Informatica 42 (2018) 265–271 265 
Probability Matrix Decomposition Based Collaborative Filtering 
Recommendation Algorithm 
Yili Tan and Huijuan Zhao 
Department of Statistics, College of Science, North China University of Science and Technology, Hebei, 063210, China 
E-mail: tangyili_tyl@163.com 
 
Yourong Wang 
Department of Basic, Tangshan College, Hebei, 063000, China 
 
Min Qiu 
Public Mathematics Department, Hubei University of Automotive Technology, Hubei, 442002, China 
 
Technical Paper 
 
Keywords: probability matrix decomposition, collaborative filtering, recommendation algorithm  
 
Received: April 8, 2018 
With the development of the society, the increased amount of information has extensively appeared on the 
Internet. It includes almost all the content we need. But information overload makes people unable to 
correctly find the information they need. Collaborative filtering recommendation algorithm can 
recommend items for users according to their demands. But traditional recommendation algorithm which 
has defects such as data sparsity needs to be improved. In this study, the collaborative filtering 
recommendation algorithm was analyzed, an improved collaborative filtering recommendation algorithm 
based on the probability matrix decomposition was put forward, and the feasibility of the algorithm was 
verified. Moreover the traditional algorithms including user based collaborative filtering algorithm, item 
based collaborative filtering algorithm, singular value decomposition based collaborative filtering 
algorithm and basic matrix based collaborative filtering algorithm were tested. The test results 
demonstrated that the proposed algorithm had a higher accuracy compared to the traditional algorithms, 
and its mean absolute error and root-mean-square error were significantly smaller than those of the 
traditional algorithms. Therefore it can be applied in the daily life. 
Povzetek: V sestavku je predstavljena dekompozicija verjetnostne matrike s priporočilnim algoritmom na 
osnovi skupinskega filtriranja.
1 Introduction
He increased amount of information which appeared due 
to the development of Internet technology increases the 
difficulty of finding the target information. Therefore, 
many recommendation algorithms were proposed. Such 
recommendation algorithms can filter information 
according to the personal preference; hence they have 
been universally applied in fields such as web browsing, 
film recommendation and e-commerce [1]. Li [2] 
analyzed the sales records in the current tea leaves sales 
system by combining Hadoop distributed system with the 
traditional collaborative filtering algorithm to obtain the 
recommendation rules which could satisfy the preference 
of customer and help users find the tea leaves they needed. 
Yu et al. [3] proposed the weighed cloud model 
attributes based service cluster algorithm and calculated 
the user score similarity using the weighed Pearson 
correlation coefficient method of service cluster algorithm 
and the user service selection index weight. They found 
that the algorithm could accurately calculate service 
recommendation credibility, satisfying the demands of 
users on service credibility, and enhance the success rate 
of the user service selection. The collaborative filtering 
algorithm has high degree of individualization and 
automation, but it exhibits a few problems such as sparsity 
and system extensibility. Therefore, in this study a 
probability matrix decomposition based collaborative 
filtering algorithm was put forward to correct up the 
defects of the traditional collaborative filtering algorithm 
and performed simulation experiments. The experimental 
results suggested that the mean absolute error (MAE), the 
root-mean-square error (RMSE) and the accuracy of the 
algorithm could reach the expected levels. This work 
provides a reference for the application of probability 
matrix decomposition based collaborative filtering 
recommendation algorithm in the searching of Internet 
information. 
2 Collaborative filtering 
recommendation algorithm 
2.1 Collaborative filtering algorithms 
based on different elements 
 

266 Informatica 42 (2018) 265–271 Y. Tan et al. 
2.1.1 Collaborative filtering algorithm based 
on users 
User based collaborative filtering algorithm focuses on 
users. It recommends using user-item score matrix. It 
firstly searches for users which are similar to the target 
users and then recommends the selection of the searched 
users to the target users. The algorithm has two functions, 
i.e. one for calculating the similarity between adjacent 
users to establish matrix and one for recommending the 
target users using algorithm evaluation method. 
2.1.2 Collaborative filtering algorithm based 
on items 
Item based collaborative filtering algorithm can provide 
recommendations to users based on evaluation data after 
establishing user-item evaluation data model. In details, it 
calculates the similarity between different items to 
determine the preference of target users and then 
recommends similar items to target users. The algorithm 
has functions for calculating the similarity between items, 
establishing similarity matrix and recommending target 
users by scoring similar items using algorithm evaluation 
method. 
2.2 Collaborative recommendation 
algorithm based on probability matrix 
decomposition 
The probability matrix decomposition can reflect the 
information of users and items to low-dimensional 
characteristic space in the aspect of probability and then 
analyze the concerns of uses about items using the linear 
combination of low-dimensional vectors [4].  
Item score matrix could be expressed as 
ij
F

