https://doi.org/10.31449/inf.v49i6.6979 Informatica 49 (2025) 175–190 175 
Balanced Generalised Tailored Approximation Point Algorithm for 
Solving Convex Optimisation Mathematical Problems in Bearing 
Vibration Signal Compressive Sensing 
Jimin Guo 
Department of Public Basic Education, Hebi Polytechnic, Hebi, 458030, China 
E-mail: gjm20231123@126.com 
Keywords: convex optimization problem, balanced-augmented lagrangian algorithm, compressive sensing, proximity 
point algorithm, bearing vibration signal 
Received: August 23, 2024 
With the continuous progress of science and technology, the transformation of compressive sensing 
problems into convex optimization problems has become a hot research topic. In this study, a novel 
algorithm, the balanced generalized customized proximal point algorithm, is proposed, which integrates 
the generalized customized proximal point algorithm with the balanced-augmented lagrangian method. 
Based on this algorithm, a compressive sensing system for bearing fault signals is designed, and the 
bearing fault signals are compressed by the universal compressive sensing model and the K-singular 
value decomposition algorithm, Then, the signals are reconstructed using the BG-CPPA. The 
experimental results showed that the BG-CPPA had a lower number of iterations and computation time 
compared with the traditional algorithm at different sparsity conditions. The reconstruction effect of the 
bearing inner ring signal was the best. Specifically, the BG-CPPA reduced the reconstruction error by 
33.33% and 20.00%, while reducing the reconstruction time by 32.46% and 52.64%. At compression 
ratios of 0.3, 0.4, and 0.5, the proposed compressive sensing system reduced the reconstruction error by 
35.39%, 44.06%, and 26.76% over the greedy algorithm, respectively. These results confirm the 
effectiveness of the BG-CPPA in improving the reconstruction accuracy and stability of bearing 
vibration signals, as well as the potential of the designed compressive sensing system in enhancing the 
observation efficiency of bearing fault vibration signals. 
Povzetek: Predlagan je izboljšan algoritem za konveksno optimizacijo v stisnjenem zaznavanju 
vibracijskih signalov ležajev, ki zmanjšuje napake rekonstrukcije in izboljšuje stabilnost ter učinkovitost 
obdelave podatkov. 
1 Introduction 
Affected by the development of science and technology, 
various problems in fields such as economy, computer 
science, and industry can be effectively transformed into 
optimization problems to solve. The convex optimization 
problem with linear equality constraints is one of the 
most common optimization problems, which is widely 
used in compressive sensing, image processing, machine 
learning, etc [1-3]. Traditional fault detection of rolling 
bearings mainly relies on bearing vibration signals. 
However, the high sampling frequency generated during 
the operation of rolling bearings can lead to a large 
amount of data, which have negative impacts on the 
detection results [4]. Compressive sensing is an abstract 
mathematical concept. It can effectively reduce the 
volume of compressed signal data obtained, thereby 
achieving bearing vibration signal compression and 
alleviating the drawbacks caused by excessive sampling 
data [5-6]. Therefore, a vibration signal compressive  
 
 
sensing system for rolling bearings is constructed to 
improve the accuracy and precision of bearing fault 
detection. Meanwhile, to improve the effectiveness of this 
method in practical applications, a convex optimization 
algorithm is designed based on the Generalized 
Customized Proximal Point Algorithm (GCPPA) and the 
Balanced-Augmented Lagrangian Method (B-ALM), 
namely the Balanced Generalized Customized Proximal 
Point Algorithm (BG-CPPA). It is expected to improve 
the compressive sensing reconstruction algorithm and 
utilize the convex optimization algorithm BG-CPPA to 
enhance the accuracy and stability of the bearing 
vibration signal compressive sensing system, promoting 
the innovation and development of bearing vibration 
signal processing technology. 
The overall structure of the study consists of five 
sections. The first section summarizes the research 
achievements and shortcomings of convex optimization 
algorithms and compressive sensing technology both 
domestically and internationally. In the second section, a 

176 Informatica 49 (2025) 175–190 J. Guo 
 
bearing vibration signal compressive sensing system 
based on the BG-CPPA is designed. In the third section, 
experiments and analysis are conducted on the proposed 
BG-CPPA and compressive sensing system. In the fourth 
section, the proposed BG-CPPA and compressive sensing 
system are compared with existing methods. In the fifth 
section, the experimental results are summarized and 
future research directions are indicated. 
2 Related works 
In modern signal processing and data analysis, the 
research on convex optimization algorithms and 
compressive sensing techniques has become a hot topic. 
Convex optimization algorithms have received 
widespread attention due to their high efficiency and 
stability in solving various mathematical problems, 
especially in dealing with optimization problems with 
complex constraints, showing unique advantages [7]. Lu 
et al. used convex optimization algorithms to re-represent 
non-convex problems in partially convex optimal control 
and optimization problems. A convex concave 
decomposition algorithm was proposed to handle 
nonlinear equality constraints, which solved the optimal 
fuel limited thrust spacecraft orbit problem [8]. For the 
finite time domain robust covariance control problem of 
partially observable linear systems, Kotsalis developed a 
computable processing framework for affine control 
strategy design based on the mean convex quadratic 
inequality and chance constrained linear inequality, 
achieving performance specifications in stochastic state 
control trajectories [9]. He et al. proposed a second-order 
continuous primal dual dynamical system with a 
time-dependent positive damping term for separable 
convex optimization problems with linear equality 
constraints. The time asymptotic properties of the system 
were verified through the Lyapunov analysis method. The 
convergence speed at different damping coefficients was 
derived [10]. A reliable and efficient trajectory generation 
method is a fundamental requirement for autonomous 
power systems. Malyuta et al. proposed a comprehensive 
tutorial on three trajectory generation methods based on 
convex optimization. The lossless convexity and two 
sequence convex programming algorithms ensured 
continuous convexity to optimize sequence trajectories. 
The convex optimization was used to generate 
non-convex trajectories [11]. To solve the collaborative 
problem where only the cost function of each node and its 
neighboring point information can be obtained, Liu et al. 
proposed a continuous time primal dual algorithm for 
constrained convex optimization problems in 
time-varying indirect connected graphs. This achieved the 
optimal solution under global convergence of the average 
state [12]. 
In addition, compressive sensing technology is gradually 
changing traditional signal processing methods due to its 
high efficiency and low-cost characteristics in signal 
acquisition and reconstruction. As a key component in 
mechanical systems, accurate vibration signals analysis is 
crucial for fault diagnosis and predictive maintenance. By 
combining convex optimization and compressive sensing 
techniques, researchers can more effectively process and 
analyze these complex signals, thereby improving the 
accuracy and efficiency of fault detection [13]. Chen 
proposed a compressive sensing method to address the 
limited data transmission capacity during remote machine 
condition monitoring, significantly reducing the 
computational complexity of vehicle fault diagnosis [14]. 
To cope with high sampling points and high sampling 
points for acoustic emission signals, Tai et al. proposed a 
compressive sensing processing framework. The wavelet 
sparse convolutional network was established to solve 
diagnosis and evaluation, thereby reducing the signal 
compression rate while ensuring acoustic reconstruction 
errors, and reducing the transmission signal data and 
pressure [15]. To promote energy perception in long-term 
vibration monitoring systems, Zonzini et al. proposed a 
model assisted variant based on the compressive sensing 
method. Sensing nail tied steel beams could retain 
reconstructed structural parameters even in defective 
configurations [16]. Although the remote wind turbine 
status monitoring system has better computing resources, 
there is data loss. Therefore, Peng et al. proposed a 
fault-tolerant missing data fault detection method based 
on compressive sensing. The compressive sensing signal 
reconstruction algorithm effectively reduced the 
probability of bearing fault detection data loss for two 
types of wind turbines [17]. Wang et al. proposed a novel 
modeling and control strategy for axial hybrid magnetic 
levitation bearings used in household flywheel energy 
storage systems to achieve effective monitoring of 
bearings. A new magnetic flux density feedback control 
was adopted instead of traditional control, achieving 
performance consistent with traditional position feedback 
control strategies [18]. Al-Chaab et al. put forward a 
medical image security compression system based on 
compressive sensing principle to solve the medical data 
security and privacy protection. The image was 
segmented and encoded using a Gaussian random number 
sensor matrix. The compression rate of the image size 
was about 30%, and the least significant bit technique 
was used to hide the data in the audio file, thereby 
improving the security and compression efficiency of the 
data [19]. 
Based on the above, current research on convex 
optimization problems with linear equality constraints 
mainly focuses on algorithm applications, while there is 
relatively little research on algorithm improvement. 
Compressive sensing technology, as a commonly used 
compression method in signal processing, has received 
less research from domestic and foreign scholars on its 
combination with convex optimization algorithms. In this 
context, a new convex optimization algorithm is proposed 
by combining GCPP algorithm and B-ALM algorithm. 
Then, a bearing vibration signal compressive sensing 

