example,we can use SRM to analyze the actor variance to This is completely different from the goal of CF which is find out whether people see others as similar in terms of in- to predict some of the missing elements in the rating matrix. telligence.Moreover,in general,it is assumed that we can Hence,SRM cannot be directly applied to CF. collect all of the data for SRM and hence no elements are missing.For CF,however,the goal is to predict some of the SRMCoFi missing ratings,making it fundamentally different from that of the original SRM. Since CF is fundamentally different from traditional appli- cations of SRM,we need to adapt SRM appropriately if we Contributions By adapting SRM,we propose in this pa- want to apply it to CF.The result is our social relations model for collaborative filtering,which we abbreviate as per a novel CF model called SRMCoFi.Some promising SRMCoFi in the sequel. properties of SRMCoFi are highlighted here: SRMCoFi formulates the CF problem under a probabilis- Model tic framework,making it very convenient to learn the SRMCoFi defines the likelihood function over the observed model parameters and subsequently control the model ca- ratings as follows: pacity automatically. SRMCoFi,which subsumes many existing model-based Fii u+ai+bi+UVi CF methods as special cases,provides a general frame- E(RijlF:j)=g-(Fu), work for CF. N M SRMCoFi offers theoretical justifications for many em- p(R F, ΠΠp(RlF,， (3) pirical findings by previous CF methods. =1j=1 SRMCoFi is very efficient and easily implementable due where U E RDxN is the latent user feature matrix and to the closed-form updates for parameter learning. V E RDxM is the latent movie feature matrix,with col- Social Relations Model umn vectors Ui and Vi denoting user-specific and movie- specific latent feature vectors respectively,E(.)is the ex- Let Fi;denote the dyadic measurement of the perception or pectation function,g()is the link function in a generalized rating from actor i to partner j.In SRM(Kenny 1994),Fij linear model (GLM)(Gelman et al.2003).o is the variance is decomposed into random effects a,b,e as follows: or dispersion parameter of the GLM(Gelman et al.2003), F)=μ+ai+b+e+ei (1) and Iij is an indicator variable which is equal to 1 if user i where u is a constant representing the overall mean,ai is rated movie j and 0 otherwise.For example,if g(Fij)= the actor effect representing the average level of rating of an exp(F）andp(RlF）=Poisson(Rilg1(Fi),the actor in the presence of a variety of partners,b;is the part- model is equivalent to a Poisson regression model(Gelman ner effect representing the average level of a rating which a et al.2003)with the log-link function,which is always used person elicits from a variety of actors,eij is the relationship to model data taking the form of counts.One representative effect representing the rating of an actor toward a particu- application of the Poisson regression model is epidemiology lar partner,above and beyond their actor and partner effects where the incidence of diseases is studied (Gelman et al. and eii is an error term representing the unstable aspect of 2003,page51). the perception.In particular,for the movie rating applica- The SRMCoFi model defined in(3)adapts SRM in two tion,actors refer to users and partners to movies. aspects: In general,a,b,e,e are assumed to be random variables .SRMCoFi adopts a bilinear term UTVi to represent the with the following distributions: a,N40,),b,N0,, relationship effect eij of SRM.Hence,SRMCoFi seam- lessly integrates the linear (ai,bi)and bilinear effects 60,. into a principled framework.This adaptation is very cru- e~N(0,o2), cial for SRMCoFi to achieve the goal of CF,which is to Cov(ai,b)≠0， Cov(e,eji)≠0.， (2) predict some missing elements in the preference matrix where Cov()denotes the covariance.In SRM.all the other After learning the model,each user i is associated with covariances are assumed to be 0.Cov(ai,bi)0 means a latent feature vector Ui and each movie j with a latent that the actor effect and partner effect of the same individual feature vector V;.Based on these learned feature vectors. ihave nonzero covariance.Since CF is a half-block design, the missing elements in the preference matrix can be pre- one specific individual can only have either actor effect or dicted easily.For example,the expected value E(RiFj) partner effect,but not both.Hence,it is unnecessary to con- computed based on(3)can be used as the predicted value sider these covariances for CF. for a missing element Rij,which is the strategy used in The focus of SRM is not on estimating the effects for spe- this paper.On the other hand,it is not easy to directly ap- cific individuals and relationships but on estimating the vari- ply SRM to predict the missing elements. ance due to effects.So,as said above,ANOVA (Li and Lo- .By exploiting different GLMs,SRMCoFi generalizes ken 2002)is the main goal of the original SRM.Moreover, SRM to accommodate different types of data,such as bi- SRM typically assumes that all of the data are observed. nary measurements and counting data.example, we can use SRM to analyze the actor variance to find out whether people see others as similar in terms of in￾telligence. Moreover, in