;  a 
matrix 
ai
M

 whose mean value was 0 and variance was 
2
M

  and a random number matrix 
aj
N

 whose mean 
value and variance were 0 and 
2
N

 respectively were 
produced by MATLAB [5], in which a refers to the 
dimension of decomposition, 
ai
M

 refers to a-
dimensional characteristic square matrix of users, and 
aj
N

 refers to the a-dimensional characteristic square 
matrix of item. Vector 
m
M
 and 
n
N
 were the 
corresponding potential characteristic vectors. In general, 
T
i j a i a j
F M N
  

. The matrix 
T
a i a j i j
M N F
  

 was 
obtained through the learning of machine training. 
Suppose the mean value of the error between actual 
score 
mn
F
 and predicted score 
mn F

 as 0 and the variable 
of 
mn
F
 and 
mn F

 as Gaussian distribution of 
2
F

, then 
the probability distribution is 
2
( |0, )
T
mn m n F
q F M N  
. 
2
( | , )
T
mn m n F
q F M N 
 was obtained through translation. 
Then the condition of the score matrix F was: 
22
11
( | , , ) ( | , )
j i
T
F mn m n F mn
mn
q F M N K F M N I 

 


 
, (1) 
Where 
mn
I
 stands for indicator function, 
1
mn
I 
 
means user m has scored item n, and 
0
mn
I 
 means user m 
has not scored item n. 
As M and N could not include each other, the mean 
value of M and N was 0, and 
2
M

 and 
2
N

 had Gaussian 
distribution, then 
22
1
( | ) ( |0, )
i
M m M
m
Q M K M I 



,  (2) 
22
1
( | ) ( |0, )
i
N n N
n
Q N K N I 



,   (3) 
Where Q stands for probability. 
The joint probability distribution of M and N can be 
obtained from equation (1), (2) and (3). 
 
2 2 2
2 2 2
1 1 1 1
( , | , , , )
( | , ) ( |0, ) ( |0, )
F M N
jj ii
T
mn m F mn m M n N
m n m n
q M N F
K F M N I K M I K N I
  
  
   
   

   
 
(4) 
The logarithm of the probability distribution of M and 
N was calculated: 
2 2 2
2
2 2 2
1 1 1 1
ln ( , | , , , )
1 1 1
()
2 2 2
F M N
jj ii
T T T
mn mn m n m n n n
m n m n F M N
q M N F
I F M N M N u N
  
  
   
   
   
.
 (5) 
The maximum solution of equation (5) was replaced 
with the minimum solution of error function containing 
normalization parameters [6]: 
22
2
min
1 1 1 1
1
( ) ( )
2 2 2
jj ii
T
mn mn m n m n
m n m n
MN
L I F M N M N

   
   
   
,    
(6) 
Where 
2
2
F
M
M




 and 
2
2
F
N
N




. As 
22
MN
 
, then target 
function was: 
22
2
min
1 1 1 1
1
( ) ( )
22
jj ii
T
mn mn m n m n
m n m n
L I F M N M N

   
   
   
 (7) 
The relationship between regularization parameter 

 
and 
2 2 2
,,
F M N
  
 can be obtained from the equation (7). 
The algorithm calculated function using stochastic 
gradient descent method [7]. It could obtain the decline 
direction of numerical values using derivatives and then 
calculate variables constantly on this direction until the 
minimal point was obtained. 
The solution of the point suggested that the updating 
formulas of 
,
mn
MN
 were transformed to the following 
formulas in each iteration: 
T
mn m n
l F M N 
,   (8) 
()
m m n m
M M l N M       
,        (9) 
()
n n m n
N N l M N       
,                   (10) 

