Learning Rating Patterns for Top-N Recommendations

Yongli Ren, Gang Li, and Wanlei ZhouSchool of Information Technology, Deakin University

221 Burwood Highway, Vic 3125, AustraliaEmail: {yongli, gang.li, wanlei}@deakin.edu.au

Abstract—Two rating patterns exist in the 𝑢𝑠𝑒𝑟×𝑖𝑡𝑒𝑚 ratingmatrix and influence each other: the personal rating patternsare hidden in each user’s entire rating history, while theglobal rating patterns are hidden in the entire 𝑢𝑠𝑒𝑟 × 𝑖𝑡𝑒𝑚rating matrix. In this paper, a Rating Pattern Subspace isproposed to model both of the rating patterns simultaneouslyby iteratively refining each other with an EM-like algorithm.Firstly, a low-rank subspace is built up to model the globalrating patterns from the whole 𝑢𝑠𝑒𝑟 × 𝑖𝑡𝑒𝑚 rating matrix;then, the projection for each user on the subspace is refinedindividually based on his/her own entire rating history. Afterthat, the refined user projections on the subspace are used toimprove the modelling of the global rating patterns. Iteratively,we can obtain a well-trained low-rank Rating Pattern Subspace,which is capable of modelling both the personal and theglobal rating patterns. Based on this subspace, we proposea RapSVD algorithm to generate Top-𝑁 recommendations,and the experiment results show that the proposed methodcan significantly outperform the other state-of-the-art Top-𝑁recommendation methods in terms of accuracy, especially onlong tail item recommendations.

Keywords-Rating Patterns, Top-𝑁 Recommendations

I. INTRODUCTION

The explosive development of E-commerce has beenleading the customers to a new era of shopping, which ischaracterized by huge volumes of products, e.g. millions ofbooks from Amazon1, or thousands of movies from Netflix2.A following challenge is that customers are facing too manydata resources so that it is not trivial to identify thoserelevant for their desires. However, manually evaluating allthose alternatives is infeasible if not impossible. As a con-sequence, how to automatically identify the most relevantitems becomes an urgent requirement in many areas. Inthe last decade, Recommender System has emerged as a so-lution to provide personalized recommendations. However,in the field of recommendations, although there is a rapiddevelopment of research on the task of rating prediction,the majority of commercial recommender systems providea list of recommended items, which is the task of Top-𝑁recommendations [1].

Over the years, various techniques have been proposed toprovide Top-𝑁 recommendations. Generally, they can pro-duce Top-𝑁 recommendations in two steps: firstly, predict

1http://www.amazon.com2http://www.netflix.com

ratings for all candidate items; secondly, sort candidate itemsaccording to their predicted ratings. Then, the Top-𝑁 rankeditems will be recommended to the active user. Recently,Cremonesi et al. provided a comprehensive evaluation ofrecommendation algorithms on the Top-𝑁 recommendationtask [2]. They found that, it is not necessary to predict theexact ratings for the Top-𝑁 recommendation task. Instead,they proposed the PureSVD algorithm, focusing on the cor-rect ranking of items rather than the exact rating prediction,which performed better on Top-𝑁 recommendations than therating prediction methods, including the well-known SVD++model that played a key role in the winning of the Netflixprogress award [3]. Moreover, it has also been pointed outthat recommending popular items to users is trivial and cannot bring in many benefits for the service providers [2]. Onthe other hand, recommending unpopular items will bringin more benefits to both the users and the providers, butthis is normally much harder [2]. According to the well-known long-tail distribution of rated items, the majority ofratings are given on a small fraction of available items [4].For example, Fig. 1 shows the rating distribution of theMovieLens3 data set, where we observe that 33% of ratingsare observed from only around 5.5% of items. We referto these items as popular items, and the other 94.5% asunpopular items or long tail items. The difficulty is howto recommend the long tail items based on limited availableratings, which is also the issue we will address in this paper.

Another important recent finding is the existence of thepersonal rating patterns and the global rating patterns.For example, in the Second Challenge on Context-awareMovie Recommendation (CAMRa), it is required to identifyuser labels from their rating history, and Jose Bento et al.provided a solution by identifying users from their ratingpatterns [5]. For the global rating pattern, an example isfound from the Netflix competition by Yehuda Koren [6], itshows that user ratings tend to increase as the age of therated movies. However, most of these existing methods aretrained only on the available ratings by going through eachavailable rating one by one. Although they achieve goodperformance on the rating prediction task, they show limitedperformance on the Top-𝑁 recommendation task [2]. Onepossible reason is that they model the rating patterns based

3http://www.grouplens.org

2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining

978-0-7695-4799-2/12 $26.00 © 2012 IEEE

DOI 10.1109/ASONAM.2012.81

465

2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining

978-0-7695-4799-2/12 $26.00 © 2012 IEEE

DOI 10.1109/ASONAM.2012.81

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0 20% 40% 60% 80% 100%0

0.1%

1%

10%

100%

% of ratings

% o

f ite

ms

[popular] [unpopular or Long Tail]

Figure 1: rating distribution over item popularity in Movie-Lens dataset

on either the global rating patterns or the personal ratingpatterns, and ignore the following facts:∙ the personal rating patterns are preserved and reflected

from a user’s entire rating history as a whole, ratherthan from every singular rating value;

∙ the personal and the global rating patterns are bothinfluenced by each other.

