78 4. MULTIMODAL COOPERATIVE LEARNING
where y
n
denotes the category label of the sample x
n
. We ultimately reach the following objective
function:
min
A;Q
1
2
N
X
nD1
k
x
n
Da
n
k
2
F
C
2
N
X
nD1
k
a
n
k
2
F
C
2
N
X
nD1
.
a
n
/
T
Q
n
a
n
: (4.27)
e alternative optimization strategy is applicable here. By fixing Q
n
, taking derivative of
the above formulation regarding a
n
, and setting it to zero, we reach
8
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
:
D
T
.
x
n
Da
n
/
C a
n
C Q
n
a
n
D 0;
D
T
D C I C Q
n
a
n
D D
T
x
n
;
a
n
D
D
T
D C I C Q
n
1
D
T
x
n
:
(4.28)
Once we obtain all the a
n
, we can easily compute Q
n
based on Eqs. (4.24) and (4.26).
Computing D with A fixed: Fixing A and taking the derivative of with respect to D,
we have
@
@D
D .DA X/A
T
: (4.29)
By setting Eq. (4.29) to zero, it can be derived that
8
<
:
D
AA
T
XA
T
D 0;
D D
XA
T
AA
T
1
:
(4.30)
It is straightforward that the above algorithm converges, because in each iteration, will
decrease, as shown in Figure 4.6. By using Algorithm 4.2, we can learn a set of dictionaries D
m
for each modality of samples X
m
and their corresponding representations A
m
.
4.5.4 ONLINE LEARNING
As analyzed in the introduction, the efficient operation and incremental learning of micro-
videos deserve our attention. To accomplish this, we present an online learning algorithm (re-
ferred to Algorithm 4.3). Generally speaking, if an incoming sample is labeled, we leverage it
to strengthen the dictionary learning. We treat the learned D over the initial training data as
D
.0/
and update it to D
.t/
at the current time t. Otherwise, we compute its sparse representation
based on the current dictionaries and classify it into the right venue category.
An Incoming Labeled Sample: At the t-th online update, a new sample x
t
with a label
y
t
is given. We can know which leaf node this micro-video is from and then use it to update
the dictionaries D
.t1/
. From Eq. (4.30), we find that the solution of D
.t/
relies on the sparse
representation A
.t/
D ŒA
.t1/
; a
t
. We thus need to compute a
t
first that is the representation
vector of x
t
. However, Eq. (4.28) tells us that a
t
is related to Q
t
computed by A
.t/
D ŒA
.t1/
; a
t
.