4.4. MULTIMODAL CONSISTENT LEARNING 67
characterize and model the task relatedness in the tree, namely, how to estimate the reasonable
weight e
v
for each node v in the tree appropriately. Although the existing tree-guided multi-
task learning approaches [57, 150] have addressed this issue by exploring the geometric structure,
they do not consider the semantic relatedness among tasks. To remedy this problem, we aim to
model the intrinsic task relatedness based on the feature space. Toward this goal, we introduce
the affinity measurement of the node group proposed in [98]. A high affinity value e
v
of the
node group G
v
indicates the dense connections and compact relations among the leaf nodes
within the given group. We hence can employ the affinity measurement to characterize the task
relatedness in the tree.
To facilitate the affinity measurement of each node group G
v
, we need to obtain the pair-
wise similarity between all leaf nodes. For simplicity, we utilize the adjacency matrix S 2 R
T T
to denote the pairwise similarity matrix and the entry S
ij
to capture the non-negative relatedness
between the i-th and j -th leaf nodes, which can be formulated as
S
ij
D exp
Nx
i
Nx
j
2
2
!
; (4.16)
where Nx
i
represents the mean feature vector of the samples belonging to the i-th venue category
which can be extracted from the training dataset; is radius parameter that is simply set as the
median of the Euclidean distances of all node pairs.
For ease of formulation and inspired by the work in [98], we define a scaled assignment
vector u
v
2 R
T
for each node of the tree over all the T leaf nodes which can be stated as
u
vt
D
8
<
:
1
p
jG
v
j
; if t 2 G
v
;
0; otherwise.
(4.17)
Based on the scaled assignment u
v
and the pairwise similarity matrix S, we can further formulate
the affinity e
v
for the node v as follows:
e
v
D u
T
v
Su
v
: (4.18)
Since the characteristics of the affinity definition, the value of the e
v
is limited within the range of
[0, 1]. More importantly, such affinity measurement can guarantee that higher nodes correspond
to weaker relatedness, and vice versa.
4.4.3 COMPLEXITY ANALYSIS
In order to analyze the complexity of our proposed TRUMANN model, we have to estimate the
time complexity for constructing A, B, and W as defined in Eqs. (4.5), (4.7), and (4.15). e
computational complexity of the training process is O.M .O
1
C O
2
C O
3
//, where O
1
, O
2
and O
3
, respectively, equal to ..D
s
/
2
N C .D
s
/
3
C .D
s
/
2
K/S, .NK
2
C NDKS C K
3
C K
2
T /
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