Research Article Open Access
Exploiting Multi-Sources Information for Location Recommendation
Henghong Yang1, Wenguo Wei1*, Huimin Zhao1 and Guiyuan Xie1
1School of Electronic and Information, Guangdong Polytechnic Normal University, Guangzhou, China
This work was supported in part by the Science and Technology Project of Guangzhou City (201802020019, 01806040010)
*Corresponding author:Wenguo Wei, School of Electronic and Information, Guangdong Polytechnic Normal University, Guangzhou, China; E-mail: @
Received: July 22, 2019; Accepted: August 19, 2019; Published: August 21, 2019
Citation: Wenguo W, Yang H, Zhao H, Xie G (2019) Exploiting Multi-Sources Information for Location Recommendation. J Comp Sci Appl Inform Technol. 4(1): 1-8. DOI: 10.15226/2474-9257/4/1/00143
AbstractTop
In location-based social networks (LBSNs), user preference, social influence and geographical influence are three major factors that affect users’ check-in behaviors. However, current studies tend to ignore the influence of the features from point-of-interest (POIs) to similar user groups. In this paper, we proposed a new approach named Base Item Attribute - Weighted Cosine Similarity (BIA-WCOS) to model social relationship of users, which consider the influence of location’s popularity and check-in frequency on user similarity. The proposed Geographical - Base Location Attribute Social Relationship (G-BLAS) framework is to exploit personalized social and geographical influences on location recommendation. We conduct a comprehensive performance evaluation of our approach using two real datasets collected from Foursquare and Gowalla. Experimental results show the effectiveness and advantages of our proposed approach.

Keywords: Matrix Factorization; Geographical Influence; Item Attribute; User Similarity; Location Recommendation;
IntroductionTop
With the advancement of mobile devices and the development of the GPS technique, we have witnessed the increasing popularity of location-based social networks (LBSNs) in recent years. In an LBSN, users can establish social links with their friends and share their experiences of visiting some locations they feel interesting, also known as POIs via “check-in”, which can reflect their preferences. Further, users can share their feeling via commenting on the locations they visit. For example, Foursquare, one of the most popular LBSNs that have over 50 million users, 105 million places and 12 billion check-ins. Facing the huge amount of data, the personalized location recommendation systems can help users to explore new locations that they are interested in.

The most widely used approach to model user preference for location recommendation is collaborative filtering (CF) technique, where user’s check-in data is modeled as user-location matrix with each entry representing the frequency of a user visiting a location. The personalized location recommendation system aims at predicting a user’s preference on unvisited locations according to user check-in data and other contextual information, such as geographical influences [1-6], social relationship [7-10] and reviews [11, 12]. The features of POIs also have a certain influence on the social relationship. For example, compared to a popular location, users’ visiting the unpopular locations often reflect the similarities between them. How to model these features more effectively becomes a research focus in our paper.

In this paper, a novel approach is proposed for determining weighted user-similarity, in which we explore the influence of POI’s popularity and visit-frequency on social relationship according to user’s historical check-in data, and then we propose a location recommendation framework by fusing multiple contextual information such as user preference, social influence, the geographical influence to social relationship and the personalized geographical influence of locations. The results show that the approach we proposed has significantly improved the recommendation accuracy.

The remaining section of this paper is structured as follows. Section 2 highlights related work and background introduction. Section 3 presents in detail our proposed approach. In Section 4, we reported and analyzed the experimental results. Finally, some concluding remarks are drawn in Section 5.
Related WorkTop
In this section, we reviewed the related work and background knowledge of our study. These are organized in three parts, i.e. geographical recommendation, social recommendation and other contextual recommendation, as detailed below.
Geographical recommendation
Geographical information is the most important factor to POIs. Characterizing user’s preference based on geographical features of locations is a widely used method in the POI recommendation system. [1] Considered that it is important to take implicit feedback characteristics of user mobility data into account as well as the location’s spatial information, thus that a scalable and flexible framework GeoMF++ is proposed to recommend locations. [2, 3] found that the geographical influence of visiting the POI should be personalized for users, thus it is unreasonable to use power-law distribution to model user’s check-in behavior. In order to prevent the fitting error caused by specific distributions, Zhang et al. proposed to use kernel density estimation method to estimate the distance distribution between any two POIs and to measure the geographical performance of the user check-in data.

In addition to the above-mentioned approach, there are also other methods proposed, such as the GeoMF model proposed in [13] to fuse geographical location information into weighted matrix factorization, and a rank-geoFM model proposed in [14] to sort the POIs based on pair wise sorting. These models can effectively model the characteristics of geographical information and can improve the accuracy of location recommendation.
Social recommendation
In addition to geographical influence, social network information also plays an important role in the location-based recommendation system. In real life, the user’s visit to POI is largely influenced by the friends around him. For example, friends may go shopping together, thus we can assume that there are similar preferences among friends, and incorporating social network information into recommendation system has a corresponding gain effect.

Modeled and analyzed [7] the trajectory of user check-in location based on the HGSM (Hierarchical-graph-based Similarity Measurement) hierarchical similarity algorithm, which measured the similarity of user’s behavior and ranked the highest ranked users as close neighbors recommended for the target user. [8] Used the two-hop random walk algorithm to exploit explicit social relationships and implicit social relationships between users based on the traditional matrix factorization model.
Other contextual information
Other context information of the user’s visit to POI may also has a corresponding influence on the recommendation effect, such as the time of user’s visit and the feature of picture published by the previous visitors. [15] Used the topic model (LDA) to exploit the topic attributes of POI based on its tag and to determine the user’s preference according to these attributes. [16] Expanded the state-of-the art Rank-GeoFM POI recommender algorithm[[14] to include some features of weather-related. [17] Proposed a location-based recommendation algorithm that fuses temporal information. The algorithm models the user’s check-in behavior as a fourth-order tensor containing time periods, and combines the geographical influence to recommend locations. [18] Believed that it is beneficial to analyze user preference by the pictures that they shared. Therefore, they proposed CNN based technique to extract feature vectors of pictures based on matrix factorization to improve the recommending accuracy.