In this paper, we aim to model both the personal ratingpatterns and the global rating patterns simultaneously. It isa fact that, the personal rating patterns are hidden in eachuser’s entire rating history, while the global rating patternsare embedded in the 𝑢𝑠𝑒𝑟 × 𝑖𝑡𝑒𝑚 rating matrix, and bothof them are influenced by each other. Based on this, wepropose the Rating Pattern Subspace to model them byiteratively refining each other with an EM-like algorithm.The basic idea is to build a low-rank subspace to modelthe global rating patterns from the whole 𝑢𝑠𝑒𝑟 × 𝑖𝑡𝑒𝑚rating matrix, then to individually refine the projection foreach user on the subspace based on his/her own entirerating history. After that, the refined user projections onthe subspace are used to improve the modelling of theglobal rating patterns. Iteratively, the well-trained low-ranksubspace will be able to model both the personal and theglobal rating patterns. Based on this model, we proposethe RapSVD algorithm to generate Top-𝑁 recommendations.Consequently, the proposed RapSVD algorithm can generatebetter Top-𝑁 recommendations in terms of accuracy, whencompared with the state-of-the-art algorithms. Particularlyon long tail item recommendations, it is observed that thesmaller the 𝑁 values, the larger the improvements.

The major contributions of this paper are as follows:∙ We propose a novel Rating Pattern Subspace method to

model both the personal rating patterns and the globalrating patterns.

∙ Based on the Rating Pattern Subspace, we propose anew RapSVD algorithm to improve Top-𝑁 recommen-dations.

∙ We conduct a large set of experiments to examine theperformance of the proposed RapSVD algorithm onvarious sparsity levels, and to compare it with 4 state-of-the-art Top-𝑁 recommendation algorithms.

The rest of the paper is organized as follows. In section II,a novel model, the Rating Pattern Subsapce, is proposed tomodel both the personal and the global rating patterns. Insection III, we present the experiment results and comparethe performance of the proposed model with other relevantmodels. Finally, we summarize this paper in section IV.

II. THE RATING PATTERN SUBSPACE

Let 𝒰 = {𝑢1, 𝑢2, ⋅ ⋅ ⋅ , 𝑢𝑥, ⋅ ⋅ ⋅ , 𝑢𝑚} denote a set of𝑚 users, 𝒯 = {𝑡1, 𝑡2, ⋅ ⋅ ⋅ , 𝑡𝑖, ⋅ ⋅ ⋅ , 𝑡𝑛} denote a set of𝑛 items, and 𝑟𝑥𝑖 denote the rating on item 𝑡𝑖 by user𝑢𝑥. The active user is the one for whom the recom-mendations are generated. ℛ denotes the 𝑢𝑠𝑒𝑟 × 𝑖𝑡𝑒𝑚rating matrix, and can be decomposed into a numberof row vectors: ℛ = [u1,u2, ⋅ ⋅ ⋅ ,u𝑥, ⋅ ⋅ ⋅ ,u𝑚]𝑇 whereu𝑥 = [𝑟𝑥1, 𝑟𝑥2, ⋅ ⋅ ⋅ , 𝑟𝑥𝑖, ⋅ ⋅ ⋅ , 𝑟𝑥𝑛], and 𝑇 denotes transpose.Similarly, ℛ can also be decomposed into a number ofcolumn vectors: ℛ = [t1, t2, ⋅ ⋅ ⋅ , t𝑖, ⋅ ⋅ ⋅ , t𝑛] and t𝑖 =[𝑟1𝑖, 𝑟2𝑖, ⋅ ⋅ ⋅ , 𝑟𝑥𝑖, ⋅ ⋅ ⋅ , 𝑟𝑚𝑖]

𝑇 , where each column vector t𝑖corresponds to the item 𝑡𝑖’s rating information by all users.

A. Latent Factor Model Formulation

Latent Factor Models have become an important approachto explore the hidden latent features of the 𝑢𝑠𝑒𝑟 × 𝑖𝑡𝑒𝑚rating matrix ℛ for recommendation purposes, such asLatent Dirichlet Allocation [7], Probability Latent SemanticAnalysis [8], and Principal Component Analysis [9]. Amongthese models, Singular Value Decomposition (SVD) hasattracted more attention in recent years. In this paper, wefocus on the SVD-based methods that model the ratingpatterns for the Top-𝑁 recommendation task.

