1 Introduction and Related Work
As the exponential growth of information generated from the website, the recommender systems have become more and more popular in both academia and industry. Recommender systems are a subclass of information filtering techniques that seek to predict the rating or preference that user would give to an item (movies, books, music, restaurant).
Although recommender systems have been widely used, such as Google, Amazon, Ebay, most of recommender systems suffer from several common drawbacks. First of all, data sparsity is the most serious problems for designing effective and efficient recommender systems. As indicated in [17, 23, 18], the density of data is less than in most of recommender systems. In other words, in typical useritem rating matrix, the number of nonzero ratings in this matrix is less than . This phenomenon is obvious and for example, customers in real life always purchase very few items in amazon compared to the whole items in amazon database. So most of collaborative filtering based recommender systems such as [1, 2, 3, 8, 10, 13, 12, 19, 21] cannot handle the users who have rated few items. What is more, traditional recommender systems ignore the social connection among users in both social based network and trust based network. It assumes that all the users are independently and identically distributed and only use the useritem matrix for recommendation task. However, in real life, we always turn to our trusted friend for suggestions for buying books, cloth, and watching movies, etc. So our preference can be easily affected by our friends. Therefore, simply ignoring social connection is not realistic when doing recommendation systems and considering the social connection information in recommender system can efficiently deal with the cold start users [25, 9].
In this paper, aiming at solving the above problems, we propose a matrix factorization framework with social network information as regularization term. The proposed model effectively handles the information from two resources, useritem matrix and user social/trust network. More specifically, the social network information is used for designing the social regularization term to constrain the matrix factorization objective function. The experimental analysis on one reallife dataset shows that our proposed model outperforms several stateoftheart algorithms.
Social based recommendation system has been studied in [15, 14, 11]. Furthermore, matrix factorization is widely used in the areas of social network analysis [6, 7, 5, 4], and name entity disambiguation [16, 24, 20, 22]. The rest of this paper is organized as follows: The problem definition and preliminary background are presented in section 2. Basic matrix factorization framework is introduced in section 3. The proposed model is described in section 4. Our experiment is reported in section 5. Finally I conclude the paper.
2 Problem Definition and Preliminary Background
In recommender system, we have a set of users , and a set of items . The useritem rating matrix is represented as and is denoted as the rating of user on item . Typically the domain of rating is the real number ranging from to .
For social network, given a directed network , wehre every user , if user is a directed neighbor of user , in other words and is a set of directed neighbors of user . For example, in trust network, if user likes the item review that user writes, then there is a outgoing edge from to , but at the same time doesn’t necessarily link to . Without losing generality, we denote the adjacent matrix of network as . In this project, the edge weight in matrix is a binary value . For instance, if user has an outgoing edge to user , then , otherwise .
Assume that a given user , an item , the recommender task in this project is to predict the missing value given matrix and . In this work I utilize the matrix factorization based model to learn the latent factors of users and items and predict the missing rating value in matrix .
3 Basic Matrix Factorization Model
In recommender system, an efficient approach for predicting missing values in useritem matrix is to employ matrix factorization method. Basically matrix factorization method is to factorize the useritem matrix and use the low ranked user and item factor matrices for missing rating value prediction.
Suppose given useritem matrix , the basic matrix factorization model is as follows:
(1) 
where is the useritem rating matrix, is the user factor matrix and is the item factor matrix, and is the number of latent space dimension for users and items in useritem matrix .
The approximation of user ’s rating on item , which is denoted by , is defined as , where is the user factor for user , is the item factor for item . And the objective function of matrix factorization model is as follows:
(2) 
where . We are going to minimize the so as to solve both and . In order to solve the optimization problem in equation 2, a gradient descent approach can be used to obtain a local minimum.
4 Social Connection Based Matrix Factorization Model
Traditional recommender systems, like collaborative filtering, only utilize the useritem rating matrix information for recommendation but ignore the social connections among users. Due to the factor that the online social network is becoming more and more popular, incorporating social network information in recommender system becomes more and more important. Particularly by embedding the social connection information in recommender system, it can efficiently solve the cold start problem. In this section, I will incorporate the user social connection information as regularization term into basic matrix factorization framework and use the gradient descent approach to solve the proposed model.
4.1 Social Regularization based Model
We formulate the objective function as the following minimization problem:
(3) 
In the formulation above, is the user factor for user (the uth column in ), is the item factor for item (the ith column in ). is the similarity function to indicate the similarity between user and user . Also we use to denote user ’s outlink friends and use the notation to represent user ’s inlink friends. In trust network, like Epinion, doesn’t necessarily equal to since trust network is directed network typically.