In summary, the existing approaches have achieved certain results on predicting a user’s preference to a location, but there are still many problems since the characteristics of locations cannot be fully utilized. In this paper, we explored the influence of the location’s characteristics to social relationship. Further, we integrate the user preference, geographical influence and the social friendship of users into one framework for unified location recommendation.
The proposed approach
In this section, we first introduce the Matrix Factorization technique with different feedback data, and then introduce the model of geographical influence and social influence that we used in this paper. Finally, we present the unified framework with all these approaches integrated together.
The model of Matrix Factorization (MF)
There are currently two types of user-history-behavior data for recommendation systems, i.e. explicit feedback data and implicit feedback data. The explicit feedback can directly represent the user’s preferences (such as the rating scores), while implicit feedback means that the feedback information does not reflect the user’s preference directly (such as click and browsing). Here, we first introduce the matrix factorization that is suitable for explicit feedback data [19], and then introduce the matrix factorization that is suitable for implicit feedback data [20, 21] as used in this paper.

Collaborative filtering (CF) is one of the most widely used approaches in location recommendation, which describes user preferences on locations [22, 23]. Given m users (u u 1 , u 2 ,..... u m ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaabw hacqGHiiIZcaqG1bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaabwha daWgaaWcbaGaaGOmaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaai Olaiaac6cacaqG1bWaaSbaaSqaaiaab2gaaeqaaOGaaiykaaaa@4495@ and n locations ( (l l 1 , l 2 ,..... l n ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaabaaa aaaaaapeGaaeiBa8aacqGHiiIZpeGaaeiBa8aadaWgaaWcbaGaaGym aaqabaGccaGGSaWdbiaabYgapaWaaSbaaSqaaiaaikdaaeqaaOGaai ilaiaac6cacaGGUaGaaiOlaiaac6cacaGGUaWdbiaabYgapaWaaSba aSqaaiaab6gaaeqaaOGaaiykaaaa@44FE@ the user’s check-in data are modeled as a user-location matrix RRmxn MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGsbGaeyicI4SaaeOuaiaad2gacaWG4bGaamOBaaaa@3C25@ by CF, where each entry of R represents the frequency of a user visiting a location. CF aims to map the users and locations into a space with dimension k< < min (m, n), and estimate users’ preferences on locations by the dot product of them, which is shown below:
R ̂ = U i L j T       (1) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqa aaaaaaaaWdbiaabkfaaSWdaeqabaWdbiablkWaKaaakiabg2da9iaa dwfapaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaamita8aadaqhaa WcbaWdbiaadQgaa8aabaWdbiaadsfaaaGcpaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqGPaaaaa@4453@
where R ̂ MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaqa aaaaaaaaWdbiaabkfaaSWdaeqabaWdbiablkWaKaaaaaa@380D@ denotes users’ preferences on locations, Ui and Lj denote the ith row in U and the jth row in L respectively, U R mxm MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGvbGaeyicI4SaaeOua8aadaahaaWcbeqaa8qacaqGTbGaaeiE aiaab2gaaaaaaa@3C6D@ denotes the user matrix, L∈Rnxn denotes the location matrix. In order to reduce the generalization error of the objective function, U and L can be used as the regularization terms. Thus, the function of minimize weighted square error loss is:
P= min U,L 1 2 ||W.(R R ̂ )| | F 2 + μ 1 2 ||U| | F 2 + μ 2 2 ||L| | F 2        (2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGqbGaeyypa0ZdamaaxababaWdbiGac2gacaGGPbGaaiOBaaWc paqaaaqabaGcdaWfqaqaaaWcbaWdbiaadwfacaGGSaGaamitaaWdae qaaOWdbmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIYaaaaiaacYha caGG8bGaam4vaiaac6cacaGGOaGaamOuaiabgkHiT8aadaWfGaqaa8 qacaqGsbaal8aabeqaa8qacqWIcmajaaGccaGGPaGaaiiFaiaacYha paWaa0baaSqaa8qacaWGgbaapaqaa8qacaaIYaaaaOGaey4kaSYaaS aaa8aabaWdbiabeY7aT9aadaWgaaWcbaWdbiaaigdaa8aabeaaaOqa a8qacaaIYaaaaiaacYhacaGG8bGaamyvaiaacYhacaGG8bWdamaaDa aaleaapeGaamOraaWdaeaapeGaaGOmaaaakiabgUcaRmaalaaapaqa a8qacqaH8oqBpaWaaSbaaSqaa8qacaaIYaaapaqabaaakeaapeGaaG OmaaaacaGG8bGaaiiFaiaadYeacaGG8bGaaiiFa8aadaqhaaWcbaWd biaadAeaa8aabaWdbiaaikdaaaGcpaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabIcacaqGYaGaaeykaaaa@6A22@
Where W R mxn MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4vaiabgI GiolaabkfadaahaaWcbeqaaiaab2gacaqG4bGaaeOBaaaaaaa@3C31@ is a weighting matrix that represents a confidence of Rij . If the value of Rij is larger than 0, we set Wij to 1, otherwise, we set Wij to 0.||.| | F 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIWaGaaiOla8aacaGG8bGaaiiFaiaac6cacaGG8bGaaiiFamaa DaaaleaacaqGgbaabaGaaGOmaaaaaaa@3DF4@ F Denotes the square of a matrix’s Frobenius norm. μ1 and μ2 are regularization parameters.

