Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 10, No. 1, 2024 

 

364 

Multi‐label	Feature	Selection	based	on	Label‐specific	
features	and	Manifold	Learning	
Wanzhu Wang, Yong Liu 

Henan University of Science and Technology, 471000, Luoyang, China 

 

Abstract: Each instance in multi-label data is associated with multiple labels, and there are irrelevant or redundant features in 
its feature space, which leads to the performance degradation of multi-label learning algorithms. Multi-label feature selection 
selects representative features from the feature space to improve the accuracy of the model. Due to the high cost of labels and 
the difficulty of data collection, there will be some missing labels in the data set, which affects the accuracy of feature selection. 
To solve this problem, a multi-label feature selection algorithm based on label-specific features and manifold learning is proposed. 
The algorithm uses the linear relationship between the features and labels in known label samples to build a linear regression 
model for learning label-specific features. By using the nonlinear relation between instances and the nonlinear relation between 
features, we can precisely learn the label-specific features. We use the Laplacian feature mapping method to construct the instance 
manifold model and the feature manifold model, which are also used as the regular term constraint weight matrix. The final 
model can not only complete the missing labels, but also select sparse and representative features. The feature selection is carried 
out by analyzing the weight of the feature given by the final model. Experiments were conducted to verify the effectiveness of 
the proposed algorithm under different label deletion rates on four evaluation indexes. 

Keywords: Multi-label learning; Manifold learning; Label-specific features; Feature selection. 
 

1. Introduction 
Multi-label learning is characterized by high efficiency and 

flexibility, and has shown wide application prospects in many 
application fields such as machine learning, data mining and 
pattern recognition [1-3], and is also widely used in various 
practical tasks, such as text classification, music classification 
and gene function prediction. For example, in the text 
category, a news story can be classified as both social and 
technological; In the music category, a piece of music may 
belong to both light music and English songs. Multiple 
concepts can be expressed by labeling each instance 
differently. However, in such cases where multi-label 
classification is required, attaching only one label to each 
instance does not express the underlying complex semantics. 
Therefore, multi-label learning is achieved by assigning 
multiple labels to an instance at the same time. 

Like single-label learning, the feature space of multi-label 
learning is often a set containing a large number of features, 
which may have up to thousands of dimensions [4]. In text 
classification, the complex semantic information of a text can 
be represented in detail by thousands of features. By 
extracting and utilizing these features effectively, we can 
understand the intrinsic meaning and context of the text more 
deeply. However, in the feature set, most of the features often 
present redundant or irrelevant features. These redundant and 
uncorrelated features not only increase the computational 
complexity, but also may negatively affect the performance of 
multi-label learning algorithms, and also lead to overfitting 
and other problems. In order to solve this problem, 
researchers proposed a multi-label feature selection algorithm, 
which is mainly used for dimensionality reduction of multi-
label data. Literature [5] proposed a filter-based feature 
selection method for multi-label classification. Based on 
random forest as a learner, a filter feature selection method 
based on statistical integration method and label space 
division based on integration method were used. Literature [6] 

proposes a multi-label feature selection based on coarse-
grained balls and label distribution. By using the pellet 
computing model and its proposed label enhancement method, 
logical labels are transformed into label distribution by 
exploring the similarities between instances and coarse-
grained balls. The method measures feature significance by 
the consistency of labels in samples within the same 
information granularity. Based on mutual information theory, 
literature [7] designed a new multi-label feature selection 
method combining dynamic correlation change, label 
redundancy and interactive information. 

