3.1 Tag-based User Similarity Measures where Sj is the tag-based similarity between user i and We introduce several possible measures for characterizing user j computed based on one of the measures defined in user similarities based on the tagging matrix Z.Here,Tij Section 3.1,C =D-S is known as the Laplacian matrix [5] denotes the index set of tags which are used by both user i with D being a diagonal matrix whose diagonal elements and user j. D=;Si,and tr()denotes the trace of a matrix. To integrate tagging information into the CF procedure 3.1.I Cosine Similarity TagiCoFi combines the criteria (9)and (10)to give the fol- The cosine similarity is defined as follows: lowing objective function for minimization: ∑kETy ZikZjk (4) √∑krZ疏V∑kerZ保 (n-u v) 3.1.2 Pearson Similarity )+(VV)] + The Pearson correlation coefficient between two users is c]. (11) defined as follows: where B is an additional regularization parameter to control p1(i,j)= ∑ker(Zk-Z)(Zk-Z) (5) the contribution from the tagging information. V∑keTy(Zk-Z)PV∑ker(Zk-Z)2 The formulation in (11)can be seen as an adaptation of relation reqularized matrir factorization(RRMF)[13]which where Zi= The Pearson similarity is then models relational data containing both relation information and content information.The main difference between Tagi- defined as: CoFi and RRMF is that TagiCoFi can handle missing data. s=1+exp(-p6,刃 (6) which is one of the key characteristics of CF. 3.1.3 Euclidean-based Similarity 3.3 Learning The objective function in (11)can be rewritten as follows: The Euclidean distance between two users is defined as follows: NM p2(i,)= (Zik-Zjk)2. (7) =1=1 The Euclidean-based similarity is then defined as: +U(aI+Bc)UT] S号e=exp [P(色，]2 +vv]. (12) 2σ2 (8) where I is the identity matrix.We use an alternating gra- where o is a user-controlled parameter. dient descent procedure to optimize (12).More specifically, each time we fix one variable (U or V)and minimize the 3.2 Model Formulation of TagiCoFi objective function with respect to the other one (V or U). Like in PMF [17].we adopt a similar MF procedure to find This procedure is repeated for several iterations until some U and V by minimizing the following criterion function: termination condition is satisfied. To learn U,we first rewrite (12)as follows: N A ∑2，(R-uvP+ruU+vv], f=g+h+C. (13) =1j=1 where C is a constant independent of U,and where a is a regularization parameter for complexity control. Furthermore,TagiCoFi employs the user similarities de- N M fined based on the tagging information to regularize the MF 9=∑∑，(B-UgVg procedure,with the goal to make the user-specific latent fea- i=1j=1 ture vectors as similar as possible if the corresponding users D have similar tagging history.We can achieve this goal by h=号∑Ua.(al+Bc)ui minimizing the following criterion function: 2 (14) d=1 NN From (14),we can see that the rows of U in h are decou- pled.Hence,we apply gradient descent to optimize one row =1=1 of U at a time with the other rows fixed. Because d=1 =(∑)a- D 1=1 =∑Ua.cUa >YijVa(Ris -Un:V.;+Ua:Vaj),(15) =tr(UCU), (10) j=13.1 Tag-based User Similarity Measures We introduce several possible measures for characterizing user similarities based on the tagging matrix Z. Here, T ij denotes the index set of tags which are used by both user i and user j. 3.1.1 Cosine Similarity The cosine similarity is defined as follows: S cos ij = P k∈T ij ZikZjk qP k∈T ij Z2 ikqP k∈T ij Z2 jk . (4) 3.1.2 Pearson Similarity The Pearson correlation coefficient between two users is defined as follows: ρ1(i, j) = P k∈T ij (Zik − Z¯i)(Zjk − Z¯j ) qP k∈T ij (Zik − Z¯i) 2 qP k∈T ij (Zjk − Z¯j ) 2 , (5) where Z¯i = P k∈T ij Zik |T ij | . The Pearson similarity is then defined as: S pea ij = 1 1 + exp(−ρ1(i, j)). (6) 3.1.3 Euclidean-based Similarity The Euclidean distance between two users is defined as follows: ρ2(i, j) = s X k∈T ij (Zik − Zjk) 2. (7) The Euclidean-based similarity is then defined as: S euc ij = exp „ − [ρ2(i, j)]2 2σ2 « , (8) where σ is a user-controlled parameter. 3.2 Model Formulation of TagiCoFi Like in PMF [17], we adopt a similar MF procedure to find U and V by minimizing the following criterion function: 1 2 XN i=1 XM j=1 Yij (Rij − U T ∗iV∗j ) 2 + α 2 h tr(U T U) + tr(V T V) i , (9) where α is a regularization parameter for complexity control. Furthermore, TagiCoFi employs the user similarities de- fined based on the tagging information to regularize the MF procedure, with the goal to make the user-specific latent fea￾ture vectors as similar as possible if the corresponding users have similar tagging history. We can achieve this goal by minimizing the following criterion function: f1 = 1 2 XN i=1 XN j=1 SijkU∗i − U∗jk 2 = 1 2 XN i=1 XN j=1 h Sij XD d=1 (Udi − Udj ) 2 i = XD d=1 Ud∗LU T d∗ = tr(ULU T ), (10) where Sij is the tag-based similarity between user i and user j computed based on one of the measures defined in Section 3.1, L = D−S is known as the Laplacian matrix [5] with D being a diagonal matrix whose diagonal elements Dii = P j Sij , and tr(·) denotes the trace of a matrix. To integrate tagging information into the CF procedure, TagiCoFi combines the criteria (9) and (10) to give the fol￾lowing objective function for minimization: f = 1 2 XN i=1 XM j=1 Yij (Rij − U T ∗iV∗j ) 2 + α 2 h tr(U T U) + tr(V T V) i + β 2 h tr(ULU T ) i , (11) where β is an additional regularization parameter to control the contribution from the tagging information. The formulation in (11) can be seen as an adaptation of relation regularized matrix factorization (RRMF) [13] which models relational data containing both relation information and content information. The main difference between Tagi￾CoFi and RRMF is that TagiCoFi can handle missing data, which is one of the key characteristics of CF. 3.3 Learning The objective function in (11) can be rewritten as follows: f = 1 2 XN i=1 XM j=1 Yij (Rij − U T ∗iV∗j ) 2 + 1 2 trh U(αI + βL)U T i + α 2 h tr(V T V) i , (12) where I is the identity matrix. We use an alternating gra￾dient descent procedure to optimize (12). More specifically, each time we fix one variable (U or V) and minimize the objective function with respect to the other one (V or U). This procedure is repeated for several iterations until some termination condition is satisfied. To learn U, we first rewrite (12) as follows: f = g + h + C, (13) where C is a constant independent of U, and g = 1 2 XN i=1 XM j=1 Yij (Rij − U T ∗iV∗j ) 2 h = 1 2 XD d=1 Ud∗(αI + βL)U T d∗. (14) From (14), we can see that the rows of U in h are decou￾pled. Hence, we apply gradient descent to optimize one row of U at a time with the other rows fixed. Because ∂g ∂Udi = “XM j=1 YijV 2 dj” Udi− XM j=1 YijVdj (Rij − U T ∗iV∗j + UdiVdj ), (15)