Tag Based Collaborative Filtering for Recommender Systems 669 Rp U: The relationship between items and their user sets. Different from Ru. P that describing each user's item set, Re u is describing each items user set. The traditional item-based collaborative filtering approach is based on this relationship. Rp U=(<Pk UplpkEP, Up cU,k=l.m). Upk is the user set of item Pk. Up=[ulu;EU,3t,eT E(u; t Pv=l), PEP, k=l.m. Rp. T: The relationship between items and their tag sets. RP. T=(<pk, Tp >l pEP, Tpk ≤T,k=1. m) Tpx is the tag set of item pk. Tp=团tteT,彐ueU,E(u,p)=1, p∈P,k=1.m From the perspective of tags, the relationship among users, tags and items is denoted as RT Up. Though it has not been used for the recommendation of items di- rectly, we still give its definition as below for the sake of helping user get a whole view of the relationships among users, tags and items. RT, UP=(<t;, UPPI tET, j=l.I] UPj is the user and item set of tag t. UP=(<u Px>luH EU, PKEP, and E(u; t- Px)=l]. The other derived two-dimensional relationships Rr. u and Rr. p are defined as below Rr, U: The relationship between tags and their user sets. Rr, U=(<t;, Ut> ET,Ut C U,j=lI]. Ut is the user set of tag t. Ut=uil u; EU, PkEP, E(u;s t; Px)=1, ET RT. P: The relationship between tags and their item sets. In this relationship, the tag collects all items that are being tagged with it by various users, which shows the result of this collaborative tagging work. RT P=It;, Pt>l tET,Pt,CP, j=l.] Pt; is the item set of tag t.P!={ppeP,彐u∈U,E(u,tp)=1 These multiple relationships can be used to recommend personalized items, virtual friends, and tags to users. But for the scope of this paper, we will only focus on how to do item recommendations in the following sections 3.3 Neighborhood Formation Neighborhood formation is to generate a set of like-minded peers for a target user. Forming neighborhood for a target user u EU with standard"best-n-neighbors""tech- nique involves computing the distances between u; and all other users and selecting the top N neighbors with shortest distances to ui Based on user profiles, the similarity of users can be calculated through various proximity measures. Pearson correlation and cosine similarity are widely used to calculate the similarity by using users'expli it rating data. However, explicit rating data is not al ways available. Unlike explicit ratings in which users are asked to supply their perceptions to items explicitly in a numeric scale, implicit ratings such as transaction histories, browsing histories, prod uct mentions. etc. are also obtainable for most e-commerce sites and communities. For online communities with the tagging facility, binary implicit ratings can be ob ained based on users'tagging information. If a user has tagged a product or item, the mplicit rating to this item by this user is set to l otherwise 0Tag Based Collaborative Filtering for Recommender Systems 669 RP, U: The relationship between items and their user sets. Different from RU, P that describing each user’s item set, RP, U is describing each item’s user set. The traditional item-based collaborative filtering approach is based on this relationship. RP, U= {<pk, Upk>|pk∈P, Upk U, k=1..m}. Upk is the user set of item pk. Upk= {ui| ui∈U, ∃tj∈T, E(ui,tj, pk) =1}, pk∈P, k=1..m. RP, T: The relationship between items and their tag sets. RP, T = {<pk, Tpk>| pk∈P,Tpk T, k=1..m} Tpk is the tag set of item pk. Tpk= {ti| tj∈T, ∃ui∈U, E(ui,tj, pk) =1}, pk∈P, k=1..m. z From the perspective of tags, the relationship among users, tags and items is denoted as RT, UP. Though it has not been used for the recommendation of items di￾rectly, we still give its definition as below for the sake of helping user get a whole view of the relationships among users, tags and items. RT, UP= {<tj,UPj>| tj∈T, j=1..l}. UPj is the user and item set of tag tj. UPj= {<ui, pk>| ui∈U,pk∈P, and E(ui,tj,pk)=1}. The other derived two-dimensional relationships RT, U and RT, P are defined as below: RT, U: The relationship between tags and their user sets. RT, U= {<tj, Utj>|tj∈T, Utj U, j=1..l}. Utj is the user set of tag tj. Utj={ui| ui∈U, ∃pk∈P, E(ui,tj, pk) =1}, tj∈T, j=1..l. RT, P: The relationship between tags and their item sets. In this relationship, the tag collects all items that are being tagged with it by various users, which shows the result of this collaborative tagging work. RT, P = {<tj, Ptj>| tj∈T, Ptj P, j=1..l} Ptj is the item set of tag tj. Ptj= {pk| pk∈P, ∃ui∈U, E(ui,tj, pk) =1}. These multiple relationships can be used to recommend personalized items, virtual friends, and tags to users. But for the scope of this paper, we will only focus on how to do item recommendations in the following sections. 3.3 Neighborhood Formation Neighborhood formation is to generate a set of like-minded peers for a target user. Forming neighborhood for a target user ui∈U with standard “best-n-neighbors” tech￾nique involves computing the distances between ui and all other users and selecting the top N neighbors with shortest distances to ui. Based on user profiles, the similarity of users can be calculated through various proximity measures. Pearson correlation and cosine similarity are widely used to calculate the similarity by using users’ expli￾cit rating data. However, explicit rating data is not always available. Unlike explicit ratings in which users are asked to supply their perceptions to items explicitly in a numeric scale, implicit ratings such as transaction histories, browsing histories, prod￾uct mentions, etc., are also obtainable for most e-commerce sites and communities. For online communities with the tagging facility, binary implicit ratings can be ob￾tained based on users’ tagging information. If a user has tagged a product or item, the implicit rating to this item by this user is set to 1 otherwise 0. ⊆ ⊆ ⊆ ⊆