BordaRank: A Ranking Aggregation Based Approach to Collaborative Filtering

BordaRank: A Ranking Aggregation Based Approach to Collaborative Filtering Yeming TANG Department of Computer Science and Technology Tsinghua University Beijing, China tym13@mails.tsinghua.edu.cn Qiuli TONG Information Technology Center Tsinghua, University Beijing, China tql@tsinghua.edu.cn Abstract Recommender systems are widely used in today s online applications. Traditional rating-oriented methods predict user ratings on items, but they fail to capture user preference among different items. This paper regards recommendation problem as a ranking task and proposes a new ranking-oriented collaborative filtering framework based on ranking aggregation methods. In this framework, recommendation lists are generated according to item rankings given by users who are similar to the target user. Then, a two-step method called BordaRank is proposed to further explain the framework. The method first uses item collaborative filtering to predict unknown ratings and then uses Borda count method to aggregate item rankings of neighbors. Finally, BordaRank is modified as a pure rankingoriented method, which could be directly applied on the sparse rating matrix without rating prediction as an intermediate step. The methods are evaluated on real world movie rating data. Experimental results show that BordaRank improves the precision and recall of original rating-oriented methods and modified BordaRank also outperforms traditional methods. Index Terms Recommender System, Ranking Aggregation, Collaborative Filtering, Information Retrieval I. INTRODUCTION Recommender systems helps people filter out the information of interest from the mass quickly and effectively. Broadly speaking, recommendation problem is defined as a rating prediction problem. Since Netflix Prize competition was held in 2006, many rating prediction algorithms were proposed to solve recommendation problems. Most of these algorithms are based on collaborative filtering techniques. One category of these algorithms uses neighborhood-based collaborative filtering methods to generate item recommendations. Such as Grouplens [1], one of the early generation of collaborative filtering algorithms, uses user rating data to calculate user similarity or item similarity. According to the calculated similarity, Grouplens predicts how well users will like new items based on similar users or items. Another category of these algorithms uses model-based collaborative filtering methods to generate item recommendations. These algorithms use user rating data to train models to predict user ratings by machine learning or data mining algorithms [2]. For example, Paterek [3] improves the set of predictors used in Netflix Cinematch by adding biases to Singular Value Decomposition (SVD) models and performs a lower error rate than the original models. Liu et al. [4] extend Restricted Boltzmann Machine (RBM) and propose Content- Based Restricted Boltzmann Machine (CBRBM); the model is applied on both rows and columns of rating matrix to predict a better result. Apart from rating-oriented recommendation algorithms, some researchers tried to combine information retrieval techniques into recommender systems, such as Learning to Rank (L2R), Normalized Discounted Cumulative Gain (NDCG) evaluation metric, etc. For example, Liu and Yang [5] propose EigenRank which improves memory based collaborative filtering methods by pairwise methods in learning to rank. In this paper, we propose an novel algorithm of rankingoriented collaborative filtering which uses ranking aggregation method to generate item recommendations. We first describe the framework of ranking aggregation based collaborative filtering algorithms. In this framework, we introduce an algorithm which uses Borda count method to aggregate item rankings derived from the predicted user-item rating matrix. We further modified our algorithm to a pure ranking-oriented collaborative filtering algorithm which could be applied directly on the original sparse rating matrix. Finally, we perform a 5-fold cross validation on MovieLens data set [6] to validate the effectiveness of our algorithms. We also discussed the impact of different neighborhood sizes and different scoring functions in section IV. II. RELATED WORKS There are mainly three parts of related works: neighborhood-based collaborative filtering, ranking-oriented collaborative filtering and ranking aggregation methods. A. Neighborhood-based collaborative filtering Neighborhood-based collaborative filtering (also called memory-based collaborative filtering) uses neighborhood to estimate the target user s ratings. The ratings are represented as a user-item rating matrix, which is highly sparse due to the huge number of users and items. The goal of neighborhoodbased collaborative filtering is to fill the user-item rating matrix. Neighborhood-based methods fall into two categories: User Collaborative Filtering (UserCF) and Item Collaborative Filtering (ItemCF). 978-1-5090-0806-3/16/$31.00 copyright 2016 IEEE ICIS 2016, June 26-29, 2016, Okayama, Japan