A local minimum of the objective function given by the objective function can be found by performing the gradient descent with respect to latent vectors
and . The derivation procedure for user and item latent factors are as follows:One of the advantages to employ this model is that the proposed model can capture the trust propagation of different users. For instance, if user has a outgoing friend , and also has a outgoing friend , but at the same time and are not friend based on the social network connection. Using this model, it can indirectly minimize the feature distance between and .
4.2 Similarity Measurement
in the proposed model, we need to quantify the similarity between users in social network. Given the knowledge of useritem matrix, we can model the user user similarity based on purchased items and corresponding ratings. In order to achieve this goal, Pearson Correlation Coefficient (PCC) and Vector Space Similarity (VSS) are proposed to define similarity between different users. The equations are as follows:
where represents the average rate of user and belongs to the subset of items which user and user both rated. is the rate user on item .
From the above definitions, we can see that and . The larger values for both VSS and PCC, the more similar between different users. Also in order to constrain the range of measurement into , I use a simple mapping function to bound its similarity range from to .
5 Experiments and Results
5.1 Epinions Trust Dataset
The dataset we employ for this work is Epinions^{1}^{1}1http://www.epinions.com. In Epinions, every user can read the reviews about a variety of items and also users can write a review for particular items. For the social network viewpoint, each member in Epinions has a trust list of other members to indicate if I trust your review or not. From my study, the dataset contains different users and different items. And the total number of rating is .
In order to verify the correctness of similarity measurement in my model, the first study is to validate my assumption that in trust network such as Epinions, social friends have similar tastes. We utilize Vector Space Similarity (VSS) as the metric to evaluate the similarity between user and user .
The analysis we conduct is to understand how does the social friends similarity compare with random peer similarity? The detailed analysis is as follows:
1. For each user in trust network, we calculate the average social friends’ similarity as follows:
(4) 
where is the trusted list of user which also means user ’s outgoing friends.
2. For each user , we also calculate the average random peer similarity as follows:
(5) 
where represents the random peer list of user , which has same size with and .
The motivation of carrying out this experiment is that we want to confirm that the social peer relation in trust network has strong positive correlation with user interest similarity. In order to reduce the noise of dataset, we only consider the users who has more than five outgoing social peers in Epinions. For those users who has less than five outgoing social peers, we simply ignore them in the evaluation.
In order to quantify the correlation between social relations and user interest similarity, we would like to measure the proportion of users whose social similarities are greater than their random similarities followed by the equation . In the Epinions dataset, based on my study, there are users whose social similarities are greater than their random similarities. So we can safely make conclusion that friends have similar taste in Epinions.
We also test PCC based similarity measurement. We observe the similar result so for simplicity, therefore we only report the result using VSS similarity function.
5.2 Evaluation Metrics
The quality of the results is measured by the the Mean Absolute Error (MAE) and the Root Mean Square Error (RMSE). The MAE and RMSE are defined as follows:
where denotes the rating user gave to item , is represents the rating user gave to item which is predicted by my proposed method, and is the total number of nonzero ratings in the test dataset.
From the definition, we can see that the smaller MAE and RMSE, the better performance of our proposed method.
5.3 Comparison Methods
Several of stateofart methods are used for comparison with the proposed approach in this work. The detailed descriptions of comparison methods are explained below:
UserMean: this method uses the mean values of every user to predict the missing values
ItemMean: the method utilizes the mean value of every item to predict the missing values.
BasicMF: Traditional Matrix Factorization and the method uses classic matrix factorization formulation without incorporating user trust network information.
Proposed Method: This method uses classic matrix factorization formulation with trust network information as regularization term which is the method proposed in this work.
The results we obtain are shown in the tables 1 and 2. In the whole process, we use PCC for user user similarity measurement. During the experiment, parameters is set to a trivial value 3.0 and in our model is set to 0.01. For the Epinions dataset, we use 90% and 80% of the original data as the training data settings and the remaining as the test set. The random selection is carried out times independently, and I report the average MAE and RMSE values in table 1 and table 2.
As we can see our proposed method performs much better than baseline methods on Epinion dataset. For instance, traditional matrix factorization method achieves and on MAE and RMSE under training setting respectively, whereas our proposed method achieves and
on MAE and RMSE respectively. And the performance of UserMean and ItemMean methods are even worse than basic matrix factorization method. Crossvalidation ttest shows that our proposed method is significantly better (pvalue 0.0042 for MAE and 0.0036 for RMSE) than traditional matrix factorization. Similar results are obtained compared with other two baseline methods and also
training setting.Method  MAE  RMSE 