On the other hand, when we try to estimate user preference according to the implicit feedback data, the user-location matrix will be modeled as a binary matrix C R mxn MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGdbGaeyicI4SaaeOuamaaCaaaleqabaGaaeyBaiaabIhacaqG Ubaaaaaa@3C3D@ . If ui has checked-in location lj at least once, the Cij is set to 1; otherwise it is set to 0. Thus the function of the minimized weighted square error loss is given as follows:
P= min U,L 1 2 ||W.(C R ̂ )| | F 2 + μ 1 2 ||U| | F 2 + μ 2 2 ||L| | F 2        (3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGqbGaeyypa0ZdamaaxababaWdbiGac2gacaGGPbGaaiOBaaWc paqaaaqabaGcdaWfqaqaaaWcbaWdbiaadwfacaGGSaGaamitaaWdae qaaOWdbmaalaaapaqaa8qacaaIXaaapaqaa8qacaaIYaaaaiaacYha caGG8bGaam4vaiaac6cacaGGOaGaam4qaiabgkHiT8aadaWfGaqaa8 qacaqGsbaal8aabeqaa8qacqWIcmajaaGccaGGPaGaaiiFaiaacYha paWaa0baaSqaa8qacaWGgbaapaqaa8qacaaIYaaaaOGaey4kaSYaaS aaa8aabaWdbiabeY7aT9aadaWgaaWcbaWdbiaaigdaa8aabeaaaOqa a8qacaaIYaaaaiaacYhacaGG8bGaamyvaiaacYhacaGG8bWdamaaDa aaleaapeGaamOraaWdaeaapeGaaGOmaaaakiabgUcaRmaalaaapaqa a8qacqaH8oqBpaWaaSbaaSqaa8qacaaIYaaapaqabaaakeaapeGaaG OmaaaacaGG8bGaaiiFaiaadYeacaGG8bGaaiiFa8aadaqhaaWcbaWd biaadAeaa8aabaWdbiaaikdaaaGcpaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabIcacaqGZaGaaeykaaaa@6A14@
Where the Wij is set as:
W ij ={ η R ij +1,  R ij >0      1,otherwise       (4) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGxbWdamaaBaaaleaapeGaamyAaiaadQgaa8aabeaak8qacqGH 9aqpdaGabaWdaeaafaqabeGabaaabaWdbiabeE7aOjaadkfapaWaaS baaSqaa8qacaWGPbGaamOAaaWdaeqaaOWdbiabgUcaRiaaigdacaGG SaGaaeiiaiaadkfapaWaaSbaaSqaa8qacaWGPbGaamOAaaWdaeqaaO Wdbiabg6da+iaaicdaa8aabaWdbiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaGymaiaacYcacaWGVbGaamiDaiaadIgacaWGLbGaamOCai aadEhacaWGPbGaam4CaiaadwgaaaaacaGL7baacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGOaGaaeinaiaabMcaaaa@5B5C@
The constant η denotes the rate of increase, in this paper we set η at 20.
Geographical influence
As aforementioned, MF can effectively estimate user preference and the relations associated with almost locations by mapping the check-in data into user-location rating matrix. However, the geographical influence plays an important role in location recommendation.