Balanced Generalised Tailored Approximation Point Algorithm for… Informatica 49 (2025) 175–190 177 
 
system is designed. Unlike current research methods, an 
innovative reconstruction algorithm for bearing vibration 
signals is designed to address the accuracy and stability 
of signal reconstruction in compressive sensing systems. 
It is expected to expand the application value of convex 
optimization algorithms and improve the effective 
observation of bearing fault vibration signals in 
compressive sensing systems. The study further 
summarizes the differences between the existing 
literature and the proposed method, as shown in Table 1. 
 
Table 1: Comparison between existing literature and proposed method 
Reference Method Problem domain Limitations 
Lu et al. [8] Convex-concave decomposition 
Nonlinear equality 
constraints 
May not scale well with 
large datasets 
Kotsalis [9] Affine control strategy 
Stochastic state 
control 
Sensitive to specific types 
of noise 
He et al. [10] 
Second-order continuous 
primal-dual dynamical system 
Separable convex 
optimization 
Strongly dependent on 
damping coefficients 
Malyuta et al. [11] 
Convex optimization-based 
trajectory generation 
Trajectory generation 
Limited applicability to 
non-convex trajectory 
generation tasks 
Liu et al. [12] 
Continuous time primal-dual 
algorithm 
Distributed consensus 
problems 
High computational 
complexity 
Chen [14] Compressive sensing approach 
Remote machine 
condition monitoring 
High demand on data 
transmission capacity 
Tai et al. [15] 
Wavelet sparse convolutional 
network 
Acoustic emission 
signal processing 
Inefficient for processing 
high sampling point data 
Zonzini et al. [16] 
Tilt-degree compressive sensing 
method 
Structural health 
monitoring 
Limited adaptability to 
defect configurations 
Peng et al. [17] 
Fault-tolerant method based on 
compressive sensing 
Wind turbine 
condition monitoring 
Sensitive to data loss issues 
Wang et al. [18] Hybrid magnetic bearing control 
Flywheel energy 
storage system 
Strong dependency on 
control strategies 
Al-Chaab et al. [19] 
CS-based medical image 
compression system 
Medical data security 
Limited hiding capability in 
audio files 
This paper GCPPA and B-ALM 
Bearing fault signal 
compressive sensing 
- 
 
3 A compressive sensing system for 
bearing vibration signals based on 
BG-CPPA 
The compressive sensing problem can be effectively 
transformed into a linear equality constrained convex 
optimization problem for solving. Therefore, a BG-CPPA 
based on GCPPA and B-ALM is proposed to solve the 
bearing fault vibration signal observation. On this basis, a 
bearing vibration signal compressive sensing system is 
designed based on the universal compressive sensing 
theory model, which is reconstructed using the 
BG-CPPA. 
3.1 BG-CPPA design 
Aiming at the low computational accuracy and poor 
stability of convex optimization algorithms in 
compressive sensing systems, a BG-CPPA suitable for 
solving linear equality constrained convex optimization 
problems is designed by combining GCPPA and B-ALM. 
The GCPPA is an extension of the Customized Proximal 
Point Algorithm (CPPA). It has great application value in 
fields such as image processing, statistics, and 
compressive sensing. Unlike the drawbacks of the CPPA 
where convergence efficiency is limited by the relaxation 
factor, GCPPA eliminates the relaxation step while 

178 Informatica 49 (2025) 175–190 J. Guo 
 
ensuring convergence efficiency [20]. The specific 
iterative expression is shown in equation (1). 
1
2
1
11
()
1
()
arg min ( )
2
(1 )
k k k
kT
k
x
kk
Ax b
t
x x A
r
r
xx




+
+

++
 
= − −





−− 


=+



 +  − 
 

 
(1) 
In equation (1), 
() x 
 represents a convex function and 
also represents the objective function. 
x
 represents a 
convex optimization problem. 

 represents a set. 

 
represents the Lagrange multiplier. 
A
 represents the 
coefficient matrix. 
b
 represent the known vector. 
r
, 
k
, 
t
 and 

 all represent parameters, 
, , 0 r k t 
, and 
(0.5,1) 
. The B-LAM algorithm is an extension of the 
Augmented Lagrangian Method (ALM). It balances the 
excessive proportions of the objective function, 
coefficient matrix, and set in the two sub-problems by 
reconstructing ALM [21]. The specific iteration is shown 
in equation (2). 
 