Conventionally, the SVD of the 𝑢𝑠𝑒𝑟×𝑖𝑡𝑒𝑚 rating matrixℛ is the factorization of the form:

ℛ = 𝒫 ⋅ Σ ⋅ 𝒱𝑇 , (1)

where 𝒫 is an 𝑚 × 𝑚 orthogonal matrix, Σ is an 𝑚 × 𝑛diagonal matrix that contains the singular values of ℛ on thediagonal, 𝒱 is an 𝑛×𝑛 orthogonal matrix. According to theEckart-Young theorem [10], the best rank-𝑘 approximation ofmatrixℛ can be achieved by SVD. However, please note thatthis conventional SVD is defined without considering theexistence of missing values. Therefore, due to the sparsityissue, it can not be directly applied to the 𝑢𝑠𝑒𝑟 × 𝑖𝑡𝑒𝑚rating matrix ℛ. Some attempts have been made to addressthis problem, e.g. filling all the missing values with anestimator [11], [12]. However, the serious sparsity problemmakes the rating patterns hidden in ℛ highly incomplete,and therefore, may limit the modelling ability of this kindof methods.

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Recently another kind of SVD-based methods have at-tracted great attention, due to its high accuracy and ef-ficiency. A representative model of this kind is the oneproposed by Simon Funk [13]. Unlike the conventionalSVD as shown in Eq. 1, Simon Funk trained the modeldirectly only on available ratings by using a gradient descentapproach. Formally, the 𝑢𝑠𝑒𝑟 × 𝑖𝑡𝑒𝑚 rating matrix ℛ isestimated by the factorization of the form:

ℛ = 𝒫 ⋅ 𝒬𝑇 , (2)

where 𝒫 is an 𝑚 × 𝑘 matrix and 𝒬 is an 𝑛 × 𝑘 matrix,denoting the user factors and item factors, respectively,and 𝑘 is the number of factors. This method leads to arapid development of recommendation techniques for therating prediction task, including NSVD [14], SVD++ [3],timeSVD [6]. However, this kind of methods is trainedonly on the available ratings by going through each ratingone by one, and ignores the existence of missing values.Although they achieve improved performance on the ratingprediction task, they show limited performance on the Top-𝑁 recommendation task [2].

In this paper, we will focus on the modelling of ratingpatterns for the Top-𝑁 recommendation task. There are twochallenges to model them accurately:∙ Due to the sparsity issue, both the personal and the

global rating patterns are highly incomplete. The mod-elling algorithm must possess the ability to recognizethese patterns from limited available data.

∙ As these two rating patterns influence each other, themodelling algorithm must treat them collectively, notseparately.

Therefore, unlike existing models, we propose a model thatcan address these two challenges by constructing a RatingPattern Subspace, as described in details in the followingsection.

B. Learning the Rating Pattern Subspace

In this section, we will build a representative subspacefor those rating patterns, including both the personal andthe global rating patterns. One of the most successful tech-niques is SVD, which can guarantee to model a matrix bestwith the output of a low rank subspace. Due to the highincompleteness of the rating matrix ℛ, SVD can not bedirectly used to model the hidden rating patterns. Therefore,we propose the Rating Pattern Subspace to model both ofthem simultaneously by iteratively refining each other witha novel EM-like algorithm.

Denote ℛ = [u1,u2, ⋅ ⋅ ⋅ ,u𝑥, ⋅ ⋅ ⋅ ,u𝑚]𝑇 , and u𝑥 =[𝑟𝑥1, 𝑟𝑥2, ⋅ ⋅ ⋅ , 𝑟𝑥𝑖, ⋅ ⋅ ⋅ , 𝑟𝑥𝑛] is the entire rating history foruser 𝑢𝑥. Due to the fact that the personal rating patternsare hidden in a user’s rating history, in this paper, forconvenience, we define the rating history u𝑥 as the ratingpattern for user 𝑢𝑥, and u𝑥 can be divided into two parts u𝑎

𝑥

and u𝑚𝑥 for the available and missing part, respectively. To

model the rating patterns hidden in ℛ, we estimate it withSVD to construct a low rank 𝑘 subspace:

ℛ = 𝒫𝑘 ⋅ Σ𝑘 ⋅ 𝒱𝑇𝑘 , (3)

where 𝒫𝑘 is an 𝑚 × 𝑘 orthogonal matrix, Σ𝑘 is a 𝑘 × 𝑘diagonal matrix that contains the first 𝑘 singular values ofℛ, 𝒱𝑘 is an 𝑛 × 𝑘 orthogonal matrix. Consequently, thereconstruction u𝑥 of u𝑥 is defined as:

u𝑥 = p𝑥 ⋅ Σ𝑘 ⋅ 𝒱𝑇𝑘 , (4)

where p𝑥 is the 𝑥th row of 𝒫𝑘, and it is treated as theprojection (or user factors) of 𝑢𝑥. Similarly, u𝑥 can also bedivided into u𝑎

𝑥 and u𝑚𝑥 parts, representing the reconstruc-

tion of u𝑎𝑥 and u𝑚

𝑥 , respectively. It is well-known that SVDguarantees to produce the best 𝑘-rank approximation of ℛwith minimal reconstruction errors. However, in this work,due to the sparsity issue, we change the goal of modellingto minimize the reconstruction error on the available ratingdata, which is defined as the squared distance between theoriginal available data and their reconstructions:

𝜀𝑎 =1

𝑚

𝑚∑𝑥=1

(u𝑎𝑥 − u𝑎

𝑥) ⋅ (u𝑎𝑥 − u𝑎

𝑥)𝑇 , (5)

where 𝑚 is the number of users.To build the representative subspace for both the personal

rating patterns and the global rating patterns, an EM-likealgorithm is proposed as follows: firstly, for each ratingpattern u𝑥 ∈ ℛ, the missing values are replaced withcorresponding averaged available rating values in 𝜂:

𝜂 =1

𝑚

𝑚∑𝑥=1

u𝑥. (6)

Then, in the 𝑗-th iteration, the standard SVD algorithm isapplied to calculate a low-rank subspace defined by 𝒫𝑘, Σ𝑘

and 𝒱𝑘. After that, we can use p𝑥, Σ𝑘 and 𝒱𝑘 to reconstructu𝑥 with Eq. 4. However, since there are a large set of missingvalues in u𝑥, p𝑥 can not be calculated from Eq. 3. Pleasenote that it is possible to estimate p𝑥 from a part of u𝑥, e.g.the available part u𝑎

𝑥, which has been widely used in thefield of multimedia research [15], [16]. Thus, we define thatp𝑥 as the least squares solution for the following equation:

(𝒱𝑘 ⋅ Σ𝑇𝑘

) ⋅ p𝑇𝑥 = ([u𝑥]

𝑎)𝑇 , (7)

where [u𝑥]𝑎 denotes u𝑥 in the current iteration step, but only

have values on the positions that correspond to u𝑎𝑥. This

implies that the projection p𝑥 is refined by the personalrating pattern u𝑥 with only available ratings. After p𝑥 isestimated, according to Eq. 4, the reconstruction u𝑥 of u𝑥

can be calculated. u𝑚𝑥 will then be updated with u𝑚

𝑥 ,

u𝑚𝑥 ← u𝑚

𝑥 , (8)

and the new 𝜂(𝑗+1) in the next (𝑗 + 1)−th iteration willbe calculated with the updated u𝑥 according to Eq. 6.

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With the updated rating matrix and the new mean vector𝜂(𝑗+1), the standard SVD algorithm is applied again tocalculate Σ𝑘 and 𝒱𝑘. This iterative process will continueuntil either the maximum iteration number is achieved, or thethe reconstruction error 𝜀𝑎 is below a pre-defined threshold.The proof of the convergence of this training algorithm isprovided as follows.

PROOF 1: In the 𝑗th iteration, for training data u𝑗𝑥, its

reconstruction is defined as

u𝑗𝑥 = p𝑥 ⋅ (Σ𝑘 ⋅ 𝒱𝑇

𝑘 )𝑗 , (9)

where (Σ𝑘 ⋅ 𝒱𝑇𝑘 )

𝑗 denote Σ𝑘 and 𝒱𝑘 in the 𝑗th iteration.We denote u𝑗

𝑥 obtained with (Σ𝑘 ⋅ 𝒱𝑇𝑘 )

𝑗 as 𝑓 𝑗u𝑥(Σ,𝒱𝑇

𝑘 )𝑗 .According to Eq. 8, 𝑓 𝑗

u𝑥(Σ,𝒱𝑇

𝑘 )𝑗 and the data in the nextiteration, u𝑗+1

𝑥 , share values on missing positions, thus thereconstruction error on u𝑗

𝑥 is represented as:

(𝜀𝑎𝑥)𝑗 = 𝑑

(𝑓 𝑗u𝑥(Σ,𝒱𝑇

𝑘 )𝑗 ,u𝑗+1𝑥

), (10)

where 𝑑(⋅, ⋅) denotes the Euclidean distance between twovectors.

If we use (Σ,𝒱𝑇𝑘 )𝑗 to calculate the reconstruction of u𝑗+1

𝑥 ,we obtain

𝑑(𝑓 𝑗+1u𝑥

(Σ,𝒱𝑇𝑘 )𝑗 ,u𝑗+1

𝑥

)≤ 𝑑

(𝑓 𝑗u𝑥(Σ,𝒱𝑇

𝑘 )𝑗 ,u𝑗+1𝑥

), (11)

because the orthogonal property of (Σ,𝒱𝑇𝑘 )𝑗 make it sure

that 𝑓 𝑗+1u𝑥

(Σ,𝒱𝑇𝑘 )𝑗 and u𝑗+1

𝑥 have the minimum euclideandistance.

In the (𝑗+1)th iteration, after applying the standard SVDon the updated training data u𝑗+1

𝑥 , we observe the minimumreconstruction error by obtaining (Σ,𝒱𝑇

𝑘 )𝑗+1:

𝑑(𝑓 𝑗+1u𝑥

(Σ,𝒱𝑇𝑘 )𝑗+1,u𝑗+1

𝑥

)≤ 𝑑

(𝑓 𝑗+1u𝑥

(Σ,𝒱𝑇𝑘 )𝑗 ,u𝑗+1

𝑥

).