In our lives, it can be discovered that users would like to visit the locations that are close to each other in geography. For example, people usually visit nearby locations such as restaurants or shopping malls after watching the movies. Therefore, we consider using geographical neighborhood characteristic on locations to improve the recommendation accuracy. In this paper we define the user’s preferences by fusing the geographical neighborhood characteristic [5] by:
min    U,L 1 2 ||W.(CU L T G)| | F 2 + μ 1 2 ||U| | F 2 + μ 2 2 ||L| | F 2         (5) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaqa aaaaaaaaWdbiGac2gacaGGPbGaaiOBaaWcpaqaa8qacaqGGaGaaeii aiaabccacaWGvbGaaiilaiaadYeaa8aabeaak8qadaWcaaWdaeaape GaaGymaaWdaeaapeGaaGOmaaaacaGG8bGaaiiFaiaadEfacaGGUaGa aiikaiaadoeacqGHsislcaWGvbGaamita8aadaahaaWcbeqaa8qaca WGubaaaOGaam4raiaacMcacaGG8bGaaiiFa8aadaqhaaWcbaWdbiaa dAeaa8aabaWdbiaaikdaaaGccqGHRaWkdaWcaaWdaeaapeGaeqiVd0 2damaaBaaaleaapeGaaGymaaWdaeqaaaGcbaWdbiaaikdaaaGaaiiF aiaacYhacaWGvbGaaiiFaiaacYhapaWaa0baaSqaa8qacaWGgbaapa qaa8qacaaIYaaaaOGaey4kaSYaaSaaa8aabaWdbiabeY7aT9aadaWg aaWcbaWdbiaaikdaa8aabeaaaOqaa8qacaaIYaaaaiaacYhacaGG8b GaamitaiaacYhacaGG8bWdamaaDaaaleaapeGaamOraaWdaeaapeGa aGOmaaaak8aacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabIcacaqG1aGaaeykaaaa@6BED@
Where G=αH+(1α) S T ,H R n×n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGhbGaeyypa0JaeqySdeMaaeisaiabgUcaRiaacIcacaaIXaGa eyOeI0IaeqySdeMaaiykaiaabofadaahaaWcbeqaaiaabsfaaaGcca qGSaGaaeisaiabgIGiolaabkfapaWaaWbaaSqabeaapeGaaeOBa8aa caqGxdWdbiaab6gaaaaaaa@4931@ is an identity matrix S R n×n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGtbGaeyicI4SaaeOua8aadaahaaWcbeqaa8qacaqGUbGaae41 aiaab6gaaaaaaa@3CCC@ for which S jk =Sim( l j , l k )/Z( l j ), MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGtbWdamaaBaaaleaapeGaamOAaiaadUgaa8aabeaak8qacqGH 9aqpcaWGtbGaamyAaiaad2gacaGGOaGaamiBa8aadaWgaaWcbaWdbi aadQgaa8aabeaak8qacaGGSaGaamiBa8aadaWgaaWcbaWdbiaadUga a8aabeaak8qacaGGPaGaai4laiaadQfacaGGOaGaamiBa8aadaWgaa WcbaWdbiaadQgaa8aabeaak8qacaGGPaGaaiilaaaa@49A0@ and α[ 0,1 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHXoqycqGHiiIZpaWaamWaaeaapeGaaGimaiaacYcacaaIXaaa paGaay5waiaaw2faaaaa@3D7D@ is a weighting parameter used to control the influence of the neighborhood locations; D(lj) is a set that represents the neighboring locations of l j .Sim( l j , l k ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGSbWdamaaBaaaleaapeGaamOAaaWdaeqaaOGaaiOla8qacaWG tbGaamyAaiaad2gacaGGOaGaamiBa8aadaWgaaWcbaWdbiaadQgaa8 aabeaak8qacaGGSaGaamiBa8aadaWgaaWcbaWdbiaadUgaa8aabeaa k8qacaGGPaaaaa@4285@ Refers to the weight of geographical influence of the location l k on  l j .Z( l j ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGSbWdamaaBaaaleaapeGaae4AaaWdaeqaaOGaae4Baiaab6ga caqGGaWdbiaadYgapaWaaSbaaSqaa8qacaWGQbaapaqabaGccaGGUa WdbiaadQfapaGaaiika8qacaWGSbWdamaaBaaaleaapeGaamOAaaWd aeqaaOGaaiykaaaa@428D@ is a normalizing factor, which is defined as Z( l j )= l k D( l j ) Sim( l j , l k ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGAbWaaeWaa8aabaWdbiaadYgapaWaaSbaaSqaa8qacaWGQbaa paqabaaak8qacaGLOaGaayzkaaGaeyypa0Zaaybuaeqal8aabaaape qab0WdaeaapeGaeyyeIu+aaSbaa4qaaiaadYgapaWaaSbaaeaapeGa am4AaaWdaeqaa8qacqGHiiIZcaWGebGaaiikaiaadYgapaWaaSbaae aapeGaamOAaaWdaeqaa8qacaGGPaaabeaaaaGccaWGtbGaamyAaiaa d2gacaGGOaGaamiBa8aadaWgaaWcbaWdbiaadQgaa8aabeaak8qaca GGSaGaamiBa8aadaWgaaWcbaWdbiaadUgaa8aabeaak8qacaGGPaaa aa@5026@ , where Sim( l j , l k ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGtbGaamyAaiaad2gacaGGOaGaamiBa8aadaWgaaWcbaWdbiaa dQgaa8aabeaak8qacaGGSaGaamiBa8aadaWgaaWcbaWdbiaadUgaa8 aabeaak8qacaGGPaaaaa@3F80@ is a Gaussian function as follows:
Sim( l j , l k )= e || x j x k | | 2 σ 2     l k D( l j )      (6) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGtbGaaeyAaiaab2gadaqadaWdaeaapeGaamiBa8aadaWgaaWc baWdbiaadQgaa8aabeaak8qacaGGSaGaamiBa8aadaWgaaWcbaWdbi aadUgaa8aabeaaaOWdbiaawIcacaGLPaaacqGH9aqpcaWGLbWdamaa CaaaleqabaWdbiabgkHiTmaalaaapaqaa8qacaGG8bGaaiiFaiaadI hapaWaaSbaaWqaa8qacaWGQbaapaqabaWcpeGaeyOeI0IaamiEa8aa daWgaaadbaWdbiaadUgaa8aabeaaliaacYhacaGG8bWaaWbaaWqabe aapeGaaGOmaaaaaSWdaeaapeGaeq4Wdm3damaaCaaameqabaWdbiaa ikdaaaaaaSGaaeiiaiaabccacaqGGaGaeyiaIiIaamiBa8aadaWgaa adbaWdbiaadUgaa8aabeaal8qacqGHiiIZcaWGebGaaiikaiaadYga paWaaSbaaWqaa8qacaWGQbaapaqabaWcpeGaaiykaaaak8aacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG2aGaaeykaaaa@6121@
Where xj and xk represent the longitude and latitude of the location lj and lk respectively. For the maximum distance between the two locations, we set a threshold as 10000, and lk will not be considered if the distance is exceeded that threshold.
Social relationship
In our reality life, it can be found that people with similar interests are more likely to form relationships, such as friendships and emotional relationship. People often go to restaurants or other places recommended by friends, which reflects that the users’ check-in behaviors are greatly affected by social relationship. Based on these observations, the social relationship has been modeled to improve the accuracy of location recommendation.