 
Start
0

G

Maximum 
number of 
iterations
,0 rt 
(0.5,1) 
Calculate new 
iteration points
Does it meet the shutdown 
criteria or the maximum 
number of iterations?
Stop
No
Yes
 
Figure 1: Iterative steps of BG-CPPA 
 
1 1 1
0
2
1
[ (2 ) ]
1
arg min ( )
2
k k k k
k k T k
xX
M A x x b
r
x x x x A x
r

  
+ − +
+

 = − − −



 
= + − − 


   
(2) 
In equation (2), 
0
M
 represents a positive definite 
matrix. The B-LAM is often transformed into an 
equivalent form in solving convex optimization problems, 
as shown in equation (3). 
1 1 1
0
1
[ (2 ) ]
1
Pr ( )
k k k k
k r k T k
M A x x b
x ox x A
r



+ − +
+
 = − − −


=+


   (3) 
However, the GCPPA has limited applicability in solving 
practical problems. The iterations of the B-ALM is 
affected by the quantity. The convergence efficiency 
decreases accordingly, which is not conducive to solving 
problems related to large-scale data. Therefore, a 
BG-CPPA is proposed by combining two algorithms. The 
specific calculation steps are shown in Figure 1. 
Firstly, the initial point, positive definite matrix, 
termination condition, and other parameters are set. A 
new iteration point is calculated based on the maximum 
number of iterations. The iterative of the BG-CPPA is 
shown in equation (4). 

1
1
2
1
()
arg min ( )
1
( )( ) (1 )
2
k k k
k
xX
T k T k k
a
Ax b
t
xx
rA A G x x A x


  
+
+

+

= − −



=+




 + − − +  −  



 
(4) 
In equation (4), 
G
 stands for the positive definite 
matrix, which can be used to ensure the convexity of the 
objective function, thus ensuring the convergence and 
uniqueness of the optimization algorithm. The shutdown 
criterion condition is shown in equation (5). 
 
 
11
max
max ,
 
k k k k
xx
or
iteration k
  
++

− − 







    (5) 
 
In equation (5), 

 represents the termination condition. 
max
 iteration
 represents the maximum number of 

Balanced Generalised Tailored Approximation Point Algorithm for… Informatica 49 (2025) 175–190 179 
 
iterations. The first-order optimality condition for the 
convergence iteration of the proposed BG-CPPA is 
shown in equation (6). 
 
1
11
1
11
( )( )
( ) ( ) ( ) 0
(1 )
( ) ( ) ( ) 0
T k k
k k T
T k k
k T k k k
rA A G x x
x x x x
A
Ax b t


   
+
++
+
++
  +−

 − + − 


 − +  − 
   

−  − + − 


(6) 
According to the optimality condition equation, the 
equivalent form of its inequality is transformed into a 
compact form of inequality, as shown in equation (7). 
 
1 1 1 1
1 1 2
( ) ( ) ( ) [ ( ) ( )]
( 1)
( ) ( ) 0
k k T k k k
k T k k
x x F Q
t
      
   
+ + + +
+ + +
− + − + −
−
+ − − 

(7) 
In equation (7), 
Q
 represents a positive definite matrix. 

 represents the iteration point. 
() F 
 denotes the 
affine monotone function, as shown in equation (8). 
( )
T
A
F
Ax b
 −
=

−

           (8) 
The lemma of algorithms is closely related to their 
properties. Therefore, two lemmas are further proposed as 
reference equations for subsequent performance 
verification of the BG-CPPA. Lemma 1: The solution of 
the first-order optimality condition inequality for convex 
optimization problems is generated based on the sequence 
generated by the BG-CPPA, as shown in equation (9). 
( )
1 1 1 1 2
1
( ) (1 )( ) ( )
T
k k k k T k k
Qt        
 + +  + + +
− −  − − −

(9) 
Lemma 2: The sequences calculated according to the 
iterative equation of the BG-CPPA satisfy the inequality 
shown in equation (10). 
22
1 1 2
2 2 2
1 1 1
2
1
1
(2 )
1
(1 )
1
(1 )
k k k k
D
k k k
QQ
k
x x t
t
t

     

+ + +
 +  +  +
+
− + − −


 − + − − − −




− − −



(10) 
According to Lemma 1 and Lemma 2, the convergence 
property of the BG-CPPA can be further effectively 
verified, thereby achieving broader application value. 
Therefore, the process of BG-CPPA to achieve 
convergence is shown in Table 2. 
3.2 A compressive sensing system design for 
bearing vibration signals based on 
BG-CPPA 
Compressive sensing technology obtains signals in actual 
engineering construction through under-sampling based 
on signal sparsity and non correlation. Combined with 
optimization algorithms, the data are reconstructed to 
directly collect natural signal data through compression. 
The sparse signal is shown in equation (11). 
y  =
              (11) 
In equation (11), 
y
 represents the sampled value or 
observed data. 

 represents a matrix that multiplies the 
number of linear measurements by the length of the 
original signal. 

 represents a sparse signal. In the 
sensing process of sparse targets, the signal itself contains 
fewer non-zero elements, i.e., the target finite coefficient 
signal contains only several non-zero elements. The 
specific sparse signal compressive sensing routine theory 
model is shown in Figure 2. 
However, in practical applications, the sampled signal 
data do not exhibit sparsity. Sparse features often need to 
be formed through changes in a certain transformation 
domain. Therefore, combined with the mathematical 
expression equation of basic compressive sensing, a 
universal compressive sensing mathematical model is 
developed, as shown in Figure 3. 
Compared with the sparse signal compressive sensing 
theory model, the target signal in the universal 
compressive sensing model is a vector with sparse 
features that has a finite length and a discrete distribution 
in the spatiotemporal dimension [22-23]. Therefore, the 
bearing vibration signal compressive sensing system 
based on the BG-CPPA proposed in the study follows a 
universal compressive sensing model, forming an 
over-complete dictionary in dictionary learning. It 
combined with the optimal measurement matrix to 
capture bearing vibration signal data. The specific 
implementation process is shown in Figure 4. 
 