(12)For the reconstruction error in the 𝑗 + 1th iteration, weobserve

(𝜀𝑎𝑥)𝑗+1 = 𝑑

(𝑓 𝑗+1u𝑥

(Σ,𝒱𝑇𝑘 )𝑗+1,u𝑗+1

𝑥

)(13)

≤ 𝑑(𝑓 𝑗+1u𝑥

(Σ,𝒱𝑇𝑘 )𝑗 ,u𝑗+1

𝑥

)

≤ 𝑑(𝑓 𝑗u𝑥(Σ,𝒱𝑇

𝑘 )𝑗 ,u𝑗+1𝑥

)

= (𝜀𝑎𝑥)𝑗 .

So,

(𝜀𝑎)𝑗+1 =1

𝑚

𝑚∑𝑥=1

(𝜀𝑎𝑥)𝑗+1 ≤ 1

𝑚

𝑚∑𝑥=1

(𝜀𝑎𝑥)𝑗 = (𝜀𝑎)𝑗 . (14)

Thus, the algorithm will converge to minimize the recon-struction error 𝜀𝑎.

It is clear that, unlike other model-based recommendationmethods, the proposed model is trained iteratively basedon the rating matrix ℛ and the entire user rating historyu𝑥, rather than by going through individual ratings one by

one. One advantage of this is that, the proposed RatingPattern Subspace can model both of the personal and theglobal rating patterns simultaneously. Specifically, accordingto Eq. 3, 𝒫𝑘 is first obtained with SVD by decomposing theentire 𝑢𝑠𝑒𝑟× 𝑖𝑡𝑒𝑚 rating matrix ℛ where the global ratingpatterns are preserved, then it is further refined accordingto Eq. 7, based on each user’s entire rating history wherehis/her personal rating patterns are preserved.

Furthermore, the training process is actually an iterativerefinement of the global rating patterns and the personalrating patterns. In our training process, the missing valuesare first initialized with the global rating patterns, and a low-rank subspace is built to capture these patterns. Then, theprojection for each user on the subspace is further refinedindividually based on his/her own entire rating history. Afterthat, the refined projections are utilized to improve the rep-resentative subspace again. Iteratively, we can obtain a well-trained representative subspace, Rating Pattern Subspace,which captures both the personal and global rating patterns.

C. Recommendation Generation

After obtaining the well-trained Rating Pattern Subspace,we propose the RapSVD algorithm to make Top-𝑁 recom-mendations.

Denote 𝒬 = Σ ⋅ 𝒱𝑇 . In Eq. 1, as 𝒫 is a unitary matrix,we obtain that:

𝒬 = Σ ⋅ 𝒱𝑇 = 𝒫𝑇 ⋅ ℛ, (15)

where ℛ is the rating matrix. Therefore, according to Eq. 4and Eq. 15,

u𝑥 = p𝑥 ⋅ Σ𝑘 ⋅ 𝒱𝑇𝑘 (16)

= p𝑥 ⋅ 𝒬= p𝑥 ⋅ 𝒫𝑇

𝑘 ⋅ ℛ.

Then, we can estimate how well user 𝑢𝑥 likes item 𝑡𝑖 by:

𝑟𝑥𝑖 = p𝑥 ⋅ 𝒫𝑇𝑘 ⋅ t𝑖, (17)

where 𝒫𝑇𝑘 is the projection of users on the Rating Pattern

Subspace, p𝑥 is the 𝑥th row of 𝒫𝑘 denoting the projectionfor 𝑢𝑥, and t𝑖 is the rating information on 𝑡𝑖 by all the users.We refer this method as the RapSVD algorithm. Please notethat 𝑟𝑥𝑖 here is not the rating on 𝑡𝑖 by user 𝑢𝑥, as we donot scale it into the correct rating range. We use it as theassociation measure on 𝑡𝑖 by user 𝑢𝑥, as in literature [2].After calculating 𝑟𝑥𝑖 for all unknown items for user 𝑢𝑥, theseitems will be sorted accordingly, then the Top-𝑁 rankeditems will form the Top-𝑁 recommendations for user 𝑢𝑥.

III. EXPERIMENT AND ANALYSIS

In this section, we conduct experiments to examine theperformance of the proposed RapSVD algorithm, and willspecifically answer the following questions:

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1) How does the proposed RapSVD algorithm perform onTop-𝑁 recommendations? For this question, since it istrivial to recommend popular items [2], we will eval-uate the performance of algorithms on recommendingun-popular (long tail) items. Specifically, we evaluatethe accuracy of Top-𝑁 recommendation on both allitems and long tail items in the following experiments.

2) How does the proposed RapSVD algorithm performcompared with the state-of-the-art algorithms? To an-swer this question, we compare the proposed RapSVDalgorithm with 4 other Top-𝑁 recommendation al-gorithms, including two Top-𝑁 oriented algorithms,PureSVD and NNcosNgbr [2], and two popular non-personalized models, Top Popular (TopPop) [2], [17]and Movie Average (MovieAvg) [3].

3) How does the proposed RapSVD algorithm performon sparse datasets? For this question, we examinethe RapSVD algorithm on a series of datasets withdifferent sparsity levels by varying the percentage ofthe user’s available ratings from 10% to 100% witha 10% step. Then, we investigate its performance onrecommending both all items and long tail items.