In this paper we believe that the social relationships between users are mutual, as shown in Figure 1. The cosine similarity [24, 25] is used here to measure the similarity between users. Given the individual user ui and uv, let L be a set of locations, and lk denotes a location belonging to the L. The definition of the user similarity is defined by:
Figure 1:Social network
sim( i,v )= l k L S ik S vk l k L S ik 2 l k L S vk 2         (7) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGZbGaaeyAaiaab2gadaqadaWdaeaapeGaaeyAaiaacYcacaqG 2baacaGLOaGaayzkaaGaeyypa0ZaaSaaa8aabaWdbmaavababeWcpa qaa8qacaWGSbWdamaaBaaameaapeGaam4AaaWdaeqaaSWdbiabgIGi olaadYeaaeqan8aabaWdbiabggHiLdaakiaadofapaWaaSbaaSqaa8 qacaWGPbGaam4AaaWdaeqaaOWdbiaadofapaWaaSbaaSqaa8qacaWG 2bGaam4AaaWdaeqaaaGcbaWdbmaakaaapaqaa8qadaqfqaqabSWdae aapeGaamiBa8aadaWgaaadbaWdbiaadUgaa8aabeaal8qacqGHiiIZ caWGmbaabeqdpaqaa8qacqGHris5aaGccaWGtbWdamaaDaaaleaape GaamyAaiaadUgaa8aabaWdbiaaikdaaaaabeaakmaakaaapaqaa8qa daqfqaqabSWdaeaapeGaamiBa8aadaWgaaadbaWdbiaadUgaa8aabe aal8qacqGHiiIZcaWGmbaabeqdpaqaa8qacqGHris5aaGccaWGtbWd amaaDaaaleaapeGaamODaiaadUgaa8aabaWdbiaaikdaaaaabeaaaa GccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabIcacaqG3aGaaeykaaaa@68B7@
Where sim(i,v) denotes the similarity between ui and uv. Sik and Svk indicate whether users ui and uv are checked in at location lk or not. We set Sik to one if the user ui has checked-in at location lk at least once, otherwise, we set Sik to zero. The same processes are used for Svk.