In the model training of multi-label feature selection 
algorithm, the correlation between labels is a key information, 
which can be effectively used to optimize the feature selection 
process. By considering the correlation between labels, the 
algorithm can more accurately capture the correlation and 
dependency between different labels, so as to select the 
features that are closely related to multiple labels. Literature 
[8] uses the co-occurrence relationship between labels to 
evaluate the similarity relationship between samples. 
According to literature [9], label correlation and label-specific 
features are two important features of multi-label learning. 
The global and local label correlation are calculated by label 
co-occurrence and neighborhood information respectively, 
and the label-specific features is learned by l_1 norm. 
Literature [10] considers the influence of two types of label 
dependencies on the algorithm. This method considers that 
second-order label dependencies and higher-order label 
dependencies are both important and complementary to 
capture label information, so second-order label dependencies 
and higher-order label dependencies are considered at the 
same time. 

Compared with traditional machine learning, multi-label 
labeling is very difficult. With the increasing complexity of 
multi-label tasks, there are also many challenges, among 
which the increase of sample dimension and data volume will 
significantly affect the labeling cost. First, because of the 



 

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large number of sample instances, the process of labeling 
them one by one is time-consuming and labor-intensive. 
Therefore, in the labeling process, manual labeling tends to 
select labels that are more important to label and discard other 
labels. The second is that human taggers can only label what 
they know according to cognitive differences. For example, 
an article is labeled machine learning, Data mining, and 
artificial neural networks, and the human tagger does not 
know about the category of artificial neural networks and only 
labels the article with the remaining two labels. A variety of 
circumstances have led to the creation of missing labels. 
However, the accuracy of the model or the efficiency of 
classification can be affected by the multi-label feature 
selection or multi-label classification on the multi-label data 
set with missing labels. In order to solve the impact of missing 
labels, many multi-label feature selection algorithms have 
been proposed to deal with missing labels based on theories 
such as multi-label information entropy [11], linear regression 
[12] and label correlation. He et al. [13] simultaneously 
considered label relevance, missing labels, and feature 
selection through a multi-classification framework. Literature 
[14] proposes an embedded packaging approach that 
combines the generation of label-specific features with 
subsequent model induction to solve the multi-label 
classification problem of missing labels. 

At the same time, it is more efficient to use the specific 
features of each label than to use all the features to classify. 
Label-specific features, refer to the subset of features most 
closely associated with each label. In the multi-label learning 
scenario, the use of label-specific features for feature 
selection can reveal the deep correlation between features and 
labels more efficiently. At the same time, from the point of 
view of the nonlinear relationship between data, consider the 
nonlinear relationship between instances and the nonlinear 
relationship between features, and study their influence on the 
relationship between labels and features. Based on the above 
considerations, this paper proposes a multi-label feature 
selection algorithm (ML-LSF) based on label-specific 
features and manifold learning that can deal with missing 
labels. 

2. Proposed Models 
In multi-label learning, 𝑋 𝑥 ，𝑥 ，…，𝑥 ∈ 𝑅  

represents a multi-label data set with n instances, where d 
represents the feature dimension; 𝑌
𝑦 ，𝑦 ，…，𝑦 ∈ 𝑅   represents the set of labels 

corresponding to n instances, where l represents the number 
of labels. The label values in set Y are set to 1, 0, and -1.When 
the label value is 1, the instance has the label. A label value of 
-1 indicates that the instance does not have the label. When 
the label value is 0, the instance is missing the label. 

2.1. Build a label-specific feature model 
In a multi-label classification task, each label in the label 

space has the most relevant and discriminating features for the 
label, which are called label-specific feature. First, the basic 
label-specific feature model is constructed by linear 
regression method. The purpose of adding 𝑙   norm is to 
ensure that the features selected by each label are sparse and 
representative. The objective function is as follows: 

 

min
1
2

‖𝑋𝑊 𝑌‖ 𝜆‖𝑊‖ 1  

Where X is the eigenmatrix of the instance, Y is the label 
matrix containing the missing label, and λ is the parameter. W 
is the coefficient matrix, 𝑤  represents the i row in the matrix 
W, the row vector represents the feature vector, and the 
column vector represents the label vector. If the value of a 
certain place in W is 0, it means that there is no correlation 
between the feature and the label at that place. 

