity matrix, and the third similarity matrix;
Obtaining eigenvalue decomposition of the transposed matrix of the low rank approximate decomposition;
Using the eigenvalue decomposition to determine the eigenvalue of the low rank approximate decomposition and eigenvector matrix data in which eigenvectors are arranged;
Calculating a Kronecker product or Kronecker sum of the eigenvector matrix;
Calculating a third-order tensor parameter from the value of the eigenvalue;
Performing the third-order tensor teacher data, the Kronecker product or Kronecker sum of the eigenvector matrix, and calculating a multi-type link prediction based on the third-order tensor parameters.
Multi-type link prediction program. When the step of calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker product, each of the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ λ _Y ^(j) λ _Z The multi-type link prediction program according to claim 16, which is determined based on the value of ^(k) . If the step of calculating the Kronecker product or the Kronecker sum of the eigenvector matrix calculates the Kronecker sum, the first similarity matrix, the second similarity matrix, and the third similarity matrix Assuming that the eigenvalues are λ _X ⁽ⁱ⁾ , λ _Y ^(j) , and λ _Z ^(k) , respectively, the i, j, and k components of the third-order tensor parameter are λ _X ⁽ⁱ⁾ + λ _Y ^(j) + The multi-type link prediction program according to claim 16, which is determined based on the value of λ _Z ^(k) .

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