Social Media Mining: An Introduction

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CUUS2079-09 CUUS2079-Zafarani 978 1 107 01885 3 January 13, 2014 17:28

9.3 Recommendation Using Social Context 261

regularization terms, we can add this term to Equation9.51. Hence, our

final goal is to solve the following optimization problem:

minU,V

1

2

∑n

i= 1

∑m

j= 1

Iij(Rij−UiTVj)^2 +β

∑n

i= 1

∑

j∈F(i)

sim(i,j)||Ui−Uj||^2 F

+

λ 1

2

||U||^2 F+

λ 2

2

||V||^2 F, (9.54)

whereβis the constant that controls the effect of social network regulariza-

tion. A local minimum for this optimization problem can be obtained using

gradient-descent-based approaches. To solve this problem, we can compute

the gradient with respect toUi’s andVi’s and perform a gradient-descent-

based method.

9.3.3 Recommendation Constrained by Social Context

In classical recommendation, to estimate ratings of an item, one determines

similar users or items. In other words,any usersimilar to the individual

can contribute to the predicted ratings for the individual. We can limit the

set of individuals that can contribute to the ratings of a user to the set of

friends of the user. For instance, in user-based collaborative filtering, we

determine a neighborhood of most similar individuals. We can take the

intersection of this neighborhood with the set of friends of the individual to

attract recommendations only from friends who aresimilar enough:

ru,i=r ̄u+

∑

v∈N∑(u)∩F(u)sim(u,v)(rv,i−r ̄v)

v∈N(u)∩F(u)sim(u,v)

. (9.55)

This approach has its own shortcomings. When there is no intersection

between the set of friends and the neighborhood of most similar individuals,

the ratings cannot be computed. To mitigate this, one can use the set ofk

most similarfriends of an individualS(i) to predict the ratings,

ru,i=r ̄u+

∑

v∈S∑(u)sim(u,v)(rv,i−r ̄v)

v∈S(u)sim(u,v)

. (9.56)

Similarly, when friends are not very similar to the individual, the pre-

dicted rating can be different from the rating predicted using most similar

users. Depending on the context, both equations can be utilized.