2.2. Build the instance manifold and feature 
manifold models 

The core idea of manifold learning is that data in high-
dimensional space is actually distributed on a low 
dimensional manifold structure, that is, although data may 
appear complex and difficult to analyze in the original high-
dimensional space, their essential structure can often be 
characterized by a low dimensional manifold. The 
characteristic makes manifold learning suitable for handling 
nonlinear relationships between data. The goal of manifold 
learning is to effectively map points on manifold M in an N-
dimensional space to a low dimensional n-dimensional space, 
where n<N. Popular regularization refers to adding manifold 
related terms to machine learning problems, with the aim of 
constraining the coefficient matrix W through manifold 
regularization terms. 

Two strongly related instances can share related subsets of 
each other [15]. If there are two strongly correlated instances, 
then the corresponding label sets or feature sets of the two 
instances are similar, and the prediction labels obtained by 
using the corresponding W are also very similar. The purpose 
of Laplacian Eigenmap in manifold learning is to ensure that 
two strongly correlated instances are as close as possible after 
dimensionality reduction, and to mine the correlation between 
the instances by using the manifold model between the 
instances. Add the instance correlation to the equation (1) to 
get the objective function: 

 

𝑚𝑖𝑛
1
2

‖𝑋𝑊 𝑌‖
𝛼
2

𝑡𝑟 𝑊 𝑋 𝐿 𝑋𝑊 𝜆‖𝑊‖ 2  

 
Where, 𝛼（𝛼 0）  is the parameter; 𝐿 𝐷 𝑆  , 𝐿  

refers to the Laplacian matrix, 𝐷  is a diagonal matrix, 𝑆 ∈
𝑅  represents the instance similarity matrix. 

For two related features of a label, the label has a high 
probability of having one feature and the other feature. 
Feature correlation considers the interaction between features. 
Add the feature correlation to the equation (2) to get the final 
objective function: 

 

min
1
2

‖𝑋𝑊 𝑌‖
𝛼
2

𝑡𝑟 𝑊 𝑋 𝐿 𝑋𝑊
𝛽
2

𝑡𝑟 𝑊 𝐿 𝑊 𝜆‖𝑊‖ 3  

 

Where, 𝛽（𝛽 0）  is a parameter; 𝐿 𝐷 𝑆  , 𝐿  
refers to the Laplacian matrix, 𝐷  is a diagonal matrix, 𝑆 ∈
𝑅  represents the feature similarity matrix. In equation (1), 
the 𝑙   norm is used to regularize the coefficient matrix to 
make it sparse and exclude some redundant features in the 
feature space. 

2.3. Fill missing labels 
In the case of missing labels, it is difficult to learn the real 

mapping relationship between features and labels, which 
leads to the unsatisfactory classification effect of multi-label 
tasks. The missing label problem is dealt with by the method 
of restoring it. The iterative process of solving coefficient 



 

366 

matrix 𝑊  and the recovery process of missing label are 
carried out simultaneously. 

According to literature [16], the known labels in the 
original label matrix are assigned to a new matrix 𝐴 , then 
𝐴 𝑌 ; And where the label is missing, 𝐴 0. 

In each iteration process of solving coefficient matrix 𝑊, 
the missing label value in 𝐴 is updated according to equation 
(4), where 𝐴 𝑋𝑊 , and the updated 𝐴  is continued for 
subsequent iterations. 

 

𝐴
1，𝐴 1

𝐴 ， 1 𝐴 1
1，𝐴 1

4  

 
The continuous values in 𝐴 are restored to discrete values, 

and the final matrix 𝐴 can be obtained according to equation 
(5). At the same time, let 𝑌 𝐴. 