Table 2: Convergence process of BG-CPPA 
Pseudocode of # BG-CPPA 
# Initialization 
x_k = initial_point_x 
y_k = initial_point_y 
G = positive_definite_matrix 
epsilon = termination_condition 

180 Informatica 49 (2025) 175–190 J. Guo 
 
max_iterations = maximum_number_of_iterations 
r = positive_real_number 
t = positive_real_number 
alpha = alpha_parameter_in_range_0_5_to_1 
# Iteration process 
for k in range(max_iterations): 
# Calculate new iteration point x_{k+1} 
x_k_plus_1 = argmin_x {theta(x) + 0.5 * ||x - y_k||_G^2 + r * ||Ax - b||^2} 
# Calculate new iteration point y_{k+1} 
y_k_plus_1 = x_k_plus_1 + alpha * (A * x_k_plus_1 - b) 
# Convergence Check 
if ||x_k_plus_1 - x_k|| <= epsilon or ||A * x_k_plus_1 - b|| <= epsilon: 
print ("Algorithm has converged.") 
break 
else: 
x_k = x_k_plus_1 
y_k = y_k_plus_1 
# Output the solution 
print ("Solution:", x_k_plus_1) 
 
y
Observations


Target 
sparse signal
 
Figure 2: Theory model of sparse signal compressive sensing 
Observations
Target sparse signal
Merge matrix Sparse vector
 
Figure 3: Universal compressive sensing theory model 

Balanced Generalised Tailored Approximation Point Algorithm for… Informatica 49 (2025) 175–190 181 
 
Sparse representation
Bearing vibration signal Measurement coding
Transmission/storage 
system
Decoding 
reconstruction
Inverse 
transformation
Reconstructing signal 
analysis
Signal compressive sensing
Signal recovery process
 
Figure 4: Implementation process of compressive sensing system for bearing vibration signals 
 
After compression detection, the BG-CPPA is used to 
reconstruct and restore the original bearing vibration 
signal. This compressive sensing system is used to 
compress signals under under-sampling conditions. 
Therefore, the negative impact of large-scale sampled 
signal data on transmission and storage systems can be 
mitigated. Meanwhile, considering the limitations of 
compressive sensing systems in signal usage, actual 
signal data have potential sparsity. Therefore, sparse 
matrices are used to represent the collected data signals, 
as shown in equation (12). 
 
1
N
ii
i
   
=
= = 

        (12) 
 
In equation (12), 
N
 represents the length of the original 
signal. 
i

 represents the coefficient atom. 

 represents 
a sparse vector in a sparse matrix. 

 represents the 
sparse basis. Sparse vectors have potential sparse features 
in sparse bases. Then, the K-Singular Value 
Decomposition (K-SVD) algorithm is used to construct 
an over-complete dictionary internally associated with 
vibration signals. The core of the K-SVD mainly includes 
two parts: iteratively updating the dictionary and sparse 
encoding [24-25]. The process of iteratively updating the 
dictionary is often accompanied by significant deviations. 
Therefore, under the sparsity control, the overall error is 
reduced by optimizing and updating each column of 
atoms. The error matrix is shown in equation (13). 
 
ii
i
Err Y d



=−

        (13) 
 
In equation (13), 
Err
 represents the error matrix. 
Y
 
represents the atomic matrix of the bearing vibration 
signal. 

 represents the sparsity. 
d
 represents the 
error value. The mathematical expression of sparse 
encoding is shown in equation (14). 
 
2
2
, arg min P O Y PO =−
      (14) 
 
In equation (14), 
P
 represents the original 
over-complete dictionary. 
O
 represents a sparse matrix. 
According to the constructed bearing vibration signal 
compressive sensing system, the termination condition of 
the BG-CPPA in numerical experiments is shown in 
equation (15). 
 
 
11
max ,
k k k k
xx   
++
− − 
      (15) 
 
The algorithm parameter selection for decompressive 
sensing problem is shown in equation (16). 
 
2
( ( ) 0.01)
0.95
0.1
0.15
TT
n
G r A A I rA A
r
t
r
  =  + −

=


 =

+

=


      (16) 

182 Informatica 49 (2025) 175–190 J. Guo 
 
In equation (16), 
n
I
 represents the identity matrix of 
n
 
dimension. 

 represents the number of repeated 
experiments. 
4 Verification analysis of bearing 
vibration signal compressive 
sensing system based on BG-CPPA 
The convex optimization algorithm BG-CPPA and the 
compressive sensing system are used for sparse signal 
simulation experiments in response to the potential 
sparsity characteristics of bearing fault vibration signals. 
The algorithm is validated in terms of function 
performance comparison, simulated signal reconstruction 
effect, and reconstructed Signal-To-Noise Ratio (SNR) at 
different compression ratios. Based on wavelet threshold 
function denoising, the fault signal observation 
verification of the compressive sensing system is carried 
out. 
4.1 BG-CPPA validation analysis 
To verify the effectiveness of the proposed BG-CPPA in 
solving convex optimization problems such as 
compressive sensing, the algorithm performance is 
verified by combining function performance comparison, 
one-dimensional simulation signal reconstruction 
efficiency, and reconstruction SNR of bearing fault 
signals at different compression ratios.  
The BG-CPPA takes the value of 0.95 for 

, 0.1 for 
r
, 
and 
2
0.15
t
r
+
=
. The number of data training samples 
is 4096 and the number of tests is 1024, with sparsity 
  1,2,...,250 k 
. Different sparsities 
Iteration 0 100 200 300 0
0.2
0.4
0.6
0.8
1.0
(a) NNDP
Accurate
Precision rate P Recall rate R
Iteration 0 100 200 300 0
0.2
0.4
0.6
0.8
1.0
(b) Proximal point algorithm
Accurate 400 400
Average fitness Optimal fitness
Precision rate P Recall rate R Average fitness Optimal fitness
 are selected and 
the average value is calculated by repeating the 
experiment 10 times.  
By controlling the increase in sparsity, the comparison 
results of BG-CPPA, GCPPA, and B-LAM at termination 
conditions of 10-8 are shown in Figure 5. 
 