User collaborative filtering uses user neighborhood to predict the target user s ratings. The user neighborhood contains users whose ratings are similar to the target user s existing ratings. User collaborative filtering estimates the target user s unknown ratings by taking the similarity weighted average ratings of users in his or her neighborhood. The recommendation list could be generated by sorting the target user s unrated items in descending order and selecting top-n recommendations. On the contrast, item collaborative filtering uses item neighborhood to predict the target user s ratings. The item neighborhood contains items which are similar to the target user s previous rated items. Item collaborative filtering estimates the item rating by taking the similarity weighted average ratings of the item s neighborhoods. Similarity measurement is the key concept in neighborhoodbased collaborative filtering. Because of the sparsity of the user-item rating matrix, one of the shortcomings of neighborhood-based collaborative filtering is that it is unlikely to find highly similar users or items. The performance of neighborhood-based collaborative filtering decreases when the rating matrix is sparse. To alleviate this problem, dimension reduction methods such as Singular Value Decomposition (SVD), Principle Component Analysis (PCA) etc. are applied to represent user vector in a reduced space. Many hybrid methods, such as content-boosted approaches [4], are also proposed. B. Ranking-oriented collaborative filtering Traditional rating-oriented recommendation algorithms is aimed at minimizing the predicted rating errors. However, rating-oriented methods are deficient in capturing the item preference of the target user, due to user ratings could be biased depending on different users. Some users prefer to use higher ratings, while others prefer to use lower ratings. Though methods that normalize user ratings when aggregate the ratings of neighborhoods are proposed, Instead of estimating user-item rating matrix, rankingoriented collaborative filtering generates recommendation lists directly. Ranking-oriented collaborative filtering methods combine thoughts from learning to rank, especially pairwise or listwise algorithms. For pairwise algorithm, the input is item pairs which indicate the target user s preference between rated items; the output is predicted preference on unrated items. One of the state of the art pairwise collaborative filtering algorithms is EigenRank [5]. It defines a preference function on item pairs, uses Kendall Rank Correlation Coefficient (KRCC) to measure similarity between users, and generate item recommendations by maximizing the loss function based on user preference. Listwise algorithm models user preference on item lists and directly learns on the lists to optimize model parameters. Unlike most of rating-oriented approaches use Root-Mean- Square Error (RMSE) or Mean Average Error (MAE) as their loss functions, listwise approaches define the loss functions on whole lists. For example, Weimer et al. [7] propose CofiRank which uses Maximum Margin Matrix Factorization (MMMF) optimizes NDCG. C. Ranking aggregation methods Ranking aggregation methods have been used in recommender systems for aggregating multiple results from different recommendation algorithms [8]. However, ranking-oriented collaborative filtering based on ranking aggregation is rarely researched due to most existing ranking-oriented collaborative filtering methods are based on ranking generation. The most representative ranking aggregation methods falls into two categories: unsupervised methods and supervised methods. For example, for unsupervised methods, Borda count method [9] is voting mechanism which considers voter s preference on all candidates; for supervised methods, Cranking which proposed in [10] uses Markov model on permutations to aggregate multiple rankings. III. RANKING AGGREGATION BASED COLLABORATIVE FILTERING In this section, we presented a new ranking-oriented collaborative filtering approach based on ranking aggregation methods. We first introduce a two-step recommendation method based on both rating-oriented and ranking-oriented method. To alleviate the rating data sparsity problem, we use ratingoriented collaborative filtering to fill the rating matrix. We applied original Borda count method described in section III-C to aggregate items rankings of the target user s neighbors. At last, we propose a pure ranking aggregation based collaborative filtering method, which could be directly applied on the original sparse rating matrix. A. Feature extraction and problem definition In general, a recommender system will collect users demographic information and items meta-data. These are raw data to be extracted and transformed to feature representations. For categorical data (gender, occupation etc.), because they are hard to compare their similarities, categorical data are represented as vectors using one-of-k coding. If a data field can take k different discrete values, the field should be represented as a k-dimension binary vector, which corresponds with k possible values. One-of-k coding representation can also be used for multi-valued data fields. For numerical data (age, date etc.), such transformation is not needed. In feature selection and extraction, we extract various features from raw data, for example, rate time in a day, rate date interval, ratings of different category items etc. For all numerical features, we further extract the standard deviation, skewness, mean, median, minimum, maximum etc. to describe the distribution attributes of the values. In order to adjusting features which are measured on different scales, all features are normalized using x x x = (1) x max x min where x is the feature vector and x is the normalized feature vector.

