Information Sciences 649 (2023) 119562
4
W.-J. He, Z. Zhang and Y. Wei
𝑠.𝑡. 𝒁
⊤
1 = 1, 𝒁
𝑖𝑗
≥ 0. (2)
In Eq. (2), 𝑨
𝑣
∈ ℜ
𝑑
𝑣
×𝑘
is the anchor set of the 𝑣-th view (𝑘 is the number of anchors, 𝑘 ≪𝑛), which could be generated by random
selection or k-means.
Although AIMSC can achieve approximately linear time complexity, its clustering result is influenced by the randomness intro-
duced
in anchor selection. Moreover, this work cannot fully employ the missing data recovery scheme for multi-view interaction.
3. Method
3.1. The proposed method
Traditional IMC methods typically measure each pairwise relationship and construct n-by-n similarity matrices, resulting in high
computational complexity, high memory consumption, and lack of scalability. Anchor-based methods eliminate these shortcomings
by reducing the number of relationships, i.e., measuring the relationships between the raw data and anchor points, to represent the
general relationships. However, current anchor-based methods still share some common defects: 1) Most existing works ignore the
recovery of missing information, which leads to insufficient usage of hidden information. 2) The selected anchors are sub-optimal
and fragile to noise.
Traditional IMC methods usually deal with the missing instances with a simple one-time filling strategy, which fills the missing
instances with 0 or the average values and never updates them during the training process. However, such measures pull all the
missing instances together since it forces them to have the same values, and thus introduce noise into the clustering result. Orthogonal
to the filling strategy mentioned above, we propose to iteratively recover the missing instances in each view:
min
𝑬
𝑖
,𝒁
𝑖
,𝒁
𝑣
𝑖=1
𝛼
𝑖
𝑿
𝑖
− 𝑨
𝑖
𝒁
𝑖
2
𝐹
+ 𝛽𝑬
𝑖
2
𝐹
+ 𝛾𝑹
𝑖
𝒁
𝑖
− 𝒁
2
𝐹
𝑠.𝑡.
𝑿
𝑖
= 𝑿
𝑖
+ 𝑬
𝑖
𝑵
𝑖
, (𝑹
𝑖
)
⊤
𝑹
𝑖
= 𝑰
𝑘
. (3)
In Eq. (3), each view learns a bipartite graph 𝒁
𝑖
∈ ℜ
𝑘×𝑛
which represents the relationship between the recovered data
𝑿
𝑖
∈ ℜ
𝑑
𝑖
×𝑛
and pre-defined anchors 𝑨
𝑖
∈ ℜ
𝑑
𝑖
×𝑘
. Specifically,
𝑿
𝑖
is the recovered data obtained by adding the justified hidden information 𝑬
𝑖
𝑵
𝑖
to the raw incomplete data 𝑿
𝑖
∈ ℜ
𝑑
𝑖
×𝑛
. The recovery of missing instances 𝑬
𝑖
∈ ℜ
𝑑
𝑖
×𝑛
𝑖
is updated as a variable to learn from the
view-specific bipartite graph 𝒁
𝑖
as well as the consensus graph 𝒁 to utilize complementary information from all views. 𝑵
𝑖
∈ ℜ
𝑛
𝑖
×𝑛
(𝑛
𝑖
is the number of missing instances in the 𝑖-th view) is an indicator matrix indicating the missing status of an instance, which is
defined by:
𝑵
𝑖
𝑚,𝑛
=
1 if the 𝑚-th missing instance in 𝑿
𝑖
is the n-th instance in
𝑿
𝑖
,
0 else.
(4)
By multiplying 𝑬
𝑖
by 𝑵
𝑖
, the recovered information is placed where the corresponding missing instance is and left the observed
positions zero. In this way, noises will not be introduced to the observed instances. Additionally, since descriptions of instances from
different views are semantically consistent, we project the bipartite graph of each view 𝒁
𝑖
onto a consensus graph 𝒁 to represent
the overall relationship between given data and the learned anchors.
In Eq. (3), the anchor set 𝑨
𝑖
is pre-defined by k-means or random selection, which is the prevalent approach taken by current
anchor-based methods. In other words, anchor selection and bipartite graph construction are separated into two stages. However,
the selected anchors of each view are independent, which weakens the interpretability of 𝒁 since each 𝒁
𝑣
denotes the similarity of
data points with different anchors. Additionally, the pre-defined anchor sets are generally not representative enough for the training
process, and the randomness introduced by anchor selection is hard to eliminate.
Contrary to the fixed anchor strategies, our method updates the anchor sets during the training process. Specifically, SIMC_ADC
learns a central consensus anchor set as well as a series of projection matrices. By projecting the central anchors to each view,
the view-specific anchors are closely connected to each other. In this way, the anchors from each view are mutually updated with
the similarity matrices, resulting in an optimal result. Also, since we have no prior knowledge of how important each view is, it
is unreasonable to set the weight of anchor learning and view-specific bipartite graph learning manually. Instead, we introduce an
adaptive weighting parameter to automatically balance the importance of each view. The objective function therefore derives:
min
𝑾
𝑖
,𝑨,𝒁
𝑖
,𝑬
𝑖
,𝒁
𝑣
𝑖=1
(𝛼
𝑖
𝑿
𝒊
− 𝑾
𝑖
𝑨𝒁
𝑖
2
𝐹
+ 𝛾𝑹
𝑖
𝒁
𝑖
− 𝒁
2
𝐹
+ 𝛽𝑬
𝑖
2
𝐹
)+𝒁
2
𝐹
𝑠.𝑡.
𝑿
𝒊
= 𝑿
𝑖
+ 𝑬
𝑖
𝑵
𝑖
, (𝑹
𝑖
)
⊤
𝑹
𝑖
= 𝑰
𝑘
, (𝑾
𝑖
)
⊤
𝑾
𝑖
= 𝑰
𝑑
,
𝑨
⊤
𝑨 = 𝑰
𝑘
, 𝒁 ≥ 0, 𝒁
⊤
1 = 1, 𝒁
𝑖
≥ 0, (𝒁
𝑖
)
⊤
1 = 1,
𝛼
𝑖
=
1
𝑿
𝒊
− 𝑾
𝑖
𝑨𝒁
𝑖
𝐹
. (5)
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