3.1 Social recommendation
The matrix factorization model represents a matrix as the product of two lower-rank matri-
ces [18, 32], which is widely used for recommendation. Let
U ={u
1
,...,u
n
} beasetof
n users and
V ={v
1
,...,v
m
} beasetofm items. We denote the feedback from user u
i
to item v
j
as R
ij
. There are different kinds of feedbacks, such as explicit feedback (e.g.,
ratings) and implicit feedback (e.g., consume, click and browse etc.), which can be used to
determine the range of R
ij
. If we observed a feedback from u
i
to v
j
,wesetI
ij
= 1; other-
wise I
ij
= 0. The matrix factorization objective function over the observed feedbacks can
be written as
min
U,V
n
i=1
m
j=1
I
ij
(R
ij
− u
T
i
v
j
)
2
+ λ(U, V) (1)
where U =[u
i
]
i∈[n]
∈ R
K×n
and V =[v
j
]
j∈[m]
∈ R
K×m
are latent matrix for users and
items, K is the number of latent factors, λ is a scalar to control relative contribution, and
(U, V) is the regularization term to avoid overfitting.
Social regularization represents the social constraints on recommender systems based
on the assumption that every user’s taste is close to the average taste of this user’s
friends [23]. Considering users’ preferences are similar or influenced by their socially con-
nected friends, social relationships are widely employed in designing social regularization
term
n
i=1
f ∈F
i
S
if
||u
i
− u
f
||
2
F
to constrain the matrix factorization objective function
of recommender systems. We use F
i
to represent the set of user u
i
’s friends. Then we
can mathematically define the social recommendation algorithm over the observed social
relationships and the observed feedbacks as:
min
U,V
n
i=1
m
j=1
I
ij
(R
ij
− u
T
i
v
j
)
2
+ α
n
i=1
f ∈F
i
S
if
||u
i
− u
f
||
2
F
+ λ(U, V) (2)
where || · ||
2
F
denotes the Frobenius norm, S
if
is the cosine similarity between feedbacks
of u
i
and u
f
on the same items, and α is the scalar to control the contribution of social
regularization.
3.2 Differential privacy
Differential privacy [5] is a popular privacy-preserving technique, which effectively per-
turbs the raw datasets by injecting noise and ensures that the output is not significantly
affected by removal/addition of a single rating [2, 36]. Considering its provable pri-
vacy guarantee with light computational overhead, we will use differential privacy in our
proposed framework.
Definition 1 -Differential Privacy [5]: A randomized algorithm f satisfies -differential
privacy, if for any two datasets D
1
and D
2
which differ at most one rating, and for any
possible anonymized output dataset
˜
D ∈ Range(f ),
Pr[f(D
1
) =
˜
D]≤e
× Pr[f(D
2
) =
˜
D] (3)
World Wide Web (2019) 22:2853–2881
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