The problem is defined as follows. Given a set of m users U = {u 1,u 2,...,u m }, a set of n items I = {i 1,i 2,...,i n }, and a user-item rating matrix R m n, for each user u find a permutation ˆπ u, which is a bijection from I to {1, 2,...,n}, to fit the users tastes best. The rating that the user u gives to item i is denoted by r u,i, while r u,i =0if the user u has not rated item i. The set of items rated by user u is denoted by I u and the set of users who have rated item i is denoted by U i. B. Similarity measurement For neighborhood-based collaborative filtering, similarity measurement is a major step. We use Euclidean distance based similarity to measure the user and item similarity. Euclidean distance based similarity is defined as 1 s x,y = (2) 1+ x y 2 where x and y are user or item feature vectors. We select k nearest neighbors as the neighborhood of user and item. The neighborhood of user u is denoted by N u and the neighborhood of item i is denoted by N i. Although our algorithm uses Euclidean distance based similarity measurement, it can be easily replaced by other similarity measurement, such as cosine similarity [11] or Pearson correlation coefficient [1] [12]. C. Borda count based collaborative filtering In this section, we introduce the basic ranking aggregation based collaborative filtering framework. A two-step method called BordaRank is introduced to further explain the proposed framework. In the first step, BordaRank uses item collaborative filtering to fill the sparse rating matrix. This step allows Borda count method to be applied for aggregating neighbors rankings in our method. Then, BordaRank derives rankings from the predicted rating matrix and generates the item recommendation to the target user. In our ranking aggregation based collaborative filtering framework, the ranking of items given by user u is denoted by π u. A recommendation algorithm should estimate a permutation ˆπ u : I {1, 2,...,n} for each user u as the recommendation list. For rating based collaborative filtering, the recommendation list ˆπ u of the target user u is generated by ranking all items according his or her estimated ratings in descending order: ˆπ u = π u where π u is the item permutation in descending order of the target user u s predicted ratings. However, for ranking aggregation based collaborative filtering algorithm, the estimated permutation ˆπ u of the target user u are generated by the items permutations of his or her neighbors: ˆπ u = f(π v1,π v2,...,π vk ) v N u where k = N u, π v is the permutation which ranks all items according previous estimated ratings of user v in descending order, function f is the ranking aggregation algorithm, for example, Borda count method in our algorithm which will be discussed later. Borda count [9] is originally an election method. In the election, each voter ranks all candidates in order of preference. Borda count gives each candidate a score according to his or her rank in each ballot. The candidate who gains the highest score wins the election. In the following part of this section, we will give a formal description of how Borda count works in our algorithm. Noting that Borda count method requires each voter to rank all the candidates, the sparse rating matrix has to be filled. The unknown ratings are predicted by item collaborative filtering: j N ˆr u,i = i I u s i,j r u,j (3) j N i I u s i,j where ˆr u,i is the estimated rating which user u will give to item i, r u,j is the actual rating which user u gives to item j, s i,j is the similarity between item i and item j, N i is the neighborhood of item i and I u is the item set rated by user i. Though user-item rating matrix is filled using item collaborative filtering, there could still be unpredicted ratings in the rating matrix, because the rated item set of the target user s neighborhood may not cover all the items: I v I v N u We adopt a simple strategy to deal with these unpredicted ratings by filling them by 0 values. The next step is aggregating neighbors permutations which derived from rating matrix and generate the estimated permutation of the target user. Inspired by Borda count method, we apply a scoring function on each item according to its ranking in the permutation. Then, for each item, we adds up the all the similarity weighted scores which the item gains among all neighbors permutations: Γ u (i) = v N u s u,v score u (i) (4) where Γ u (i) is the similarity weighted score sum of item i for the target user u, s u,v is the similarity between user u and user v. The most common formula of the scoring function [13] is linear function score u (i) =n π u (i)+1 (5) where n is the total number of items, π v (i) is the ranking of item i in permutation π v. Our algorithm uses this scoring function as well. The estimated permutation ˆπ u for user u is generated by sorting the similarity weighted score sum Γ u (i) of each item i in descending order. Other variants of scoring function formula could also be applied. For example, the General Election in Nauru held on June 19, 2010 used reciprocal function formula; the election of the most valuable NBA player uses piecewise function formula. We will further compare and discuss the selection of scoring function formula in section IV.