Probability Matrix Decomposition Based… Informatica 42 (2018) 265–271 267 
Where 

 stands for the learning rate of the stochastic 
gradient descent.  
3 Experiment 
3.1 Experimental data 
A 100k data set originated from the movies provided by 
GroupLens project team from University of Minnesota 
were used in the experiment, denoted as data set A.  
Data set A included 100,000 scores for 1,682 movie 
items given by 943 users. Each user scored 20 movie items 
at least. The score was an integer between 0 and 5. The 
more the user liked the movie the higher was the score. 
The sparseness of the data set A suggested the percentage 
of the movie items which were not scored by the users, i.e. 
1-100,000/(943×1682) ≈ 0.937. 
The 100,000 scores in data set A were randomly 
divided into two disjoint sets, the training and the testing 
set. The training set which included 80% of the data was 
expressed as S1, while the testing set which included 20% 
of the data was expressed as S2. The data set A was 
divided 10 times to perform cross validation on the 
algorithm. 
To enhance the recommendation efficiency of the 
algorithm, batch processing module was added. The 
100000 scores were divided into 10 batches. 10000 scores 
were processed every time.  This way, the computational 
quantity of the system and the convergence instability of 
the model produced in calculation could be reduced. 
The collaborative filtering recommendation algorithm 
based on probability matrix decomposition performed as 
follows. 
Input: training set and testing set 
Output: Predicted score and square root error 
Data such as regularization parameter were set 
The number of movies and users were set. 
If the iteration epoch < max epoch, then the 100000 
scores were divided into 10 groups, 10,000 in each group, 
for separate processing. 
If the patch processing was lower than 10, then the 
loss function q was calculated, and then matrix calculation 
was performed. 
End 
The predicted scores in the testing set were revised to 
positive integers through rounding off, and then square 
root error was calculated. 
End 
3.2 Scoring criteria 
3.2.1 MAE 
The MAE measure included  the calculation  of the 
absolute and average values of the difference between a 
predicted score and a real score [8]; hence it could be used 
for detecting the average difference between a predicted 
score and a real score. The smaller the value of MAE was, 
the more accurate the algorithm was. 
1
ek ek
MAE e M and k N d x
c
   

, (11) 
where 
ek
d
 stands for the predicted score of the user e 
on item k, 
ek
x
 stands for the real score of the user e on the 
item k, set M and N stand for the sets of users and items 
in the testing set, and c stands for the number of 
ek
d
 or 
ek
x
. 
3.2.2 Root-mean-square error 
Root-mean-square error refers to the average value of 
quadratic sum of the error between the two scores. The 
smaller the root-mean-square error was, the more accurate 
the prediction was [9].  
2
1
()
ek ek
RMSE e M and k N d x
c
   

, (12) 
where 
ek
d
 stands for the predicted score of the user e 
on item k, 
ek
x stands for the real score of user the e on 
item k, set M and N stand for the sets of users and items 
in the testing set, and c stands for the number of 
ek
d
 or 
ek
x . 
3.2.2 Accuracy 
Accuracy could be expressed as: 
X
Accuracy
R

,    (13) 
Where
  |
ek ek ek
X d d x 
, i.e. set X was the set of the 
predicted scores which were equal to the real scores in the 
testing set, 
ek
dD 
 (D was the set of the predicted scores), 
and 
ek
xR 
 (R was the set of the real scores). 
Both, the corrected probability of an item and the 
prediction accuracy, could be recommended to users. 
3.3 Design of experiment 
The specific content of the experiment was as follows. 
To analyze the application performance of probability 
matrix decomposition based collaborative filtering 
algorithm in the experimental aspect, the movie evaluation 
mentioned in the preceding text was taken as the data set, 
and the user based collaborative filtering algorithm, the 
item based collaborative filtering algorithm and the 
probability matrix decomposition based collaborative 
filtering algorithm were compared. To better analyze the 
application performance of the probability matrix 
decomposition based collaborative filtering algorithm, the 
other two algorithms, i.e. the basic matrix based 
collaborative filtering algorithm and the singular value 
decomposition based collaborative filtering algorithm, 
were also tested. The parameter setting of the algorithms 
is shown in Table 1. 
The user based collaborative filtering algorithm and 
the item based collaborative filtering algorithm were 
tested six times. The algorithm itself corresponds to the six 
characteristic factor numbers (dimension k) of the 
probability matrix decomposition collaborative filtering 
algorithm and the singular value decomposition based 
collaborative filtering algorithm. The basic matrix 