A. Data Set

The data set we experiment with is the popular MovieLens4, which is collected by the GroupLens Group from theUniversity of Minnesota, and has been widely used inthe recommendation research. MovieLens includes around 1million ratings collected from 6, 040 users on 3, 900 movies.

The data set is split into two subsets, the training setand the test set. The test set only contains 5-star ratings.Following literature [3], [2], [17], we reasonably assume thatthe 5-star rated items are relevant to the active user, and usea similar methodology to conduct experiments. Specifically,we randomly select 2% ratings and use all the 5-star ratingsto form the test set, but make sure that at least one 5-star rating exists for each individual user. The remainingratings in the data set form the training set. To examinethe performance of algorithms on sparse dataset, we keepthe test set the same, but vary the percentage of observedratings for each user, from 10% to 100% with a 10%step. This set of configurations is represented as Given10%,Given20%, Given30%, Given40%, Given50%, Given60%,Given70%, Given80%, Given90%, Given100%, respectively.In addition, after training the model on the training set, werandomly select 1000 additional items that are not rated bythe active user, then predict ratings on the test item and theseadditional 1000 selected items, then rank them and select thetop ranked 𝑁 items as the Top-𝑁 recommendation list forthe active user. This testing methodology is popular in theresearch of Top-𝑁 recommendations [3], [2], [17].

4http://www.grouplens.org

0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

N

reca

ll

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Figure 2: Recall at 𝑁 on long tail

B. Comparison and Evaluation

To examine the performance of the proposed RapSVDalgorithm, we compare it with 4 other state-of-the-art Top-𝑁recommendation algorithms, including two Top-𝑁 orientedalgorithms, PureSVD and NNcosNgbr [2], and two popularnon-personalized models, Top Popular (TopPop) [2], [17]and Movie Average (MovieAvg) [3]. Specifically, the numberof the nearest neighbours in NNcosNgbr is set to 200; thenumber of factors in PureSVD is set to 50. For the proposedalgorithm RapSVD, the dimension of Rating Pattern Sub-space is set to 50, the max iteration of training is set to 50,and the error threshold is set to 10−6.

The quality of Top-𝑁 recommendations is measured bythe Recall, which is defined based on the overall numberof hits. For the active user, if the test item is in the Top-𝑁recommendation list, we call this a hit. The overall Recallis defined as follows:

𝑟𝑒𝑐𝑎𝑙𝑙 =𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ℎ𝑖𝑡𝑠

∣𝑋∣ , (18)

where 𝑋 is the test set. A higher recall value means betterTop-𝑁 recommendations.

C. Comparison on long tail Item Recommendations

As it is trivial to recommend popular items and a few toppopular items can skew the performance of algorithms [2],

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Table I: Recall when 𝑁 = 10 and 𝑁 = 20 on long tail Items

𝐺𝑖𝑣𝑒𝑛10% 𝐺𝑖𝑣𝑒𝑛20% 𝐺𝑖𝑣𝑒𝑛30% 𝐺𝑖𝑣𝑒𝑛40% 𝐺𝑖𝑣𝑒𝑛50%Algorithm 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20RapSVD 0.0378 0.0940 0.0663 0.1452 0.0991 0.1990 0.1355 0.2560 0.1665 0.2968PureSVD 0.0299 0.0818 0.0525 0.1311 0.0848 0.1815 0.1176 0.2353 0.1474 0.2751NNcosNgbr 0.0230 0.0598 0.0371 0.0785 0.0552 0.1141 0.1037 0.1949 0.1156 0.2257TopPop 0.0000 0.0000 0.0000 0.0001 0.0000 0.0001 0.0000 0.0001 0.0000 0.0001MovieAvg 0.0036 0.0232 0.0073 0.0367 0.0095 0.0431 0.0165 0.0586 0.0169 0.0560

𝐺𝑖𝑣𝑒𝑛60% 𝐺𝑖𝑣𝑒𝑛70% 𝐺𝑖𝑣𝑒𝑛80% 𝐺𝑖𝑣𝑒𝑛90% 𝐺𝑖𝑣𝑒𝑛100%Algorithm 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20RapSVD 0.2021 0.3464 0.2331 0.3812 0.2725 0.4252 0.3132 0.4640 0.3622 0.5086PureSVD 0.1801 0.3198 0.2091 0.3568 0.2515 0.4035 0.2988 0.4502 0.3546 0.4987NNcosNgbr 0.1429 0.2531 0.1872 0.3034 0.2272 0.3562 0.2666 0.3889 0.3293 0.4518TopPop 0.0000 0.0002 0.0000 0.0002 0.0001 0.0003 0.0001 0.0029 0.0013 0.0096MovieAvg 0.0245 0.0673 0.0281 0.0716 0.0289 0.0745 0.0303 0.0755 0.0318 0.0818

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Figure 3: Improvement on long tail Recommendations

we will focus on evaluating the performance of algorithmson recommending long tail items.