 

𝐴
1，𝐴 0.1

1，𝐴 0.1 5  

 

2.4. Model Optimization 
It can be seen that the objective function belongs to the non-

smooth convex optimization problem, so the method of 
Accelerated Proximal Gradient can be used to deal with the 
optimization problem. Equation (3) is divided into two parts: 

 
min 𝐹 𝑊 𝑓 𝑊 𝑔 𝑊 6  

 
Among them, 
 

𝑓 𝑊
1
2

‖𝑋𝑊 𝑌‖
𝛼
2

𝑡𝑟 𝑊 𝑋 𝐿 𝑋𝑊
𝛽
2

𝑡𝑟 𝑊𝑊 𝐿 7  

 
𝑔 𝑊 𝜆‖𝑊‖ 8  

 
Each element in the coefficient matrix 𝑊  is calculated 

using a soft threshold, which is: 
 

𝑊 𝑝𝑟𝑜𝑥 𝐺 9  
 

The method proposed by Lin et al[17] is adopted to 
accelerate the solution of 𝑊, which can be set 

 

𝑊 𝑊
𝑏 1

𝑏
𝑊 𝑊 10  

 
Significant runtime savings. 
At the same time, when solving the optimization problem 

of equation (7), Lipschitz continuity is satisfied: ‖∇𝑓 𝑊
∇𝑓 𝑊 ‖ 𝐿 ‖𝑊 𝑊 ‖ ,𝐿   is Lipschitz constant, and 𝐿  
can be obtained by proving Lipschitz continuity. The proof 
process is as follows: 

The gradient of 𝑊 in equation (7) is calculated as 
 

∇𝑓 𝑊 𝑋 𝑋𝑊 𝑋 𝑌 𝛼𝑋 𝐿 𝑋𝑊 𝛽𝐿 𝑊 11  
 

Assuming that there are parameters 𝑊   and 𝑊  , 
substituting parameters 𝑊   and 𝑊   into equation (11) 
respectively, we can get 

 
∇𝑓 𝑊 𝑋 𝑋𝑊 𝑋 𝑌 𝛼𝑋 𝐿 𝑋𝑊 𝛽𝐿 𝑊 12  

∇𝑓 𝑊 𝑋 𝑋𝑊 𝑋 𝑌 𝛼𝑋 𝐿 𝑋𝑊 𝛽𝐿 𝑊 13  
 

According to equation (12) and equation (13), and ∆𝑊
𝑊 𝑊 , can be obtained 

 
‖∇𝑓 𝑊 ∇𝑓 𝑊 ‖

       𝑋 𝑋∆𝑊 𝛼𝑋 𝐿 𝑋∆𝑊 𝛽𝐿 ∆𝑊

       2‖𝑋 𝑋∆𝑊‖ 2‖𝛼𝑋 𝐿 𝑋∆𝑊‖ 2 𝛽𝐿 ∆𝑊

       2‖𝑋 𝑋‖ ‖∆𝑊‖ 2‖𝛼𝑋 𝐿 𝑋‖ ‖∆𝑊‖ 2 𝛽𝐿 ‖∆𝑊‖

       2 ‖𝑋 𝑋‖ ‖𝛼𝑋 𝐿 𝑋‖ 𝛽𝐿 ‖∆𝑊‖ 14

 

 

Therefore 
 

‖∇𝑓 𝑊 ∇𝑓 𝑊 ‖ 2 ‖𝑋 𝑋‖ ‖𝛼𝑋 𝐿 𝑋‖ 𝛽𝐿 ‖∆𝑊‖ 15  

 

Therefore 
 

𝐿 2 ‖𝑋 𝑋‖ ‖𝛼𝑋 𝐿 𝑋‖ 𝛽𝐿 16  

 
 

ML-LSF algorithm as shown in algorithm 1. 
Algorithm 1: Multi-label Feature Selection based on Label-

specific Feature and Manifold Learning  
Input: training matrix 𝑋 ∈ 𝑅  , label matrix 𝑌 ∈ 𝑅  

with missing labels, parameters 𝜆, 𝛼, 𝛽; 
Output: coefficient matrix 𝑊∗ 
1）Initialization: 𝑊 𝑊 𝑋 𝑋 𝛾𝐼 𝑋 𝑌 , 𝑏

