28 3. MULTIMODAL TRANSDUCTIVE LEARNING
difference is that the lowest-rank representation is adaptively obtained by a low-rank constraint,
which is approximated by the trace norm rather than being specified in advance.
3.5 NOTATIONS AND PRELIMINARIES
We first declare several notations. In particular, we employ bold capital letters (e.g., X) and bold
lowercase letters (e.g., x) to denote matrices and vectors, respectively. We use non-bold letters
(e.g., x) to represent scalars, and Greek letters (e.g., ˇ) as parameters. If not clarified, all vectors
are in column form. Moreover, given a matrix M 2 R
N D
, its i-th row and the i-th column of
matrix M are denoted by m
i
and m
i
, respectively. e L
p;q
-norm of matrix M is defined as
k
M
k
p;q
D
"
X
N
iD1
X
D
j D1
ˇ
ˇ
M
ij
ˇ
ˇ
p
q=p
#
1=q
; (3.1)
where M
ij
is the .i; j /-th element of matrix M. By assigning different values to p and q, there
are several regularization terms, which are stated as follows.
e L
1
-norm is defined when p D q D 1,
k
M
k
1
D
N
X
iD1
m
i
1
: (3.2)
e Frobenius norm L
F
is defined when p D q D 2,
k
M
k
F
D
v
u
u
t
N
X
iD1
D
X
j D1
M
ij
2
: (3.3)
e trace norm of matrix M is defined as
k
M
k
D
X
i
ı
i
.
M
/
; (3.4)
where
X
i
ı
i
.
M
/
is the sum of singular values of matrix M.
Our proposed model targets at reasoning from observed training micro-videos to testing
ones. Such prediction belongs to transductive learning, in which both labeled samples as well as
unlabeled samples are available for training. It hence obtains better performance. In contrast, in-
ductive model is reasoning from observed training cases to general rules, which are then applied
to the test cases.
3.6 MULTIMODAL TRANSDUCTIVE LEARNING
Without loss of generality, suppose we have N labeled samples and M unlabeled samples with
K > 2 modalities. It is worth noting that the unlabeled samples also serve as testing sam-
ples. Z
k
stands for the number of features extracted from the k-th modality. en the k-th
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