UserMean  0.9415  1.2361 
ItemMean  1.2236  1.7985 
MF  0.8641  1.1071 
Proposed Method  0.8324  1.0756 
Method  MAE  RMSE 

UserMean  0.9545  1.2489 
ItemMean  1.2663  1.8575 
MF  0.8722  1.1254 
Proposed Method  0.8462  1.0964 
5.4 Study of Parameter Sensitivity
In our proposed method, the parameter plays the important role for how much our method should incorporate the social network information. Large value of indicates more impact of social connection information in the propsoed model and smaller value of indicates less impact of social connection information and zero value makes the proposed model close to basic matrix factorizaton method. In this experiment, we can see how the performance of model changes as we vary the value of parameter. The result of this experiment is shown in Figure 1(a) and 1(b). As we can see that the performance in terms of both MAE and RMSE degrades for the choice of .
5.5 Impact of Similarity Function
The similarity function measures how similar of user and in the social network. In the proposed method, two similarity functions PCC and VSS are used and in this section, in order to measure how similarity functions contribute the proposed model, besides PCC and VSS, two other similarity frameworks will be used for comparison:
1. Setting all the similarities between users as 1.
2. Assigning a random similarity between and to any pair of friendship.
Similarity  MAE  RMSE 

Sim=1  0.8415  1.1061 
Sim=Random  0.8636  1.1085 
Sim=VSS  0.8351  1.0789 
Sim=PCC  0.8324  1.0756 
Similarity  MAE  RMSE 

Sim=1  0.8515  1.1121 
Sim=Random  0.8536  1.1145 
Sim=VSS  0.8491  1.0989 
Sim=PCC  0.8462  1.0964 
As we can see from the result that the performance of unweighted similarity and random similarity measurements are worse than VSS and PCC. Among these four similarity functions, PCC performs best. So this observation demonstrates that the similarity function plays important role in my proposed model.
5.6 Performance on cold start users
In this section we study how performance for our proposed model for the cold start users in recommender systems. Cold start users are users who have rated or purchased few items and traditional collaborative filtering methods are not able to deal with this kind of users since cold start users don’t have enough rating/purchase history. In this experiment, we consider users who have rated less than 5 items as cold start users. In Epinions, more than 55% of users are cold start users. So proposing effective recommender systems for coping with cold start users are becoming more and more important. For conducting this experiment, we only test on the cold start users and put one of their rated items into test set and other rated items into training set. The result is shown in table 5.
Method  MAE  RMSE 

UserMean  1.071  1.502 
ItemMean  1.082  1.582 
MF  0.982  1.261 
Proposed Method  0.892  1.121 
As we can see from the result, our proposed method could deal with the cold start users and the performance is much better than other three baseline methods.
6 Conclusion
In this paper, we leverage the social network information in the matrix factorization framework to obtain a better recommender system, especially for the cold start users. The experimental analysis on the reallife dataset shows that our proposed approach performs significantly better than several stateoftheart methods.
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