268 Informatica 42 (2018) 265–271 Y. Tan et al. 
collaborative filtering algorithm and the singular value 
decomposition based collaborative filtering algorithm 
used in the performance comparison were developed by 
referring to the relevant literature and revised according to 
the data which needed to be detected. The characteristic 
factor number of singular value decomposition based 
collaborative filtering algorithm was fixed, 6. The setting 
of characteristic factor number of the basic matrix 
collaborative filtering algorithm was the same as the 
probability matrix decomposition collaborative filtering 
algorithm, [10, 60], and the unit stepping was set to10. 
4 Experimental results and analysis 
4.1 MAE and RMSE 
It could be noted from the Figure 1 and 2 that the 
predictive recommendation performance of the user based 
collaborative filtering algorithm was the poorest, and the 
prediction performance of the singular value 
decomposition based collaborative filtering algorithm was 
not affected  by characteristic factor number, but was 
poorer than that of the user based collaborative filtering 
algorithm. The performance of the user based 
collaborative filtering algorithm was far worse than that of 
the probability matrix decomposition collaborative 
filtering algorithm and the basic matrix based 
collaborative filtering algorithm. The performance of the 
probability matrix decomposition collaborative filtering 
algorithm and the basic matrix based collaborative 
filtering algorithm was similar, but the probability matrix 
decomposition collaborative filtering algorithm was still 
superior. The reason why there was a significant 
difference between the performance of the user based 
collaborative filtering algorithm and the item based 
collaborative filtering algorithm is the score given by a  
scorer was probably affected by the view of other scorers 
who had the same interests. The  reason why the 
performance of  the singular value decomposition based 
collaborative filtering algorithm in the prediction and 
recommendation was significantly poorer than that of the 
basic matrix collaborative filtering algorithm and the 
probability matrix decomposition collaborative filtering 
algorithm was the fact that the singular value 
decomposition based collaborative filtering algorithm was 
actually an improved version of the item based 
collaborative filtering algorithm and therefore had similar 
shortcomings as the original algorithm, i.e., the matrix 
obtained after the preprocessing had data distortion 
compared to the original matrix, which could have 
affected the accuracy and the similarity of the score 
prediction. But the performance of the singular  value 
decomposition based collaborative filtering algorithm was 
better than of the  item based collaborative filtering 
algorithm, indicating the  improved accuracy of the 
singular  value decomposition based collaborative 
filtering algorithm. 
It could be noted from the Figure 1 that the values of 
the root-mean-square error (RMSE) corresponding to the 
basic matrix based collaborative filtering algorithm and 
the probability matrix decomposition collaborative 
filtering algorithm gradually decreased with the increase 
of the characteristic factor number; the larger the 
characteristic factor number, the smaller the decrease 
amplitude. When the characteristic factor number was 50, 
the value of RMSE was the minimum, and the prediction 
accuracy was the highest; when the characteristic factor 
number was between 10 and 20, the decrease amplitude of 
RMSE of  the basic matrix based collaborative filtering 
algorithm and the probability matrix decomposition 
collaborative filtering algorithm was large, around  1.14% 
and 0.700% respectively. It was found that the values of 
the RMSE of the two algorithms were lowly sensitive to 
the characteristic factor number, especially of the 
probability matrix decomposition collaborative filtering 
algorithm. When the characteristic factor number was 
larger than 40,  the fluctuation of the RMSE was quite 
small. 
Similar to Figure 1, the MAE corresponding to the 
basic matrix based collaborative filtering algorithm and 
Algorithm User based 
collaborative 
filtering 
algorithm 
Item based 
collaborative 
filtering 
algorithm 
Probability 
matrix 
decomposition 
collaborative 
filtering 
algorithm 
Basic matrix 
collaborative 
filtering 
algorithm 
Singular value 
decomposition 
based 
collaborative 
filtering 
algorithm 
Neighbourhood 
or model 
Neighbourhood Neighbourhood Probability 
matrix 
decomposition 
collaborative 
filtering model 
Basic matrix 
collaborative 
filtering model 
Singular value 
decomposition 
based 
collaborative 
filtering model 
Characteristic 
factor number 
(dimension k) 
\ \ [10,60] [10,60] 6 
Learning rate \ \ 0.02 0.02 \ 
Table 1: The parameter setting of the five algorithms. 
 