Generally, traditional similarity calculations suggest that each item has the same weights to influence each other, while in reality we discover that the higher frequency that users visit the same location, the greater impact on user similarity between them. For example, for location lk, let users A, B and C visit it for 5, 2 and 6 times, respectively. Based on the theory, the higher frequency of visits represents the greater the user preference. Compared to user B, the user similarity between user A and user C should be larger because they have a higher degree of common visits [26]. On the other hand, we think that the factors that users visit the location can be divided into subjective factors and objective factors. The subjective factor is the user’s own preference for the type of location, and the objective factor is the popularity of the location. For the location assuming that it has a high popularity, the number of users who visit the location will continue to increase, so the user’s access to the location is not highly relevant to the type of itself. Then when the location lk has been visited by many people, it is difficult to find a project similar to it. Therefore, the higher the popularity of the location that users access together, the weaker the influence on the similarity between users should be, and the corresponding weight should also be smaller. According to these observations, we proposed a novel approach to estimate the user similarity base on the cosine similarity as follow:
λ k =( r ik + r vk 2 )ln| m I k |       (8) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaH7oaBpaWaaSbaaSqaa8qacaWGRbaapaqabaGcpeGaeyypa0Za aeWaa8aabaWdbmaalaaapaqaa8qacaWGYbWdamaaBaaaleaapeGaam yAaiaadUgaa8aabeaak8qacqGHRaWkcaWGYbWdamaaBaaaleaapeGa amODaiaadUgaa8aabeaaaOqaa8qacaaIYaaaaaGaayjkaiaawMcaai GacYgacaGGUbGaaiiFamaalaaapaqaa8qacaWGTbaapaqaa8qacaWG jbWdamaaBaaaleaapeGaam4AaaWdaeqaaaaak8qacaGG8bGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG4aGa aeykaaaa@5225@
where rik and rvk indicate the frequency of user ui and uv visiting the location lk respectively. In addition, lk denotes the number of people who checked in at the location lk, m denotes the total number of users. In this paper we use the ratio to denote the influence of the location popularity. We combined the two location characteristics as weighting parameter λk and fused it into the user similarity, which is modeled as follows:
si m new ( i,v )= l k L λ k S ik S vk l k L λ k S ik 2 l k L λ k S vk 2           (9) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGZbGaamyAaiaad2gapaWaaWbaaSqabeaapeGaamOBaiaadwga caWG3baaaOWaaeWaa8aabaWdbiaabMgacaGGSaGaaeODaaGaayjkai aawMcaaiabg2da9maalaaapaqaa8qadaqfqaqabSWdaeaapeGaamiB a8aadaWgaaadbaWdbiaadUgaa8aabeaal8qacqGHiiIZcaWGmbaabe qdpaqaa8qacqGHris5aaGccqaH7oaBpaWaaSbaaSqaa8qacaWGRbaa paqabaGcpeGaam4ua8aadaWgaaWcbaWdbiaadMgacaWGRbaapaqaba GcpeGaam4ua8aadaWgaaWcbaWdbiaadAhacaWGRbaapaqabaaakeaa peWaaOaaa8aabaWdbmaavababeWcpaqaa8qacaWGSbWdamaaBaaame aapeGaam4AaaWdaeqaaSWdbiabgIGiolaadYeaaeqan8aabaWdbiab ggHiLdaakiabeU7aS9aadaWgaaWcbaWdbiaadUgaa8aabeaak8qaca WGtbWdamaaDaaaleaapeGaamyAaiaadUgaa8aabaWdbiaaikdaaaaa beaakmaakaaapaqaa8qadaqfqaqabSWdaeaapeGaamiBa8aadaWgaa adbaWdbiaadUgaa8aabeaal8qacqGHiiIZcaWGmbaabeqdpaqaa8qa cqGHris5aaGccqaH7oaBpaWaaSbaaSqaa8qacaWGRbaapaqabaGcpe Gaam4ua8aadaqhaaWcbaWdbiaadAhacaWGRbaapaqaa8qacaaIYaaa aaqabaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabIcacaqG5aGaaeykaaaa@767C@
Unified framework: G-BLAS
In this section, we introduce the integrated model of our approach. According to the Eq.(5) and Eq.(9), the proposed framework to combined three factors is shown as formula (10) below:
P= min U,L 1 2 ||W.(CU L T G)| | F 2 + μ 1 2 ||U| | F 2 + μ 2 2 ||L| | F 2 + μ 3 2 i=1 m vu si m new (i,v)|| u i u v | | F 2     (10) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGqbGaeyypa0ZdamaaxababaWdbiGac2gacaGGPbGaaiOBaaWc paqaa8qacaWGvbGaaiilaiaadYeaa8aabeaak8qadaWcaaWdaeaape GaaGymaaWdaeaapeGaaGOmaaaacaGG8bGaaiiFaiaadEfacaGGUaGa aiikaiaadoeacqGHsislcaWGvbGaamita8aadaahaaWcbeqaa8qaca WGubaaaOGaam4raiaacMcacaGG8bGaaiiFa8aadaqhaaWcbaWdbiaa dAeaa8aabaWdbiaaikdaaaGccqGHRaWkdaWcaaWdaeaapeGaeqiVd0 2damaaBaaaleaapeGaaGymaaWdaeqaaaGcbaWdbiaaikdaaaGaaiiF aiaacYhacaWGvbGaaiiFaiaacYhapaWaa0baaSqaa8qacaWGgbaapa qaa8qacaaIYaaaaOGaey4kaSYaaSaaa8aabaWdbiabeY7aT9aadaWg aaWcbaWdbiaaikdaa8aabeaaaOqaa8qacaaIYaaaaiaacYhacaGG8b GaamitaiaacYhacaGG8bWdamaaDaaaleaapeGaamOraaWdaeaapeGa aGOmaaaakiabgUcaRmaalaaapaqaa8qacqaH8oqBpaWaaSbaaSqaa8 qacaaIZaaapaqabaaakeaapeGaaGOmaaaacqGHris5daqhaaWcbaGa amyAaiabg2da9iaaigdaaeaacaWGTbaaaOGaeyyeIu+aaSbaaSqaai aadAhacqGHiiIZcaWG1baabeaakiaadohacaWGPbGaamyBa8aadaah aaWcbeqaa8qacaWGUbGaamyzaiaadEhaaaGccaGGOaGaamyAaiaacY cacaWG2bGaaiykaiaacYhacaGG8bGaamyDa8aadaWgaaWcbaWdbiaa dMgaa8aabeaak8qacqGHsislcaWG1bWdamaaBaaaleaapeGaamODaa WdaeqaaOGaaiiFaiaacYhadaqhaaWcbaWdbiaadAeaa8aabaWdbiaa ikdaaaGcpaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaae imaiaabMcaaaa@8F2A@
Where μ1 and μ1 are the weighting parameters that control the U and L respectively, and μ3 denotes the weighting parameter that controls the influence of social relationship on recommendation. ||U|| F 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG8bGaaeiFaiaabwfacaqG8bGaaeiFa8aadaqhaaWcbaWdbiaa bAeaa8aabaWdbiaaikdaaaaaaa@3CDA@ And ||L|| F 2 MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG8bGaaeiFaiaabYeacaqG8bGaaeiFa8aadaqhaaWcbaWdbiaa bAeaa8aabaWdbiaaikdaaaaaaa@3CD1@ are used as regularization terms to prevent over-fitting.

In this paper we used the gradient descent algorithm [15] to obtain the optimal solution of Eq. (10), the partial derivative of U and L is given by:
P L =W.( U L T GC )GU+ μ 2 l j          (11) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaamiuaaWdaeaapeGaeyOaIyRaamit aaaacqGH9aqpcaWGxbGaaiOlamaabmaapaqaa8qacaWGvbGaamita8 aadaahaaWcbeqaa8qacaWGubaaaOGaam4raiabgkHiTiaadoeaaiaa wIcacaGLPaaacaWGhbGaamyvaiabgUcaRiaabY7apaWaaSbaaSqaa8 qacaaIYaaapaqabaGcpeGaamiBa8aadaWgaaWcbaWdbiaadQgaa8aa beaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabIcacaqGXaGaaeymaiaabMcaaaa@5425@
P U =W.( U L T GC ) G T L+ μ 1 u i + μ 3 vU si m new (i,v)( u i u v )   (12) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaeyOaIyRaamiuaaWdaeaapeGaeyOaIyRaamyv aaaacqGH9aqpcaWGxbGaaiOlamaabmaapaqaa8qacaWGvbGaamita8 aadaahaaWcbeqaa8qacaWGubaaaOGaam4raiabgkHiTiaadoeaaiaa wIcacaGLPaaacaWGhbWdamaaCaaaleqabaWdbiaadsfaaaGccaWGmb Gaey4kaSIaaeiVd8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaWG 1bWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiabgUcaRiaabY7apa WaaSbaaSqaa8qacaaIZaaapaqabaGcpeWaaybuaeqal8aabaWdbiaa dAhacqGHiiIZcaWGvbaabeqdpaqaa8qacqGHris5aaGccaWGZbGaam yAaiaad2gapaWaaWbaaSqabeaapeGaamOBaiaadwgacaWG3baaaOGa aiikaiaadMgacaGGSaGaamODaiaacMcacaGGOaGaamyDa8aadaWgaa WcbaWdbiaadMgaa8aabeaak8qacqGHsislcaWG1bWdamaaBaaaleaa peGaamODaaWdaeqaaOWdbiaacMcacaqGGaGaaeiiaiaabccacaqGOa GaaeymaiaabkdacaqGPaaaaa@6BBE@
Experimental Results and AnalysisTop
In this section we first introduce the dataset, performance metrics and parameter settings that we used in the experiments, followed by detailed comparison of the results and analysis of the performance of each approach.
Dataset Description
The experimental data used in this study are collected from Foursquare and Gowalla , which are the two most popular LBSNs. Foursquare encourages users to share information such as their current locations, which contains 1,196,248 check-ins for 24,941 users to 28,593 POIs. Gowalla is a second check-in website after Foursquare. Users can share information about places, activities, travel routes, etc. among friends on it. It contains 6, 941,890 checkin data for 196,591 users to 950,327 POIs. Each of the check-in data includes userID, locationID, and the coordinate of POI and check-in time. We extracted some data for experimentation and the detailed statistics of the check-in data in the datasets are summarized in Table 1.