Table. I shows the recall performance of examined al-gorithms on all Given data sets, when 𝑁 equals to 10and 20, respectively. We observe that RapSVD significantlyoutperforms all the compared algorithms on all sparsitylevels. Specifically, when 𝑁 = 10, on Given60% data set,RapSVD achieves a recall of 0.2021 that outperforms thebest compared result 0.1801 (from PureSVD) by 12.22%;when 𝑁 = 10, on Given40% data set, RapSVD obtains arecall of 0.1355 that outperforms the best compared result0.1176 (from PureSVD) by 15.22%. Furthermore, whenplotting the improvements across all sparse levels (all Givendata set), we observe that the sparser the training data thelarger the improvement. As shown in Fig. 3, it is clearthat the improvement increases as the training data becomesparser, e.g. when 𝑁 = 10, this improvement on 𝐺𝑖𝑣𝑒𝑛80%is 8.35%, and it increases to 26.29% on 𝐺𝑖𝑣𝑒𝑛20% data set.The main reason is that, when the training data is sparser,both the personal and the global rating patterns will behighly incomplete, and solely modelling either one of themcan not generate good results. In this case, the proposediterative refinement process between them will bring in morebenefits to the modelling of rating patterns. Therefore, thewell-trained Rating Pattern Subspace can produce betterresults. Moreover, it is also observed that the improvements

when 𝑁 = 10 are larger than those when 𝑁 = 20, whichmeans RapSVD can achieve larger improvements on smaller𝑁 values. This is very practical, since the research on howusers scan ranked results shows that, the first several rankeditems in the results attract much more attention than theothers [18], [19]. Therefore, the significant improvement onsmaller 𝑁 values means larger improvement in terms of usersatisfaction.

Next, in order to examine the performance of RapSVDthoroughly, we vary the value of 𝑁 from 1 to 20, and reportthe results on 𝐺𝑖𝑣𝑒𝑛30% and 𝐺𝑖𝑣𝑒𝑛60% datasets as shownin Fig. 2. It is observed that RapSVD can outperform all theother compared algorithms on all of 𝑁 values. On the otherdatasets, RapSVD can also outperform all the comparedalgorithms again on all 𝑁 values. This indicates that theproposed RapSVD algorithm is robust, and can get betterperformance on all sparsity levels. This is very helpful,because the sparsity issue is one of the biggest challenges inthe research of recommendations, and normally the densityof the 𝑢𝑠𝑒𝑟 × 𝑖𝑡𝑒𝑚 rating matrix is around 1% [20].

D. Comparison on all items Recommendations

To fully examine the performance of the RapSVD algo-rithm, we also conduct experiments on recommending allitems, including popular and unpopular (long tail) items.

Table. II reports the performance of compared algorithmson Given data sets. We observe that RapSVD achieves thebest results on all data sets, except on 𝐺𝑖𝑣𝑒𝑛10% and𝐺𝑖𝑣𝑒𝑛20% data sets where the non-personalized algorithmTopPop shows unexpected good performance. The reasonof this might be that when the training set is too sparse, itwould be hard for personalized algorithms to be trained well.Therefore, the popular items in the test set will skew the per-formance of algorithms [2], and make the non-personalizedTopPop achieve good results. However, although TopPopachieves good performance on all items when the trainingset is sparse, it performs much worse on recommending longtail items on the same data set as shown in Table. I, whichis consistent with the findings from Literature [2].

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Table II: Recall when 𝑁 = 10 and 𝑁 = 20 on all items

𝐺𝑖𝑣𝑒𝑛10% 𝐺𝑖𝑣𝑒𝑛20% 𝐺𝑖𝑣𝑒𝑛30% 𝐺𝑖𝑣𝑒𝑛40% 𝐺𝑖𝑣𝑒𝑛50%Algorithm 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20RapSVD 0.1436 0.2272 0.1920 0.2927 0.2395 0.3564 0.2878 0.4175 0.3322 0.4685PureSVD 0.1310 0.2102 0.1716 0.2701 0.2115 0.3205 0.2471 0.3698 0.2840 0.4131NNcosNgbr 0.2027 0.2990 0.2052 0.3095 0.2096 0.3248 0.2532 0.3709 0.2617 0.3818TopPop 0.2088 0.3080 0.2247 0.3222 0.2256 0.3251 0.2332 0.3320 0.2413 0.3410MovieAvg 0.0036 0.0443 0.0152 0.0812 0.0235 0.1003 0.0353 0.1198 0.0451 0.1253

𝐺𝑖𝑣𝑒𝑛60% 𝐺𝑖𝑣𝑒𝑛70% 𝐺𝑖𝑣𝑒𝑛80% 𝐺𝑖𝑣𝑒𝑛90% 𝐺𝑖𝑣𝑒𝑛100%Algorithm 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20 𝑁 = 10 𝑁 = 20RapSVD 0.3867 0.5283 0.4346 0.5757 0.4808 0.6199 0.5260 0.6576 0.5703 0.6912PureSVD 0.3267 0.4592 0.3721 0.5090 0.4275 0.5640 0.4939 0.6226 0.5534 0.6709NNcosNgbr 0.2796 0.3905 0.3112 0.4156 0.3374 0.4426 0.3558 0.4553 0.4036 0.4962TopPop 0.2484 0.3480 0.2559 0.3564 0.2661 0.3658 0.2717 0.3724 0.2860 0.3857MovieAvg 0.0585 0.1456 0.0710 0.1631 0.0691 0.1629 0.0691 0.1613 0.0742 0.1734