Probability Matrix Decomposition Based… Informatica 42 (2018) 265–271 269 
the probability matrix decomposition based  collaborative 
filtering algorithm  also decreased with the increase of the 
characteristic factor number and reached the minimum 
values, 0.675 and 0.666 respectively, when the 
characteristic factor number was 60. Moreover it was 
noted that when MAE was taken as the evaluation index, 
the curves of the basic matrix collaborative filtering 
algorithm and the probability matrix decomposition based 
collaborative filtering algorithm nearly coincided, and the 
prediction performance was also close. Moreover, since 
the RMSE is more sensitive to the measurement error, the 
probability matrix decomposition based  collaborative 
filtering algorithm had an advantage over the basic matrix 
collaborative filtering algorithm due to the addition of  the 
regularization term. 
 
Figure 1: Variation of the RMSE of the five algorithms with the increase of the characteristic factor number. 
 
Figure 2: Variation of the MAE of the five algorithms with the increase of the characteristic factor number. 
4.2 Accuracy 
It was found from the comparison of the RMSE and the 
MAE between the five algorithms that the prediction 
performance of the CF-User, the item based collaborative 
filtering algorithm and the singular value decomposition 
based collaborative filtering algorithm was significantly 
different from the basic matrix based collaborative 
filtering algorithm and the probability matrix 
decomposition based collaborative filtering algorithm. 
Therefore, only the accuracy of  the basic matrix based 
collaborative filtering algorithm and the probability 
matrix decomposition based collaborative filtering 
algorithm were considered. The results are shown in Table 
2. 
It could be noted from Figure 3 that the tendency of 
the accuracy of the basic matrix collaborative filtering 
algorithm and the probability matrix decomposition based 
collaborative filtering algorithm was opposite to the 
tendencies of the MAE and the RMSE. When the 
characteristic factor number was small, the accuracy was 
low; the accuracy increased first and then tended to be 
stable with the increase of the characteristic factor number 
Characteristic 
factor number 
PMF BMF 
10 0.40321 0.34666 
20 0.41321 0.36022 
30 0.41403 0.36142 
40 0.41300 0.36134 
50 0.41299 0.36132 
60 0.41298 0.36132 
Figure 3: The accuracy of the basic matrix based 
collaborative filtering algorithm and the probability matrix 
decomposition based collaborative filtering algorithm 
under different characteristic factor numbers. 
 

270 Informatica 42 (2018) 265–271 Y. Tan et al.  
 
and nearly had no fluctuation when the characteristic 
factor number was larger than 30. It was because the 
effective information increased with the increase of the 
characteristic factor number. The accuracy of the 
probability matrix decomposition based collaborative 
filtering algorithm was much higher than that of the basic 
matrix based collaborative filtering algorithm. Therefore 
the proposed algorithm could improve the searching speed 
and preciseness. 
 
Figure 4: Comparison of the accuracy of the probability 
matrix decomposition based collaborative filtering 
algorithm (PMF) and the basic matrix based 
collaborative filtering algorithm (BMF). 
The recommendation system which can filter diversified 
data is an effective filtering approach [10]. It can 
recommend individual information to users according to 
users’ requirements. Therefore it can be convenient for 
information collection and has been extensively applied 
on the Internet. Wei et al. [11] put forward project 
category similarity and interestingness measure based 
collaborative filtering recommendation algorithm which 
could recommend information to users through 
calculating project categories and interestingness and had 
high prediction preciseness. In a study of Chen et al. [12], 
a mixed recommendation system was put forward to 
recommend users with learning projects searching. In the 
test, the algorithm effectively collected information, 
suggesting a favorable performance. 
5 Conclusion 
In conclusion, the probability matrix decomposition based 
collaborative recommendation algorithm was put forward 
in this study, and then it was developed for data searching 
recommendation. Afterwards the MAE, the RMSE and the 
accuracy of the algorithm were tested. Moreover the MAE 
values, the RMSE values and the accuracy of the CF-User, 
the item based collaborative filtering algorithm, the 
singular value decomposition based collaborative filtering 
algorithm and the basic matrix based collaborative 
filtering algorithm were compared. The testing results 
suggested that the improved collaborative 
recommendation algorithm had the highest preciseness 
and accuracy, and the preciseness and the accuracy 
became the largest and stable when the characteristic 
factor number was more than 40. Therefore it could be 
applied in a computer searching system. This work 
provides a reference for the progress of the collaborative 
recommendation algorithm. 
6 Acknowledgement 
This study was supported by the Fund of Natural Science 
of Hebei (A2015209040). 
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