Information Technology Reference
In-Depth Information
,
i
|T
+
˜
(
˜
)
,
i
|
Note that the number of positive and negative tags in the post
u
,
and
u
|T
−
˜
,
i
|
, are constant once the tag relevances are determined. For simplicity, we denote
u
=
)
∈P
O
|T
+
,
i
|·|T
−
N
,
i
|
and further define
,
i
(
˜
u
u
˜
˜
u
⊛
⊞
.
⊝
⊠
T
−
˜
T
+
˜
1
|T
u
,
i
|
−
1
|T
u
,
i
|
)
C
×
u
u
×
i
i
i
×
t
(
,
i
↗
,
i
↗
p
=
u
˜
u
u
.
p
is a long row vector of length
)
∈P
O
|T
+
,
i
|·|T
−
,
i
|
. Therefore, with our novel
,
i
(
˜
u
u
˜
u
˜
ranking optimization scheme, the tucker decomposition model amounts to minimiz-
ing:
p
)
×
(
f
1
N
(2.13)
Note that the work in [
31
,
32
] provided similar ranking schemes for recommender
systems, while the main difference is that we explicitly consider the incomplete and
ambiguous characteristics of the user-generated tagging data and filter out the quasi-
positive tags. In their formulation, given a post
, all the tags that are not
annotated by
user u
to
image i
will be treated as negative tags, and the corresponding
negative set is:
(
u
,
i
)
∈ P
O
T
u
,
i
=
t
1
|
(
u
,
i
)
∈ P
O
∧
y
u
,
i
,
t
=
(2.14)
Apparently, this formulation ignores the issues of missing tags and noisy tags, which
cannot be directly applied to the social tagging problems. In addition, Rendle et al.
employed l-1 norm for regularization, while in the proposed RMTF, additional mul-
tiple intrarelations are utilized as the smoothness constraints, which are detailed in
the following subsection.
2.3.2 Multicorrelation Smoothness Constraints
In addition to the ternary interrelations, we also collect multiple intrarelations among
users, images, and tags. These intrarelations constitute the user, image, and tag-
affinity graphs
W
U
∈ R
|T|×|T|
, respectively.
Two objects with high affinities should be mapped close to each other in the learnt
subspaces. Therefore, the intrarelations are employed as the smoothness constraints
to preserve the affinity structure in the low dimensional factor subspaces. In this
subsection, we first introduce how to construct the affinity graphs, and then incorpo-
rate them into the tensor factorization framework.
∈ R
|U|×|U|
,
W
I
∈ R
|I|×|I|
and
W
T