𝑏 1, 𝑡 1 
2）Repeat: 
3）   Calculate the Laplacian matrix 𝐿 , 𝐿  
4）   Calculate 𝐿  according to equation(16) 

5）   𝑊 𝑊 𝑊 𝑊  

6）   𝐺 𝑊 ∇𝑓 𝑊  

7）   𝑊 𝑝𝑟𝑜𝑥 𝑊 ∇𝑓 𝑊  

8）   𝑊 𝑊  
9）   Update 𝑌 by 𝑌 𝑋𝑊 , according to equation (4) 
10）  𝑡 𝑡 1 

11）  𝑏  

12）until convergence 
13 ） Restore the label matrix 𝑌  with missing labels 

according to equation (5) 
14）𝑊∗ 𝑊  
15）Compute ‖𝑤 ‖, 𝑖 ∈ 1, 𝑑  
16）Sort and select the first 𝑙 features 

2.5. Time Complexity 
In step 1, 𝑊  needs to be initialized and the time 

complexity is 𝑂 𝑛𝑑𝑙 𝑛𝑑 𝑑  ; In step 4, we need to 
compute 𝐿 , and the time complexity is 𝑂 𝑑 𝑙 ; In step 
6, we need to compute ∇𝑓 𝑊  , the time complexity is 
𝑂 𝑑 𝑙 𝑑𝑙 , and after 𝑡 iterations, the time complexity is 
𝑂 𝑡 𝑑 𝑙 𝑑𝑙 . As a result, the algorithm's time complexity 
is 𝑂 𝑛𝑑𝑙 𝑛𝑑 𝑑 𝑂 𝑑 𝑙 𝑂 𝑡 𝑑 𝑙 𝑑𝑙 . 



 

367 

3. Experimental Design and Results 

3.1. Data set and experiment setup 
In order to verify the effectiveness of ML-LSF, Mulan 

public data sets from multi-label library 
(http://mulan.sourceforge.net/datasets.html) to choose five 
classic data set, as shown in table 1. 

 
Table 1. Multi-label Datasets 

Data set Domain Sample Feature Class 
Arts Text 5000 462 26 

Entertain Text 5000 640 21 
Emotions Music 593 72 6 
Reference Text 5000 793 33 

Scene Image 2407 294 6 
 

For the algorithm ML-LSF in this paper, there are three 
parameters 𝜆 , 𝛼 , 𝛽 , and the parameter 𝜆  is set to 
10 ，10 ，…，10   range; Parameter 𝛼
2 ，2 ，…，2  , parameter 𝛽
2 ，2 ，…，2 , K-nearest neighbor probability graph 

model 𝐾 10  for instance correlation and feature 
correlation. Set the label missing rate to 0%, 10%, and 20%. 
As with all comparison algorithms, the classical ML-KNN 
classifier is used, where the parameters Num and Smooth are 
set to 10 and 1, respectively. 

3.2. Experimental results and analysis 
In order to verify the performance of ML-LSF algorithm, 

PMU, MFML, MLMLFS and LSF-CI algorithms are selected 
to compare with ML-LSF algorithm. The first three 
algorithms are popular multi-label feature selection 
algorithms and the last one is label-specific features multi-
label learning algorithm. The best results are highlighted in 
bold in the table. 