D. Modified Borda count based collaborative filtering Although Borda count based collaborative filtering performs a good result, it still needs a rating prediction step to alleviate data sparsity problem. In this section, we describe a pure ranking-oriented collaborative filtering algorithm which applies modified Borda count method directly on the original sparse rating matrix without any rating prediction. Modified Borda count method does not require users to rank all items. It allows that each user could only rank his or her top-k favorite items, where 1 k I. In this section, π u is represented as the partial permutation of user u on items: π u : I {1, 2,...,k u } where k u is the total number of items which user u have actually ranked. The scoring function needs to be modified to score u (i) =k u π u (i) (6) where π u (i) is the rank position of item i in the partial permutation π u. Substitute the scoring function into equation (4), we can obtain the similarity weighted score sum as the following equation: Γ u (i) = s u,v (k u π u (i)) (7) v N u Our ranking-oriented collaborative filtering algorithm generates the estimated permutation ˆπ u of the target user u by sorting the score sum Γ u (i) in descending order. IV. EXPERIMENTS We use MovieLens data set [6] to evaluate the proposed method. This data set consists of 100,000 ratings by 943 users on 1682 movies. The rating ranges from 1 to 5, where 1 means the worst and 5 means the best. Each user has rated at least 20 movies. The data set also contains basic information of users and items, such as user gender, movie release date. A. Comparison with other algorithms We perform a 5-fold cross validation to evaluate BordaRank and modified BordaRank algorithm. The data is equally split into five disjoint sets, from u1 to u5. Each set is used for test data while others are used for training data. We also implement user collaborative filtering and item collaborative filtering as baseline algorithms. In the evaluation, a user gives an item rating larger than 3, i.e. the user rates 4 or 5, is regarded as the user likes the item. F 1 score and coverage are used to measure all the algorithms and the results are shown in table I and table II. According to the result, BordaRank has a higher F 1 score than item collaborative filtering, which indicates that ranking aggregation in BordaRank could have a better recommendation effort than simply sorting the item ratings of the target user. The modified BordaRank performs a better result than other rating prediction based algorithms, which indicates that ranking-oriented collaborative filtering can capture user preference on items more precisely. However, the item coverages of BordaRank and modified Borda Rank are lower than the coverage of item collaborative filtering. 0.18 0.16 0.14 0.12 0.10 0.45 0.40 0.35 0.30 0.25 0.20 0.15 BordaRank modified BordaRank 10 20 30 40 50 60 70 80 90 100 Fig. 1. F1 score of BordaRank and modified BordaRank 0.10 BordaRank modified BordaRank 0.05 10 20 30 40 50 60 70 80 90 100 Fig. 2. Coverage of BordaRank and modified BordaRank B. Impact of neighborhood size The item coverage of a recommender system indicates how many items are able to be recommended by the system. In collaborative filtering, the item coverage is mainly affected by the neighborhood size. We compare the impact of different neighborhood size on BordaRank and modified BordaRank. The result is shown in figure 1 and figure 2. In figure 1, the F1 score of both BordaRank and modified BordaRank increases with neighborhood size increases. The F1 score tends to converge when neighborhood size is larger than 50. In figure 2, the coverage of BordaRank and modified BordaRank decreases with neighborhood size increases. The coverage decreases sharply while neighborhood size is smaller than 30.