The Foursquare dataset for this experiment contains 496,488 check-in data for 13,805 users to 19,587 POIs. The Gowalla dataset contains 161, 553 check-ins for 5, 433 users to 9, 687 POIs. In our experiment, the data needs to be pre-processed due to the sparseness, we filtered out the users whose check-in times are less than 10 and the locations are visited by less than 10 users. Finally, based on the five-fold cross-validation method, we randomly split each dataset into training set and the testing set and the average of the test results is taken as the experimental result.
Table 1: Statistics of the two datasets

Datasets

No. of users

No. of locations

No. of check-ins

user-location matrix density

Foursquare

13805

19587

496488

1.83×10-3

Gowalla

5433

9687

161553

3.06×10-3

Performance Metrics
In this work, we use two widely used metrics (precision and recall) to evaluate the performance of the approach that we proposed. The precision and recall of the top-K location recommendations to a target user are denoted by P@K and R@K respectively. P@K defines the ratio of the discovered locations to the K recommended locations, and R@K defines the ratio of discovered locations to the set of locations that the target user has visited in the testing data. Generally, the higher the precision and recall values are, the better the performance is. P@K and R@K are defined as follows:
Precision@K= 1 |T| i=1 r |R( u i ) E( u i )| K        (13) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGqbGaaeOCaiaabwgacaqGJbGaaeyAaiaabohacaqGPbGaae4B aiaab6gacaqGabGaae4saiabg2da9maalaaapaqaa8qacaaIXaaapa qaa8qacaGG8bGaamivaiaacYhaaaWaaybCaeqal8aabaaabaaaneaa peGaeyyeIuoaaOWaa0baaSqaaiaadMgacqGH9aqpcaaIXaaabaGaam OCaaaakmaalaaapaqaa8qacaGG8bGaamOuamaabmaapaqaa8qacaWG 1bWdamaaBaaaleaapeGaamyAaaWdaeqaaaGcpeGaayjkaiaawMcaa8 aadaqfGaqabSqabeaacaaMb8oaneaapeGaeyykICmaaOGaamyramaa bmaapaqaa8qacaWG1bWdamaaBaaaleaapeGaamyAaaWdaeqaaaGcpe GaayjkaiaawMcaaiaacYhaa8aabaWdbiaadUeaaaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaae4mai aabMcaaaa@62C9@
Precision@K= 1 |T| i=1 r |R( u i ) E( u i )| |R( u i )|        (14) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqGqbGaaeOCaiaabwgacaqGJbGaaeyAaiaabohacaqGPbGaae4B aiaab6gacaqGabGaae4saiabg2da9maalaaapaqaa8qacaaIXaaapa qaa8qacaGG8bGaamivaiaacYhaaaWaaybCaeqal8aabaaabaaaneaa peGaeyyeIuoaaOWaa0baaSqaaiaadMgacqGH9aqpcaaIXaaabaGaam OCaaaakmaalaaapaqaa8qacaGG8bGaamOuamaabmaapaqaa8qacaWG 1bWdamaaBaaaleaapeGaamyAaaWdaeqaaaGcpeGaayjkaiaawMcaa8 aadaqfGaqabSqabeaacaaMb8oaneaapeGaeyykICmaaOGaamyramaa bmaapaqaa8qacaWG1bWdamaaBaaaleaapeGaamyAaaWdaeqaaaGcpe GaayjkaiaawMcaaiaacYhaa8aabaWdbiaacYhacaWGsbGaaiikaiaa dwhadaWgaaWcbaGaamyAaaqabaGccaGGPaGaaiiFaaaacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG 0aGaaeykaaaa@6848@
Where ui denotes a user, R(ui) denotes the set of locations that the user ui have visited in the testing data, E(ui) denotes the set of locations which is recommended to user ui, T denotes the set of users in the testing data. In particular, users are more inclined to the results of high recommendation rankings; therefore we choose P@5, P@10, R@5, and R@10 as evaluation metrics in our experiments. For the regularization parameters μ1 and μ2 we set at 0.03. Furthermore, we set the instance weighting parameter α at 0.4. The weighting parameter that controls the social relationship μ3 is set at 0.01 by cross-validation.
Benchmarking Algorithms
1) Base Matrix Factorization [27] (BaseMF): This is the based matrix factorization approach designed for explicit feedback datasets, which is used to predict a user’s preference by considering the check-in data.

