LLE与LE驱动的异构多任务多视图聚类算法：数据挖掘新趋势
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在当前数据挖掘领域，多任务多视图聚类算法作为一种新兴的研究热点，近年来备受瞩目。本文由Yiling Zhang、Yan Yan和Tianrui Li等人合作，针对异构场景提出了一个创新的多任务多视图聚类算法，该算法结合了局部线性嵌入（Local Linear Embedding, LLE）和拉普拉斯嵌入（Laplacian Eigenmaps, LE）技术。研究背景源自现实世界中，随着数据集复杂性的增加，任务间的相互关联性和数据的多维度特性使得单一任务或单一视角的处理方法已无法满足需求。
传统的多任务多视图学习方法通常应用于分类问题，假设所有任务共享相同的类别标签集。然而，实际应用中，任务之间的标签集合往往存在差异，这就要求算法具备对多样性和异构性的适应能力。本文旨在解决这一问题，通过将LLE和LE的优势相结合，设计了一种能够在处理不同任务和多视角数据时，考虑到每个任务独特性以及数据间潜在联系的聚类算法。
LLE通过保持邻域数据的局部线性关系，有效地捕获数据的低维结构，而LE则通过拉普拉斯矩阵的谱分析，揭示数据的全局结构。作者可能在这篇文章中探讨了如何利用这两种技术来构建一个联合模型，该模型能够同时考虑不同任务间的协同信息和各视图数据的独立特征，从而实现更准确的聚类结果。
在论文的具体实现部分，可能包含了数据预处理步骤，如特征选择和降维，以及如何融合来自不同任务和视图的数据。同时，为了处理异构情况，可能会引入某种形式的集成策略，比如加权平均或者通过共享的低维表示空间进行连接。文章还可能探讨了算法的优化策略，如模型参数的选择和迭代方法，以确保在处理多任务多视图数据时具有良好的收敛性和稳定性。
此外，实验部分可能展示了算法在多个真实世界数据集上的性能评估，包括对比其他多任务多视图聚类方法的准确性、鲁棒性和效率。通过对复杂异构场景的实验验证，该算法有望提供一种新的多任务多视图聚类框架，为数据挖掘和机器学习领域的实际应用提供有价值的解决方案。
这篇研究论文不仅介绍了基于LLE和LE的多任务多视图聚类算法的基本原理，而且强调了其在处理异构场景中的优势和实用性，为理解数据中潜在的复杂关系提供了新的视角。对于数据挖掘和机器学习领域的研究人员和工程师来说，这篇论文提供了有价值的新思路和技术参考。
778 Y. Zhang et al. / Knowledge-Based Systems 163 (2019) 776–786
features. Suppose X
2
, X
3
, X
4
, X
5
are four samples from the locally
neighborhood of X
1
, then X
1
can be represented as follows:
X
1
= W
2
X
2
+ W
3
X
3
+ W
4
X
4
+ W
5
X
5
(1)
where W = {W
1
, W
2
, . . . , W
n
} is the weight matrix and the
dimension of the matrix is same as that of object matrix X. The
ideology of LLE is that this locally linear structure in the locally
neighborhood does not change after dimension reduction. There-
fore, we obtain the relationship after data mapping and dimension
reduction as follows:
X
′
1
= W
2
X
′
2
+ W
3
X
′
3
+ W
4
X
′
4
+ W
5
X
′
5
(2)
where X
′
= {X
′
1
, X
′
2
, . . . , X
′
n
} is the object matrix after mapping. As
we know, the linear structure only affects the relationship nearby
the current sample and it does not have an influence on the objects
staying away from the sample. The objective function of LLE is
partitioned to two parts:
J(W ) =
n
i=1
∥X
i
−
k
j=1
W
j
X
j
∥
2
2
(3)
J(X
′
) =
n
i=1
∥X
′
i
−
k
j=1
W
j
X
′
j
∥
2
2
(4)
Eq. (3) is to obtain the weight matrix W of each object to learn
the relationship between the objects and their neighborhoods.
And Eq. (4) exploits the matrix from Eq. (3) to get the results of
dimension reduction, a new object matrix X
′
.
2.2. Laplacian Eigenmaps (LE)
Laplacian Eigenmaps (LE) is similar to LLE, and it also utilizes the
locality to construct the relationship between objects and preserve
the manifold structure of data [40]. The main idea of LE is that if
object i and object j are similar in original space, then they will be
very similar after dimension reduction. Assume that there are n ob-
jects with m
1
attributes, X = {X
1
, X
2
, . . . , X
n
} ∈ R
n∗m
1
is the object
matrix, and Y = {Y
1
, Y
2
, . . . , Y
i
, . . . , Y
n
} ∈ R
n∗m
2
is the matrix with
m
2
attributes after dimension reduction, i = 1, 2, . . . , n, m
1
>
m
2
, W
i
= {W
i,1
, W
i,2
, . . . , W
i,j
, . . . , W
i,k
} represents the adjacent
matrix of the ith object with k samples, W
i,j
is the jth entry of
W
i
. LE consists of three steps, including graph construction, weight
decision and eigenmapping.