TABLE I THE F1 SCORE OF EACH METHOD IN 5-FOLD CROSS VALIDATION UserCF 0.0206 0.0144 0.0129 0.0122 0.0125 0.01452 ItemCF 0.0321 0.0238 0.0187 0.0186 0.0221 0.02306 BordaRank 0.1255 0.0971 0.0782 0.0776 0.0806 0.09180 modified BordaRank 0.1770 0.1451 0.1189 0.1162 0.1176 0.14469 TABLE II THE ITEM COVERAGE OF EACH METHOD IN 5-FOLD CROSS VALIDATION UserCF 0.5820 0.5856 0.5832 0.5672 0.5904 0.58168 ItemCF 0.9417 0.9441 0.9334 0.9370 0.9388 0.93900 BordaRank 0.1336 0.1570 0.1570 0.1623 0.1647 0.15492 modified BordaRank 0.1504 0.1522 0.1480 0.1510 0.1552 0.15136 C. Impact of different scoring functions We designed four different scoring functions for Borda count based collaborative filtering: linear score u (i) =n π u (i)+1 reciprocal score u (i) = 1 π u (i) logarithmic score u (i) =log 2 (n π u (i)+1) polynomial score u (i) =(n +1) 2 2(n +1)π u (i)+π u (i) 2 The results of the methods using these scoring functions are shown in table III, table IV and table V. According to the result, the logarithmic scoring function performs better than linear and reciprocal scoring functions, though the latter are more common. We further implements and evaluates modified BordaRank using the logarithmic scoring function. In the pure ranking-oriented collaborative filtering, the precision and recall of modified BordaRank using logarithmic scoring function is lower than using linear scoring function, while the coverage of modified BordaRank using logarithmic scoring function is better. V. CONCLUSIONS AND FUTURE WORK In this paper, we proposed a new ranking-oriented collaborative filtering framework based on ranking aggregation methods. Differs from traditional rating-oriented collaborative filtering and ranking generation based collaborative filtering, our algorithm generates item recommendations from item rankings which are given by the neighbors of the target user. In the proposed framework, we use Borda count method as ranking aggregation method and provide BordaRank algorithm. BordaRank uses rating predict results from item collaborative filtering method and aggregates neighborhoods rankings to give item recommendations. Experimental results shows BordaRank improves precision and recall of item collaborative filtering method on a considerable scale. We further proposed modified BordaRank which is a pure ranking-oriented collaborative filtering method. The modified BordaRank could be directly applied on the sparse rating matrix. Finally, we evaluated both methods on MovieLens data set and compared the impact of different neighborhood sizes and different selections of scoring function. Both BordaRank and modified BordaRank have higher precision and recall than other baseline methods. For future work, we would like to investigate different ranking aggregation methods for our ranking aggregation based collaborative filtering framework. 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TABLE III THE PRECISION OF METHODS USING DIFFERENT SCORING FUNCTION linear 0.1017 0.0705 0.0517 0.0512 0.0536 0.0657 reciprocal 0.0213 0.0149 0.0131 0.0113 0.0135 0.0148 logarithmic 0.1031 0.0736 0.0535 0.0524 0.0552 0.0676 polynomial 0.0990 0.0669 0.0505 0.0485 0.0515 0.0633 modified linear 0.1406 0.1017 0.0786 0.0764 0.0770 0.0949 modified logarithmic 0.1020 0.0723 0.0539 0.0542 0.0551 0.0675 TABLE IV THE RECALL OF METHODS USING DIFFERENT SCORING FUNCTION linear 0.1640 0.1560 0.1601 0.1595 0.1620 0.1603 reciprocal 0.0457 0.0459 0.0470 0.0453 0.0330 0.0434 logarithmic 0.1677 0.1706 0.1692 0.1632 0.1694 0.1680 polynomial 0.1556 0.1458 0.1538 0.1518 0.1519 0.1518 modified linear 0.2388 0.2529 0.2448 0.2423 0.2486 0.2455 modified logarithmic 0.1636 0.1634 0.1702 0.1640 0.1671 0.1657 TABLE V THE COVERAGE OF METHODS USING DIFFERENT SCORING FUNCTION linear 0.1356 0.1570 0.1570 0.1623 0.1647 0.1553 reciprocal 0.3609 0.3686 0.3781 0.3674 0.3633 0.3677 logarithmic 0.1593 0.1831 0.1718 0.1795 0.1867 0.1761 polynomial 0.1385 0.1564 0.1534 0.1712 0.1653 0.1570 modified linear 0.1504 0.1522 0.1480 0.1510 0.1552 0.1514 modified logarithmic 0.1373 0.1647 0.1570 0.1659 0.1736 0.1597 [10] J. D. Lafferty, G. Lebanon, and J. D. Lafferty, Cranking: Combining Rankings Using Conditional Probability Models on Permutations, in Proceedings of the Nineteenth International Conference on Machine Learning, ser. ICML 02. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., 2002, pp. 363 370. [Online]. Available: http://dl.acm.org/citation.cfm?id=645531.655830 [11] S. Wang, J. Sun, B. J. Gao, and J. Ma, VSRank: A Novel Framework for Ranking-Based Collaborative Filtering, ACM Transactions on Intelligent Systems and Technology (TIST), vol. 5, no. 3, pp. 51:1-51:24, 2014. [Online]. Available: http://doi.acm.org/10.1145/2542048 [12] J. Herlocker, J. Konstan, and J. Riedl, An Empirical Analysis of Design Choices in Neighborhood-Based Collaborative Filtering Algroithms, Information retrieval, pp. 287 310, 2002. [Online]. Available: http://link.springer.com/article/10.1023/a:1020443909834 [13] P. Emerson, The original Borda count and partial voting, Social Choice and Welfare, vol. 40, no. 2, pp. 353 358, 2013.