2) Weighting Matrix Factorization [22] (WMF): This is the weight matrix factorization approach designed for implicit feedback datasets. It predicts a user’s preference without considering the geographical Influence, social relationship of users and other context information.

3) The approach in [28] is defined as USG in this paper: This approach is based on the observation that a user visiting a location follows the power law distribution. A unified location recommendation framework is used to linearly combine the geographical influence and the user’s preference.

4) The approach in [29] is defined as NCPD in this paper: This approach classifies the POIs by geographical neighborhood characteristics and fuses the location and popularity of POIs to predict a user’s preference based on the Non-negative Matrix Factorization (NMF).
Experimental Results
Figure 2 and Figure 3 depict the performance of the recommendation techniques based on the real dataset collected from Foursquare and Gowalla, using precision and recall measurements. Experimental results show that our framework G-BLAS outperforms the baseline matrix factorization model (i.e., BaseMF, WMF), which has not used any other contextual information, especially when compared to the approaches that utilize geographical characteristics or social relationship (i.e., USG, NCPD). The details are demonstrated as follows.
Figure 2:The performance of the recommendation techniques on Foursquare
Figure 3:The performance of the recommendation techniques on Gowalla
Figure 2(a) and 2(b) show the performance of a variety of recommendation techniques on the foursquare dataset. Taking P@5 as an example, it can be observed that the performance of WMF model is 19.23% which is higher than that of BaseMF model. This shows that the WMF model can more effectively model the user check-in data by assigning appropriate weights, which improves the recommendation performance. However, both WMF and BaseMF only consider the users’ check-in data and the models are relatively simple, so the precision and recall rates of them are lower than other approaches. The USG fuses user preference, geographical influence, and social relationship to predict the users’ preference on unvisited location, which outperforms BaseMF and WMF by 53.85% and 29.03% respectively. Similarly, NCPD also considers the user preference information and geographical influence of POIs, but its performance is only higher than the USG model by 15.14%. One possible reason is that USG considers that the power-law distribution is satisfied between the distance and the check-in probability, which is inconsistent with reality. Not all data sets are applicable to a specific distribution, and it also indicates that the influence of the user’s social relationship is weaker than the geographical influence. In addition, from the result we can find that our approach always achieves the best result on the evaluation metrics, in terms of P@5, where the average improvements of G-BLAS over BaseMF, WMF, USG and NCPD are 95.15%, 61.52%, 27.51%, 10.87% respectively. It has demonstrated that the proposed G-BLAS can effectively mitigate the data sparsity and significantly improve the recommendation accuracy by fusing user preference and incorporating multiple context information.

Figure 3(a) and 3(b) show the performance of a variety of recommendation techniques on the Gowalla dataset. Because of the lower sparseness of the check-in data, the performance of each approach on the Gowalla dataset is better than that on the foursquare dataset. In general, compared with other approach that we have benchmarked with, the performances of our proposed approach again show significant improvements, which validate the effectiveness of our approach.
Parameter Tuning
In our proposed approach, the parameter α denotes the weight that the influence of geographical neighborhood characteristics to user preference, and the impact of social relationship on location recommendation is controlled by the parameter μ3. In Figure 4 we tested the effect of different parameter α on recommended performance (setting the number of recommended locations as 5). As seen from the two figures, we have derived interesting findings: (1) our approach achieves the better results when the parameter α is set at 0.4; (2) if we don’t consider the influence of geographical neighborhood characteristic, in other words the parameter α is set at 0, the recommendation accuracy will be degraded. It validates the importance of geographical influence in the process of location recommendation.

In Figure 5 we investigate the performance of our approach under different values of μ3. A lager μ3 indicates that the social relationship of users is more closely linked to location recommendation. As seen, μ3 = 0.01 is the most suitable setting in the two datasets. Furthermore, when parameter μ3 is larger than 1, the fluctuation of performance becomes less obvious.
Figure 4:Effects of parameter α on recommendation accuracy (μ3= 0)
Figure 5:Effects of parameter μ3 on recommendation accuracy(α= 1 )
Table 2: Performance comparison of recommended technologies on Foursquare

Other model performance improvements compared to BaseMF

Foursquare
(P@5)

WMF

USG

NCPD

G-BLAS

19.23%

53.85%

76.92%

95.15%

Other model performance improvements compared to WMF

Foursquare
(P@5)

USG

NCPD

G-BLAS

29.03%

48.39%

64.52%

Other model performance improvements compared to USG

Foursquare
(P@5)

NCPD

G-BLAS

15.14%

27.51%

Other model performance improvements compared to NCPD

Foursquare
(P@5)

G-BLAS

10.87%

ConclusionsTop
In this paper, we have explored the influence of the location characteristic on social relationship and then proposed a novel approach to measure the similarity between users. Furthermore, we proposed a framework to more accurately model user’s preferences on location by fusing the geographical neighborhood characteristics and the social relationship of users. From the results, we can see that our approach achieves significantly higher recommendation performance than the state-of-the-art approaches, including Based, WMF, USG and NCPD.

There are two directions for future investigation:(1) how to fuse the influence of temporal information on user’s preference to further extend our framework, and (2) how to capture more relationship between locations to improve the recommendation accuracy.
Author ContributionsTop
Conceptualization and Methodology, HH.Y, WG.W; software, HH.Y. and GY.X; writing–original draft preparation, HH.Y; writing– review and editing, HM.Z and HH.Y.; supervision, WG.W.
Conflicts of InterestTop
The authors declare no conflict of interest
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