The first step, graph construction, is achieved by some simple
methods, such as k-nearest-neighbors (KNN) algorithms. Secondly,
LE decides the weight between objects by exploiting the con-
structed relationship from the first step and using fundamental
functions, like thermonuclear function, to get the adjacent matri-
ces. Finally, it computes the eigenvalues and eigenvectors of Lapla-
cian matrix L by utilizing the adjacent matrices from the second
step to achieve eigenmapping, where Laplacian matrix L
i
= D
i
−W
i
and D
i
=
j
W
ij
is the diagonal matrix, W = {W
1
, W
2
, . . . , W
n
}
contains the adjacent matrix of each object.
In summary, the objective function of LE is shown as follows:
J(Y ) = min
i,j
∥Y
i
− Y
j
∥
2
W
i,j
(5)
The difference between LE and LLE is that LLE uses the neighbors
to reconstruct a test sample and it transforms original data from
high dimensional space into separable low dimensional space,
while LE considers keeping the similar distance from origin space
to a new feature space and it makes related samples to be as close
as possible in reduced space. They both preserve the locality and
achieve dimension reduction effectively, and play a vital role on
clustering or classifying high-dimension data. However, neither of
them is able to achieve two times of dimension reduction, because
the weight matrix or adjacent matrix is not changed and it makes
no sense on the second time of dimension reduction. Therefore,
we combine the two algorithms to solve multi-task multi-view
clustering.
It is necessary for the two dimension because of the complexity
of original data. For multi-task multi-view problems, especially
in heterogeneous situation, various tasks and views may have
different structures. We explore the first dimension reduction to
transform the original data from multiple views into the common
intermediate space (view space) and obtain the more separable
data structures, and LLE is suitable for this step because it keeps
the locally linear structures and makes the data be separable si-
multaneously; The second dimension reduction is to transform
the data from the intermediate space (view space) into task space
and extract the shared and complementary features among various
tasks. We want to obtain the samples that are relate to each other
to be close such that sufficient shared and complementary features
are learned, and LE is more appropriate for this step.
3. Methodology
For multi-task multi-view clustering, we usually consider that
there is similar relationship between several views of each task.
However, it is difficult to directly learn knowledge through the
original data. In order to facilitate information extracting, we col-
lectively learn the feature transformation matrices for all views
from each task by LLE. Besides, some similar features exist in
multiple tasks, we want to keep the distances between tasks in
the process of mapping and solve the multi-task problems by LE.
In this section, we introduce the detail of our proposed method,
L3E-M2VC.
As previous mentioned, LLE and LE are for learning the struc-
tures in views of each task and multiple tasks, respectively. There-
fore, the process of multi-task multi-view clustering for knowledge
sharing is partitioned to two steps. First step is mapping the views
of each task to a common space called view space. Second is
extracting the features in multiple tasks and mapping them to a
discriminant space called task space, which is shown in Fig. 1.
Firstly, through the first transformation step, the data samples
of the vth view from the tth task in original space is transformed
into the view space, which just makes samples be more separable
and each transformation matrix is dependent on various tasks and
views. The view space can be seemed as a common intermediate
latent space that is shared by all the views from each task [38].
Then through the transformation in the second step, the sample
is mapped from view space to task space. Assume the second
transformation matrix is R
t
, which consists of two parts, including
the shared feature R which is common in multiple tasks, and the
complementary feature Rc
t
that belongs to only one task. Of course,
the second transformation step for the different views in the same
task is identical, because they are mapped to the same space in
the first step. In other words, the view space offers a common
plane for various views in each task which uniforms the features
in different views; then the task space considers the shared and
complementary features in multiple tasks in order to facilitate
knowledge sharing in multi-task multi-view clustering.
3.1. Problem definition
Assume that there are T clustering tasks with V
t
views, and
X
v
t
= {x
v
t,1
, x
v
t,2
, . . . , x
v
t,i
, . . . , x
v
t,n
t
} ∈ R
n
t
∗d
v
t
, i = 1, 2, . . . , n
t
, t =
1, 2, . . . , T , and v = 1, 2, . . . , V
t
, where n
t
represents the number
of objects in tth task, and d
v
t
is the dimension of feature of the vth
view in the tth task. Obviously, each task is clustered to c
t
classes,
and it is different for various tasks about the label sets because
of the heterogeneity. The structures of data in view space Y
v
t
and
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