general, it is assumed that we can collect all of the data for SRM and hence no elements are missing. For CF, however, the goal is to predict some of the missing ratings, making it fundamentally different from that of the original SRM. Contributions By adapting SRM, we propose in this pa￾per a novel CF model called SRMCoFi. Some promising properties of SRMCoFi are highlighted here: • SRMCoFi formulates the CF problem under a probabilis￾tic framework, making it very convenient to learn the model parameters and subsequently control the model ca￾pacity automatically. • SRMCoFi, which subsumes many existing model-based CF methods as special cases, provides a general frame￾work for CF. • SRMCoFi offers theoretical justifications for many em￾pirical findings by previous CF methods. • SRMCoFi is very efficient and easily implementable due to the closed-form updates for parameter learning. Social Relations Model Let Fij denote the dyadic measurement of the perception or rating from actor i to partner j. In SRM (Kenny 1994), Fij is decomposed into random effects a, b, e as follows: Fij = µ + ai + bj + eij + ij , (1) where µ is a constant representing the overall mean, ai is the actor effect representing the average level of rating of an actor in the presence of a variety of partners, bj is the part￾ner effect representing the average level of a rating which a person elicits from a variety of actors, eij is the relationship effect representing the rating of an actor toward a particu￾lar partner, above and beyond their actor and partner effects, and ij is an error term representing the unstable aspect of the perception. In particular, for the movie rating applica￾tion, actors refer to users and partners to movies. In general, a, b, e,  are assumed to be random variables with the following distributions: ai iid∼ N (·|0, σ2 a ), bj iid∼ N (·|0, σ2 b ), eij ∼ N (·|0, σ2 e ), ij iid∼ N (·|0, σ2  ), Cov(ai , bi) 6= 0, Cov(eij , eji) 6= 0, (2) where Cov() denotes the covariance. In SRM, all the other covariances are assumed to be 0. Cov(ai , bi) 6= 0 means that the actor effect and partner effect of the same individual i have nonzero covariance. Since CF is a half-block design, one specific individual can only have either actor effect or partner effect, but not both. Hence, it is unnecessary to con￾sider these covariances for CF. The focus of SRM is not on estimating the effects for spe￾cific individuals and relationships but on estimating the vari￾ance due to effects. So, as said above, ANOVA (Li and Lo￾ken 2002) is the main goal of the original SRM. Moreover, SRM typically assumes that all of the data are observed. This is completely different from the goal of CF which is to predict some of the missing elements in the rating matrix. Hence, SRM cannot be directly applied to CF. SRMCoFi Since CF is fundamentally different from traditional appli￾cations of SRM, we need to adapt SRM appropriately if we want to apply it to CF. The result is our social relations model for collaborative filtering, which we abbreviate as SRMCoFi in the sequel. Model SRMCoFi defines the likelihood function over the observed ratings as follows: Fij = µ + ai + bj + U T i Vj , E(Rij |Fij ) = g −1 (Fij ), p(R|F, φ) = Y N i=1 Y M j=1 [p(Rij |Fij , φ)]Iij , (3) where U ∈ R D×N is the latent user feature matrix and V ∈ R D×M is the latent movie feature matrix, with col￾umn vectors Ui and Vj denoting user-specific and movie￾specific latent feature vectors respectively, E(·) is the ex￾pectation function, g(·) is the link function in a generalized linear model (GLM) (Gelman et al. 2003), φ is the variance or dispersion parameter of the GLM (Gelman et al. 2003), and Iij is an indicator variable which is equal to 1 if user i rated movie j and 0 otherwise. For example, if g −1 (Fij ) = exp(Fij ) and p(Rij |Fij ) = Poisson(Rij |g −1 (Fij )), the model is equivalent to a Poisson regression model (Gelman et al. 2003) with the log-link function, which is always used to model data taking the form of counts. One representative application of the Poisson regression model is epidemiology where the incidence of diseases is studied (Gelman et al. 2003, page 51). The SRMCoFi model defined in (3) adapts SRM in two aspects: • SRMCoFi adopts a bilinear term U T i Vj to represent the relationship effect eij of SRM. Hence, SRMCoFi seam￾lessly integrates the linear (ai , bj ) and bilinear effects into a principled framework. This adaptation is very cru￾cial for SRMCoFi to achieve the goal of CF, which is to predict some missing elements in the preference matrix. After learning the model, each user i is associated with a latent feature vector Ui and each movie j with a latent feature vector Vj . Based on these learned feature vectors, the missing elements in the preference matrix can be pre￾dicted easily. For example, the expected value E(Rij |Fij ) computed based on (3) can be used as the predicted value for a missing element Rij , which is the strategy used in this paper. On the other hand, it is not easy to directly ap￾ply SRM to predict the missing elements. • By exploiting different GLMs, SRMCoFi generalizes SRM to accommodate different types of data, such as bi￾nary measurements and counting data