 
170
190
210
230
250
270
50 0 150 100 250 200
Sparsity
Iterations
GCPPA
B-ALM
BG-CPPA
(a) Comparison of iteration times of three 
algorithms at different sparsity levels
0.4
0.5
0.7
0.8
1.0
1.1
50 0 150 100 250 200
Sparsity
Time (s)
GCPPA
B-ALM
BG-CPPA
(b) Comparison of computation time for three 
algorithms at different sparsity levels
0.6
0.9
 
Figure 5: Comparison of algorithm performance at different sparsity variations 
 
From Figure 5 (a), the B-ALM exhibited significant 
fluctuations with increasing sparsity. The iterations of 
BG-CPPA and GCPPA were relatively more stable. The 
number of iterations of BG-CPPA was lower than the 
other two methods in the whole sparsity change, which 
shows that the algorithm can maintain consistent 
performance and better robustness when dealing with 
data with different sparsity. The sparsity of detection 
algorithm is low, which is very important for fault 
detection system. In bearing fault detection, the reliability 
is directly related to the accuracy and timeliness of fault 
detection, thus affecting the maintenance and operation 
safety of equipment. Figure 5 (b) shows the computation 
time of three algorithms at different sparsity levels. The 
calculation time of BG-CPPA was less affected by 
sparsity, which fluctuated around 0.60s. The calculation 
time of B-ALM was greatly affected by sparsity. When 
the sparsity was greater than 150, the calculation time 
increased by 12.13% -30.47%. This shows that 
BG-CPPA can maintain high efficiency at different 
sparsity. In contrast, the calculation time of B-ALM 
algorithm increases significantly with the increase of 
sparsity, which indicates that it is inefficient when 
dealing with high sparsity data. Overall, the BG-CPPA 
performs significantly better than the other two 
algorithms when the termination condition is 10-8. 
The BG-CPPA is to transform the compressive sensing 
signal of bearing vibration into a convex optimization 

Balanced Generalised Tailored Approximation Point Algorithm for… Informatica 49 (2025) 175–190 183 
 
problem for reconstruction. Therefore, to further verify 
the performance of the BG-CPPA, a reconstruction 
verification is conducted using simulated signals. 
Considering that the signal data obtained in practical 
applications often contains external noise disturbances, an 
additional 15 decibel Gaussian white noise is added to the 
simulated signal for performance testing. 15dB is a 
common SNR level. After repeated experiments, it is 
found that 15 dB is a balance point between algorithm 
performance and computing resources, which can ensure 
sufficient performance and avoid excessive computing 
burden. The comparison results of the three methods are 
shown in Figure 6. 
From the signal reconstruction results of the three 
algorithms, the three convex optimization algorithms 
were not sensitive to the signal sparsity. There was no 
distortion in the reconstruction results. This indicates that 
when the convex optimization algorithm performs 
reconstruction, it can automatically predict sparsity and 
accurately recover the original signal. Overall, the 
GCPPA shown in Figure 6 (a) has the worst simulation 
effect, followed by Figure 6 (b). The simulation effect of 
the BG-CPPA proposed in the study is superior among 
the three algorithms. The average reconstruction time, 
reconstruction error, SNR and Peak Signal-to-Noise Ratio 
(PSNR) comparisons of the three methods are shown in 
Table 3. 
In Table 3, the BG-CPPA had the lowest reconstruction 
error and reconstruction time among the three algorithms. 
In terms of reconstruction error, BG-CPPA decreased by 
32.46% and 52.64% compared with GCPPA and B-ALM, 
respectively. In terms of reconstruction time, the 
BG-CPPA reduced 33.33% and 20.00% respectively 
compared with the two algorithms. From the 
reconstruction results, it is demonstrating that the 
BG-CPPA has significant advantages in reconstruction 
speed, noise resistance, and reconstruction accuracy, 
making it more effective in solving convex optimization 
problems in compressive sensing technology. From the 
SNR and PSNR of the three algorithms, the SNR and 
PSNR of the proposed method are better than the other 
two methods. This indicates that the reconstructed signal 
quality of BG-CPPA is better, with less difference from 
the original signal and higher recovery accuracy. The 
reconstruction SNR results of three algorithms at 
different compression ratios are shown in Figure 7. From 
Figure 7, the reconstruction SNR of the three algorithms 
increased with the increase of the compression ratio. The 
reconstruction SNR of BG-CPPA had the best increase in 
amplitude and speed compared with the other two 
algorithms. This indicates that BG-CPPA has advantages 
in reconstruction processing, which has the best 
reconstruction effect. The performance of three 
algorithms in reconstruction testing is shown in Figure 8. 
 
 
-1.5
-0.5
0
1.0
1.5
2.5
50 0 150 100 250 200
Time (s)
Amplitude
-1
0.5
2.0
Original signal GCPPA
(a) Simulation effect of GCPPA algorithm under 15dB 
gaussian white noise
-1.5
-0.5
0
1.0
1.5
2.5
50 0 150 100 250 200
Time (s)
Amplitude
-1
0.5
2.0
Original signal B-ALM
(b) Simulation effect of B-ALM algorithm under 15dB 
gaussian white noise
-1.5
-0.5
0
1.0
1.5
2.5
50 0 150 100 250 200
Time (s)
Amplitude
-1
0.5
2.0
Original signal BG-CPPA
(c) Simulation effect of BG-GCPPA algorithm under 
15dB gaussian white noise
 
Figure 6: Comparison of reconstruction effects at 15 dB Gaussian white noise 
 

184 Informatica 49 (2025) 175–190 J. Guo 
 
Table 3: Comparison of analog signal reconstruction performance at 15dB Gaussian white noise 
Targets GCPPA B-ALM BG-CPPA 
Err 3.05 4.35 2.06 
Time (s) 0.06 0.05 0.04 
SNR (dB) 21.50 28.13 29.76 
PSNR (dB) 22.04 28.97 30.53 
 
0
5
10
15
20
25
30
35
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Compression ratio (N/M)
Reconstruct signal-to-noise ratio
BG-CPPA
GCPPA
B-ALM
0.8
 
Figure 7: Reconstruction signal-to-noise ratio at different compression ratios 
 
-4
0
4
-2
2
0 50 100 150 200 250 300 350 400
Time (s)
Amplitude
(a) Inner ring fault signal
-4
0
4
-2
2
0 50 100 150 200 250 300 350 400
Time (s)
Amplitude
(b) Sparse representation of inner circle fault signals
-4
0
4
-2
2
0 50 100 150 200 250 300 350 400
Time (s)
Amplitude
(c) Original sparse signal
-4
0
4
-2
2
0 50 100 150 200 250 300 350 400
Time (s)
Amplitude
(d) GCPPA sparse signal reconstruction effect
-4
0
4
-2
2
0 50 100 150 200 250 300 350 400
Time (s)
Amplitude
(e) B-ALM sparse signal reconstruction effect
-4
0
4
-2
2
0 50 100 150 200 250 300 350 400
Time (s)
Amplitude
(f) BG-CPPA sparse signal reconstruction effect
 
Figure 8: Comparison of three algorithms for reconstructing sparse signal of bearing inner ring fault 
 

Balanced Generalised Tailored Approximation Point Algorithm for… Informatica 49 (2025) 175–190 185 
 