The experimental results from Table 2 to Table 5 were 

analyzed: 
When the label missing rate is 10%, the performance of 

ML-LSF algorithm is significantly improved compared with 
the other four algorithms. When a label is missing, the method 
of first completing the missing label can effectively improve 
the performance of the algorithm. MFML algorithm uses 
feature interaction to select more valuable features that may 
be ignored due to incomplete label space. In this way, the 
multi-label feature selection problem under missing labels 
can be solved. MLMLFS algorithm uses linear regression 
model to recover missing labels, which improves the accuracy 
of its modeling. The experimental results show that these two 
algorithms can also alleviate the performance problems 
caused by missing labels. However, the PMU algorithm can 
effectively handle multi-label feature selection when the label 
space is complete. However, when there are missing labels, 
the label space is damaged to a certain extent, resulting in the 
algorithm being unable to accurately model the relationship 
between features and labels. 

In general, the classification performance of ML-LSF 
algorithm is better than other algorithms. 
 

Table 2. Comparison results under 10% missing labels 
Data set PMU MFML MLMLFS LSF-CI ML-LSF 

Arts 0.0625 0.0590 0.0589 0.0610 0.0588 
Entertain 0.0631 0.0587 0.0580 0.0617 0.0576 
Emotions 0.2533 0.2458 0.2436 0.2564 0.2389 
Reference 0.0312 0.0280 0.0283 0.0305 0.0278 

Scene 0.1121 0.1097 0.1092 0.1103 0.1091 
Average 0.1044 0.1002 0.0996 0.1040 0.0984 

 
Table 3. Comparison results under 10% missing labels 

Data set PMU MFML MLMLFS LSF-CI ML-LSF 
Arts 0.4630 0.4987 0.5012 0.5011 0.5197 

Entertain 0.5512 0.5678 0.5519 0.5612 0.5803 
Emotions 0.7236 0.6830 0.7326 0.7130 0.7401 
Reference 0.6153 0.6055 0.5918 0.6011 0.6183 

Scene 0.7989 0.6497 0.6501 0.7991 0.8137 
Average 0.6304 0.6009 0.6055 0.6351 0.6544 

 
Table 4. Comparison results under 10% missing labels 

Data set PMU MFML MLMLFS LSF-CI ML-LSF 
Arts 5.5996 5.4869 5.5016 5.4756 5.3817 

Entertain 3.4021 3.2659 3.2854 3.3520 3.1978 
Emotions 2.3016 2.2356 2.2217 2.2231 2.0356 
Reference 3.5325 3.5023 3.4958 3.5124 3.4726 

Scene 0.7326 0.6989 0.6807 0.7012 0.6695 
Average 3.1137 3.0379 3.0370 3.0529 2.9514 

 
 



 

368 

Table 5. Comparison results under 10% missing labels 
Data set PMU MFML MLMLFS LSF-CI ML-LSF 

Arts 0.6658 0.6271 0.6411 0.6315 0.6154 
Entertain 0.6119 0.5903 0.6123 0.6124 0.5189 
Emotions 0.3772 0.4298 0.3967 0.4128 0.3121 
Reference 0.5120 0.5109 0.5116 0.5056 0.4713 

Scene 0.3210 0.5397 0.3530 0.3498 0.3115 
Average 0.4976 0.5396 0.5029 0.5024 0.4458 

 

 

 
Figure 1. Results of four classification indicators when the label missing rate is 10% 

in Arts dataset 

4. Conclusions 
The missing label problem in the multi-label feature 

selection is analyzed, and a multi-label feature selection 
algorithm ML-LSF is proposed to deal with the missing label 
problem. Based on the linear relationship between label-
specific features and labels and the nonlinear relationship 
between data, we learn the label-specific features and use the 
linear relationship between label-specific features and labels 
to fill the missing labels. The process of feature selection and 
label fill is iterative synchronously. The final solution is a 
non-smooth, convex optimization problem. The accelerated 
near-end gradient method is chosen to solve this problem. 
When comparing ML-LSF with other algorithms, the case of 
missing label rate of 10% is simulated, which can effectively 
verify the effectiveness of ML-LSF algorithm in processing 
multi-label data with missing label. Experimental results 
show that the proposed algorithm has good performance and 
is an effective multi-label feature selection algorithm. 

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