Information Technology Reference
In-Depth Information
The negative tag set is constituted as:
t
∈ N
T
u
,
i
T
u
,
i
=
|
(
u
,
i
)
∈ P
O
∧
y
u
,
i
,
t
=
1
∧
t
(2.10)
where
N
T
u
,
i
indicates the set of tags relevant to the annotated tags in post
(
u
,
i
)
.
t
5
∈ T
u
1
,
i
1
, presumably
tag
1 and
tag
2 are relevant to
tag
3. The final tagging
data representation for the running example is illustrated in Fig.
2.2
b. The triplets
corresponding to tags
t
Then
t
4
,
∈ N
T
u
,
i
are also removed from the learning process and filled
by plain question marks. The minus signs indicate the filtered negative triplets.
Any tag
t
∈ T
u
,
i
is considered a better description for image
i
than all the tags
∈ T
u
,
i
. The pairwise ranking relationships can be denoted as:
t
t
1
∈ T
u
,
i
∧
t
2
∈ T
u
,
i
y
u
,
i
,
t
1
>
ˆ
ˆ
y
u
,
i
,
t
2
⃔
(2.11)
The optimization criterion is to minimize the violation of the pairwise ranking rela-
tionships in the reconstructed tensor
Y
, which leads to the following objective:
min
(
f
(
ˆ
y
t
−
−ˆ
y
t
+
))
(2.12)
,
i
,
i
˜
u
,
˜
u
,
,
C
U
,
I
,
T
t
+
∈T
u
,
i
t
−
∈T
u
,
i
(
u
,
i
)
∈P
O
where
f
is a monotonic increasing function (e.g., the logistic sigmoid
function or Heaviside function). Through necessary algebra manipulation, we derive
the matrix form of the objective function:
: R ₒ[
0
,
1
]
⊛
⊞
.
⊝
⊠
T
−
˜
T
+
˜
1
|T
u
,
i
|
−
1
|T
u
,
i
|
)
C
×
u
u
×
i
i
i
×
t
(
,
i
↗
,
i
↗
min
f
u
˜
u
u
U
,
I
,
T
,C
.
1
(
u
,
×
)
∈P
O
|T
+
u
,
i
|·|T
−
˜
i
u
,
i
|
where
↗
is the cross product,
f
switches to a component-wise function and
1
D
∈
T
+
˜
1
×
D
R
is 1-vector with all the elements
1
d
=
1.
is the positive tag set for the
,
i
u
,
i
post
(
˜
u
)
:
t
(
u
,
i
)
+
1
t
(
u
,
i
)
+
|
T
+
T
+
˜
,
i
=
,...,
u
u
,
i
|
R
|
T
u
,
i
|×
r
T
T
+
˜
T
+
˜
∈
is the tag vector matrix composed by the positive tags in
,
i
:
,
i
u
u
t
(
˜
.Here
t
is
t
(
u
,
i
)
+
t
T
+
˜
t
(
˜
=
1
,...,
-th row vector of the tag
,
i
)
+
:
t
,
i
,
i
,
i
)
+
:|
T
+
˜
(
˜
u
)
+
:
u
u
u
,
i
|
u
factor matrix.