From Figures 8 (a) and 8 (b), the trained over-complete 
dictionary showed that the bearing vibration signal had 
potential sparsity. The sparsity effect was superior, which 
met the experimental requirements of the proposed 
compressive sensing system. The sparse representation 
signal reconstruction effects of the three algorithms at a 
compression ratio of 0.5 are shown in Figures 8 (d) - (f). 
Figure 8 (c) shows the original sparse signal. The 
reconstruction effects of the three algorithms showed that 
the BG-CPPA had the best reconstruction effect. The 
convex optimization algorithm BG-CPPA performed the 
best in reconstructing sparse signals. It is suitable as a 
reconstruction algorithm for the compressive sensing 
system of bearing vibration signals, thereby improving 
the reconstruction processing performance of the 
compressive sensing system. 
4.2 Verification analysis of compressive 
sensing system for bearing vibration 
signals 
To further verify the superiority of the convex 
optimization algorithm BG-CPPA in the bearing 
vibration signal compressive sensing system, the 
proposed bearing vibration signal compressive sensing 
system is analyzed. The vibration signals generated by 
bearing failure operation are complex and diverse, with a 
large amount of noise. Therefore, the wavelet threshold 
function is used to denoise it. The pre- and post noise 
reduction effects of bearing vibration signals are shown 
in Figure 9. 
By comparing before and after signal denoising, the 
signal in Figure 9 (a) was affected by noise. The 
threshold function showed more fluctuations, making it 
impossible to directly compress and sense the signal data. 
Figure 9 (b) shows the threshold function after denoising. 
The overall signal was smoother after denoising. On this 
basis, the compressive sensing system is validated for 
bearing fault signal compression. The testing and 
reconstruction results of the bearing fault inner ring 
signal at a compression ratio of 0.5 are shown in Figure 
10.  
From Figure 10 (a) and Figure 10 (b), the improved 
K-SVD sparsely represented the bearing fault signal, 
fully demonstrating its potential sparse features. Based on 
the Gaussian random observation matrix with a 
compression ratio of 0.5 in Figure 10 (c), sparse signals 
were compressed through matrix compression to achieve 
signal data compression. Figure 10 (d) shows the 
reconstruction effect of the proposed convex optimization 
algorithm. It effectively reconstructed and recovered 
bearing signals. Meanwhile, the Greedy Algorithm (GA) 
is introduced to compare the reconstruction performance 
of bearing inner ring fault signals with three methods at 
different compression ratios. The specific comparison 
data is shown in Table 4. 
 
 
-4
0
4
-2
2
0 100 200 300 400 500 600 700 800
Amplitude
900 1000
Time (s)
(a) Original signal
-4
0
4
-2
2
0 100 200 300 400 500 600 700 800
Amplitude
900 1000
Time (s)
(b) Noise reduction
 
Figure 9: Simulation analysis of noise reduction effect on noise signals 

186 Informatica 49 (2025) 175–190 J. Guo 
 
-4
0
4
-2
2
0 200 400 600 800 1000 1200 1400 1600
Time (s)
Amplitude
(a) Inner ring fault signal
2000
-4
0
4
-2
2
0 500 1000 1500 2000
Time (s)
Amplitude
(b) Sparse representation of fault signals
2500
-2
0
2
0 200 400 600 800 1000 1200 1400
Time (s)
Amplitude
(c) Sparse length of the original signal with a compression ratio of 0.5
-4
0
4
-2
2
0 200 400 600 800 1000 1200 1400 1600
Time (s)
Amplitude
(d) BG-CPPA sparse signal reconstruction effect
2000
 
Figure 10: Signal reconstruction effect of compressive sensing system at 0.5 compression ratio 
Table 4: Comparison of reconstruction performance for bearing fault inner ring signal at different compression ratios 
Algorithm 
M/N=0.3 M/N=0.4 M/N=0.5 
Err 
Reconstruct 
signal-to-noise 
ratio (dB) 
Time 
(s) 
Err 
Reconstruct 
signal-to-nois
e ratio (dB) 
Time 
(s) 
Err 
Reconstruct 
signal-to-nois
e ratio (dB) 
Time 
(s) 
GCPPA 21.34 0.14 3.76 9.21 3.81 4.37 6.99 11.32 3.83 
B-ALM 19.27 0.33 2.15 8.76 5.66 2.28 6.34 13.54 2.67 
BG-CPPA 9.22 0.73 0.29 4.45 13.18 0.54 3.01 18.04 0.79 
GA 14.27 0.46 1.70 5.49 23.56 1.04 4.11 17.43 1.28 
 
From Table 4, the reconstruction performance of 
BG-CPPA was significantly better than GCPPA and 
B-ALM at different compression ratios. At a compression 
ratio of 0.4, the reconstruction SNR of the GA increased 
by 10.38dB compared with BG-CPPA. However, at 
compression ratios of 0.3 and 0.5, the BG-CPPA 
performed better. In terms of the running time, BG-CPPA 
showed better ability. It had the lowest average signal 
reconstruction time among all algorithms at three 
compression ratios. The BG-CPPA reduced time 
consumption by 82.94%, 48.08%, and 38.28% compared 
with the GA at three compression ratios, respectively. As 
the compression ratio increases, the BG-CPPA requires 
more time to reconstruct the SNR and signal, while the 
Err decreases. This indicates that BG-CPPA can 
effectively improve the signal reconstruction accuracy 
and efficiency of the bearing vibration signal compressive 
sensing system. It also confirms the effectiveness and 
reliability of the proposed compressive sensing system in 
bearing fault vibration signals and compression 
processing. The mean, Standard Deviation (SD) and 95% 
confidence intervals of the reconstruction errors of the 
bearing fault signals at different compression ratios are 
shown in Table 5. 
From Table 5, the mean value of the reconstruction error 
decreased with the increase of the compression ratio, 
which indicates that the compressive sensing system can 
still maintain high reconstruction accuracy at higher 
compression ratios. Meanwhile, the smaller standard 
deviation and confidence interval width indicate the 
consistency and reliability of the experimental results.
Table 5: Means, standard deviations and 95% confidence intervals of reconstruction errors (Err) for bearing fault 
signals 
Compression ratio Mean SD 95% confidence 
interval 
0.3 9.22 0.73 (8.76, 9.68) 
0.4 4.45 0.29 (4.16, 4.74) 
0.5 3.01 0.54 (2.47, 3.55) 
 

Balanced Generalised Tailored Approximation Point Algorithm for… Informatica 49 (2025) 175–190 187 
 