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Figure 4: Recall at 𝑁 on all items

On the other hand, we can also observe that as thedensity of the training set increases, the personalized al-gorithm begins to show their recommendation ability. Forexample, when 𝑁 = 20, TopPop achieves a recall at0.3080 and 0.3857 on 𝐺𝑖𝑣𝑒𝑛10% and 𝐺𝑖𝑣𝑒𝑛100%, re-spectively; however, RapSVD achieves a recall at 0.2272and 0.6912 on 𝐺𝑖𝑣𝑒𝑛10% and 𝐺𝑖𝑣𝑒𝑛100%, respectively.Moreover, RapSVD can outperform all the other comparedalgorithms in this case. Specifically, on 𝐺𝑖𝑣𝑒𝑛30% when𝑁 = 20, RapSVD obtains a recall 0.3564, and outperformsthe best compared result 0.3251 (from TopPop) by 9.63%;on 𝐺𝑖𝑣𝑒𝑛60% when 𝑁 = 20, RapSVD obtains a recall

0.5283, and outperforms the best compared result 0.4592(from PureSVD) by 15.05%. This indicates that the proposedRapSVD algorithm can also achieve better performance onrecommending all items than those state-of-the-art person-alized algorithms.

Clearly, as shown in Table. II, we can see that RapSVDoutperforms the other algorithms on 8 out of 10 datasets.Specifically, RapSVD achieves the best recalls on data setsfrom 𝐺𝑖𝑣𝑒𝑛30% to 𝐺𝑖𝑣𝑒𝑛100%, while TopPop obtainsbest performance on 𝐺𝑖𝑣𝑒𝑛10% and 𝐺𝑖𝑣𝑒𝑛20% datasets.However, as shown in Table I, Fig. 2a and Fig. 2b, TopPopbecomes almost useless in recommending long tail items.This indicates that TopPop is not a capable algorithm,considering the fact that it can only recommend popularitems. Again, to examine the performance of algorithmsthoroughly, we vary the 𝑁 values from 1 to 20, and reportthe results on 𝐺𝑖𝑣𝑒𝑛60% and 𝐺𝑖𝑣𝑒𝑛70% datasets in Fig. 4.We observe that RapSVD can obtain better performance thanall compared algorithms across all the 𝑁 values, e.g. on the𝐺𝑖𝑣𝑒𝑛60% data set, RapSVD obtains larger recalls than allthe other algorithms from 𝑁 = 1 to 𝑁 = 20. For the otherdata sets from 𝐺𝑖𝑣𝑒𝑛30% to 𝐺𝑖𝑣𝑒𝑛100%, RapSVD can alsoachieve better performance than all compared algorithms onall of 𝑁 values. This indicates that the recommendations onthe popular items can also benefit from the proposed RatingPattern Subspace.

IV. CONCLUSION

In this paper, a novel model named as Rating PatternSubspace, is proposed to model both the personal and theglobal rating patterns simultaneously. The method works inan iterative fashion between modelling the personal ratingpatterns and the global rating patterns, and the quality ofmodelling is improved in successive iterations. It is provedthat the training process for the Rating Pattern Subspaceconverges. The basic idea is that, the sparsity issue makesboth the personal and the global rating patterns incomplete,so modelling either one of them will be limited. Due to afact that they actually influence each other, we propose anEM-like algorithm to refine them iteratively. Based on the

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proposed Rating Pattern Subspace, the RapSVD algorithmis further proposed for Top-𝑁 recommendations. The ex-periment results show that the RapSVD algorithm achievesvery good performance on long tail item recommendation-s, as well as on all items recommendations, significantlyoutperforming the state-of-the-art Top-𝑁 recommendationalgorithms. Particularly on long tail item recommendations,we observe that for smaller 𝑁 values, the improvement ismore obvious. Moreover, RapSVD shows its robustness onthe sparsity issue which is one of the biggest challengesto the recommendation techniques. The success of RapSVDalgorithm and the Rating Pattern Subspace model has con-tributed to a number of new techniques and ideas that areintroduced in this paper. In particular, three contributions arelisted as follows:

∙ For the first time, the personal rating patterns that arehidden in a user’s entire rating history are used to refinethe modelling of the global rating patterns.

∙ An effective training algorithm is proposed to learn arepresentative subspace that captures both the personaland the global rating patterns simultaneously.

∙ Based on the Rating Pattern Subspace, an efficientalgorithm RapSVD is proposed to make Top-𝑁 rec-ommendations.

In the future, we plan to investigate how to improve thequality of Rating Pattern Subspace further to make betterTop-𝑁 recommendations.

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