4.3 Time complexity analysis 
In the BG-CPPA, the most time-consuming part is the 
iterative process, especially solving the optimization 
problem and updating the iteration points in each 
iteration. The time complexity of each iteration is 
assumed to be O(f), where f represents the computational 
complexity of a single iteration. The initialization step 
has a time complexity of O(1), and the algorithm 
performs up to N iterations with a time complexity of 
O(f) per iteration. Therefore, the time complexity of the 
entire iterative process is O(N·f). A convergence check is 
performed after each iteration, which also has a time 
complexity of O(1). Therefore, the overall time 
complexity of the BG-CPPA is O(N·f). Although the time 
complexity of the BG-CPPA is proportional to the 
number of iterations, experimental verification has shown 
that the algorithm can maintain a low number of 
iterations at different sparsity levels. This indicates that 
the BG-CPPA has high efficiency and good scalability in 
handling compressive sensing problems. 
5 Discussion 
The BG-CPPA introduced in this study outperforms 
existing methods in the compressive sensing of bearing 
fault signals, particularly in stability, efficiency, and error 
reduction. It exhibits robustness across varying data 
sparsity, maintaining low iteration counts and 
computation time with minimal fluctuations, indicating 
its effectiveness in handling data size changes and 
adapting to practical uncertainties. The computation time 
of B-ALM increases with the increase of sparsity, while 
BG-CPPA remains efficient. Due to its optimized 
computation and fast convergence, the average 
reconstruction time of BG-CPPA is only 0.04s, 
significantly faster than GCPPA and B-ALM. In addition, 
the reconstruction SNR of BG-CPPA is superior to other 
algorithms, especially when the compression ratio 
increases. The accuracy in representing and recovering 
signal sparsity through the K-SVD-based over-complete 
dictionary contributes to its superior reconstruction 
precision. 
The novelty of the BG-CPPA is that it combines the 
advantages of GCPPA and B-ALM while overcoming the 
limitations of both in practical applications. Although 
GCPPA eliminates relaxation steps and improves 
convergence efficiency, its applicability is limited when 
dealing with large-scale data problems. Although the 
B-ALM algorithm balances the objective function, 
coefficient matrix, and set proportion by reconstructing 
ALM, its iteration times are affected by the size of the 
problem, and the convergence efficiency is 
correspondingly reduced. The B-CPCA not only 
improves the applicability and convergence efficiency of 
the algorithm by combining the advantages of both, but 
also significantly improves the computational efficiency  
 
and reconstruction accuracy of the algorithm through 
optimizing the iterative process. 
6 Conclusion 
A convex optimization algorithm BG-CPPA combining 
GCPPA and B-ALM is proposed to address the poor CS 
accuracy and reconstruction effect of bearing fault 
vibration signals. Therefore, a bearing vibration signal 
compressive sensing system combining universal 
compressive sensing model and K-SVD over-complete 
dictionary is designed. The results showed that the 
BG-CPPA had fewer iterations and time than traditional 
algorithms at different sparsity levels. Compared with 
GCPPA and B-ALM, BG-CPPA reduced the 
reconstruction error ratio by 32.46% and 52.64%, 
respectively. At different compression ratios, the 
reconstruction performance of BG-CPPA was superior to 
the other two algorithms. According to the BG-CPPA, the 
compressive sensing system had the best reconstruction 
effect compared with different reconstruction algorithms. 
Compared with GA, BG-CPPA reduced reconstruction 
time by 82.94%, 48.08%, and 38.28% respectively at 
compression ratios of 0.3, 0.4, and 0.5. The results 
indicate that the proposed convex optimization algorithm 
BG-CPPA has great application value in solving linear 
equality constrained convex optimization problems. The 
compressive sensing system based on BG-CPPA has 
certain feasibility, which can effectively improve the 
reconstruction accuracy and effectiveness of bearing 
signal compressive sensing. However, the convex 
optimization algorithm proposed in the study only solves 
the compressive sensing technology. It has not been 
validated in other fields. It has certain limitations. 
Therefore, future research work will be carried out in the 
following areas: 
Further explore the potential applications of BG-CPPA in 
other fields, such as image processing and machine 
learning. 
Optimize the BG-CPPA to improve its performance in 
large-scale data processing. 
Investigate the application of BG-CPPA in real-time 
systems, such as online monitoring and fault diagnosis. 
Develop more efficient sparse representation and 
reconstruction techniques to enhance signal processing. 
Explore the combination of BG-CPPA with other 
optimization techniques, such as deep learning or 
evolutionary algorithms. 
Track and evaluate the performance of the BG-CPPA in 
real-world applications over time to gather feedback and 
guide future improvements. 

188 Informatica 49 (2025) 175–190 J. Guo 
 
References 
[1] J. Zhang, B. Chen, R. Xiong, and Y. Zhang, 
“Physics-inspired compressive sensing: Beyond 
deep unrolling,” IEEE Signal Processing Magazine, 
vol. 40, no. 1, pp. 58-72, 2023. 
https://doi.org/10.1109/MSP.2022.3208394 
[2] M. Gams, and T. Kolenik, “Relations between 
electronics, artificial intelligence and information 
society through information society rules,” 
Electronics, vol. 10, no. 4, pp. 514, 2021. 
https://doi.org/10.3390/electronics10040514 
[3] S. S. Chandra, L. M. Bran, X. Liu, S. Liu, S. 
Bollmann, and S. Crozier, “Deep learning in 
magnetic resonance image reconstruction,” Journal 
of Medical Imaging and Radiation Oncology, vol. 
65, no. 5, pp. 564-577, 2021. 
https://doi.org/10.1111/1754-9485.13276 
[4] D. Liu, L. Cui, W. Cheng, D. Zhao, and W. Wen, 
“Rolling bearing fault severity recognition via data 
mining integrated with convolutional neural 
network,” IEEE Sensors Journal, vol. 22, no. 6, pp. 
5768-5777, 2022. 
https://doi.org/10.1109/JSEN.2022.3146151 
[5] G. Ye, M. Liu, and M. Wu, “Double image 
encryption algorithm based on compressive sensing 
and elliptic curve,” Alexandria Engineering Journal, 
vol. 61, no. 9, pp. 6785-6795, 2022. 
https://doi.org/10.1016/j.aej.2021.12.023 
[6] C. Li, G. Li, and P. K. Varshney, 
“Communication-efficient federated learning based 
on compressed sensing,” IEEE Internet of Things 
Journal, vol. 8, no. 20, pp. 15531-15541, 2021. 
https://doi.org/10.1109/JIOT.2021.3073112 
[7] A. M. Elbir, K. V. Mishra, S. A. Vorobyov, and R. 
W. Heath, “Twenty-five years of advances in 
beamforming: From convex and nonconvex 
optimization to learning techniques,” IEEE Signal 
Processing Magazine, vol. 40, no. 4, pp. 118-131, 
2023. https://doi.org/10.1109/MSP.2023.3262366 
[8] P. Lu, “Convex–concave decomposition of 
nonlinear equality constraints in optimal control,” 
Journal of Guidance, Control, and Dynamics, vol. 
44, no. 1, pp. 4-14, 2021. 
https://doi.org/10.2514/1.G005443 
[9] G. Kotsalis, G. Lan, and A. S. Nemirovski, “Convex 
optimization for finite-horizon robust covariance 
control of linear stochastic systems,” SIAM Journal 
on Control and Optimization, vol. 59, no. 1, pp. 
296-319, 2021. 
https://doi.org/10.1137/20M135090X 
[10] X. He, R. Hu, and Y. P. Fang, “Convergence rates 
of inertial primal-dual dynamical methods for 
separable convex optimization problems,” SIAM 
Journal on Control and Optimization, vol. 59, no. 5, 
pp. 3278-3301, 2021. 
https://doi.org/10.1137/20M1355379 
[11] D. Malyuta, T. P. Reynolds, M. Szmuk, T. Lew, R. 
Bonalli, M. Pavone, B. Açıkmeşe, “Convex 
optimization for trajectory generation: A tutorial on 
generating dynamically feasible trajectories reliably 
and efficiently,” IEEE Control Systems Magazine, 
vol. 42, no. 5, pp. 40-113, 2022. 
https://doi.org/10.1109/MCS.2022.3187542 
[12] H. Liu, W. X. Zheng, and W. Yu, “Continuous-time 
algorithm based on finite-time consensus for 
distributed constrained convex optimization,” IEEE 
Transactions on Automatic Control, vol. 67, no. 5, 
pp. 2552-2559, 2021. 
https://doi.org/10.1109/TAC.2021.3079192 
[13] C. Wang, C. Liu, M. Liao, and Q. Yang, “An 
enhanced diagnosis method for weak fault features 
of bearing acoustic emission signal based on 
compressed sensing,” Mathematical Biosciences 
and Engineering, vol. 18, no. 1, pp. 1670-1688, 
2022. https://doi.org/10.3934/mbe.2021086 
[14] Z. Chen, “Rolling bearing fault diagnosis with 
compressed signals based on hybrid compressive 
sensing,” Journal of Vibroengineering, vol. 24, no. 
1, pp. 18-29, 2022. 
https://doi.org/10.21595/jve.2021.22061 
[15] J. Tai, C. Liu, X. Wu, and J. Yang, “Bearing fault 
diagnosis based on wavelet sparse convolutional 
network and acoustic emission compression 
signals,” Mathematical Biosciences and 
Engineering, vol. 19, no. 8, pp. 8057-8080, 2022. 
https://doi.org/10.3934/mbe.2022377 
[16] F. Zonzini, M. Zauli, M. Mangia, N. Testoni, and L. 
De Marchi, “Model-assisted compressed sensing for 
vibration-based structural health monitoring,” IEEE 
Transactions on Industrial Informatics, vol. 17, no. 
11, pp. 7338-7347, 2021. 
https://doi.org/10.1109/TII.2021.3050146 
[17] Y. Peng, W. Qiao, and L. Qu, “Compressive 
sensing-based missing-data-tolerant fault detection 
for remote condition monitoring of wind turbines,” 
IEEE Transactions on Industrial Electronics, vol. 
69, no. 2, pp. 1937-1947, 2021. 
https://doi.org/10.1109/TIE.2021.3057039 
[18] H. Wang, Z. Wu, K. Liu, J. Wei, and H. Hu, 
“Modeling and control strategies of a novel axial 
hybrid magnetic bearing for flywheel energy storage 
system,” IEEE/ASME Transactions on 
Mechatronics, vol. 27, no. 5, pp. 3819-3829, 2022. 
https://doi.org/10.1109/TMECH.2022.3145705 
[19] W. Al-Chaab, Z. A. Abduljabbar, E. W. Abood, V. 
O. Nyangaresi, H. M. Mohammed, and J. Ma, 
“Secure and low-complexity medical image 
exchange based on compressive sensing and lsb 
audio steganography,” Informatica, vol. 47, no. 6, 
pp. 65-74, 2023. 
https://doi.org/10.31449/inf.v47i6.4628 
[20] H. M. Chen, X. J. Cai, and L. L. Xu, “Approximate 
customized proximal point algorithms for separable 
convex optimization,” Journal of the Operations 

Balanced Generalised Tailored Approximation Point Algorithm for… Informatica 49 (2025) 175–190 189 
 
Research Society of China, vol. 11, no. 2, pp. 
383-408, 2023. 
https://doi.org/10.1007/s40305-022-00412-w 
[21] S. Xu, “A dual-primal balanced augmented 
Lagrangian method for linearly constrained convex 
programming,” Journal of Applied Mathematics and 
Computing, vol. 69, no. 1, pp. 1015-1035, 2022. 
https://doi.org/10.1007/s12190-022-01779-y 
[22] M. Wang, Q. Wang, J. Chanussot, and D. Hong, 
“L₀-l₁ hybrid total variation regularization and its 
applications on hyperspectral image mixed noise 
removal and compressed sensing,” IEEE 
Transactions on Geoscience and Remote Sensing, 
vol. 59, no. 9, pp. 7695-7710, 2021. 
https://doi.org/10.1109/TGRS.2021.3055516 
[23] S. Li, Z. Dai, M. Liu, and M. Cai, “Leak 
identification method of water supply pipeline based 
on compressed sensing and least squares twin 
support vector machine,” IEEE Sensors Journal, vol. 
23, no. 7, pp. 7115-7128, 2022. 
https://doi.org/10.1109/JSEN.2022.3211343 
[24] S. Hamada, K. Ichige, K. Kashiwagi, N. Arakawa, 
and R. Saito, “Accurate source-number estimation 
using denoising preprocessing and singular value 
decomposition,” IEICE Transactions on 
Communications, vol. 105, no. 6, pp. 766-774, 
2022. 
https://doi.org/10.1587/transcom.2021ISP0003 
[25] R. Chen, D. Pu, Y. Tong, and M. Wu, 
“Image‐denoising algorithm based on improved 
K‐singular value decomposition and atom 
optimization,” CAAI Transactions on Intelligence 
Technology, vol. 7, no. 1, pp. 117-127, 2022. 
https://doi.org/10.1049/cit2.12044 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

190 Informatica 49 (2025) 175–190 J. Guo