A Comprehensive Survey of Neighborhood-Based Recommendation Methods

Abstract

Among collaborative recommendation approaches, methods based on nearest-neighbors still enjoy a huge amount of popularity,
due to their simplicity, their efficiency, and their ability to produce accurate and personalized recommendations. This chapter
presents a comprehensive survey of neighborhood-based methods for the item recommendation problem. In particular, the main
benefits of such methods, as well as their principal characteristics, are described. Furthermore, this document addresses
the essential decisions that are required while implementing a neighborhood-based recommender system, and gives practical
information on how to make such decisions. Finally, the problems of sparsity and limited coverage, often observed in large
commercial recommender systems, are discussed, and a few solutions to overcome these problems are presented.

Full text

A comprehensive survey of neighborhood-based
recommendation methods
Christian Desrosiers and George Karypis
Abstract Among collaborative recommendation approaches, methods based on
nearest-neighbors still enjoy a huge amount of popularity, due to their simplicity,
their efficiency, and their ability to produce accurate and personalized recommenda-
tions. This chapter presents a comprehensive survey of neighborhood-based meth-
ods for the item recommendation problem. In particular, the main benefits of such
methods, as well as their principal characteristics, are described. Furthermore, this
document addresses the essential decisions that are required while implementing a
neighborhood-based recommender system, and gives practical information on how
to make such decisions. Finally, the problems of sparsity and limited coverage, of-
ten observed in large commercial recommender systems, are discussed, and a few
solutions to overcome these problems are presented.
1 Introduction
The appearance and growth of online markets has had a considerable impact on
the habits of consumers, providing them access to a greater variety of products and
information on these goods. While this freedom of purchase has made online com-
merce into a multi-billion dollar industry, it also made it more difficult for con-
sumers to select the products that best fit their needs. One of the main solutions
proposed for this information overload problem are recommender systems, which
provide automated and personalized suggestions of products to consumers. Rec-
ommender systems have been used in a wide variety of applications, such as the
Christian Desrosiers
Computer Science & Eng., University of Minnesota, Twin Cities, e-mail: desros@cs.umn.edu
George Karypis
Computer Science & Eng., University of Minnesota, Twin Cities e-mail: karypis@cs.umn.
edu
1

2 Christian Desrosiers and George Karypis
recommendation of books and CDs [47, 53], music [45, 70], movies [31, 51, 55],
news [6, 41, 76], jokes [23], and web pages [3, 52].
The recommendation problem can be defined as estimating the response of a user
for new items, based on historical information stored in the system, and suggesting
to this user novel and original items for which the predicted response is high. The
type of user-item responses varies from one application to the next, and falls in one
of three categories: scalar, binary and unary. Scalar responses, also known as rat-
ings, are numerical (e.g., 1-5 stars) or ordinal (e.g., strongly agree, agree, neutral,
disagree, strongly disagree) values representing the possible levels of appreciation
of users for items. Binary responses, on the other hand, only have two possible
values encoding opposite levels of appreciation (e.g., like/dislike or interested/not
interested). Finally, unary responses capture the interaction of a user with an item
(e.g., purchase, online access, etc.) without giving explicit information on the appre-
ciation of the user for this item. Since most users tend to interact with items that they
find interesting, unary responses still provide useful information on the preferences
of users.
The way in which user responses are obtained can also differ. For instance, in a
movie recommendation application, users can enter ratings explicitly after watching
a movie, giving their opinion on this movie. User responses can also be obtained im-
plicitly from purchase history or access patterns [41, 76]. For example, the amount
of time spent by a user browsing a specific type of item, can be used as an indicator
of the user’s interest for this item type. For the purpose of simplicity, from this point
on, we will call rating any type of user-item response.
1.1 Formal definition of the problem
In order to give a formal definition of the item recommendation task, we need to
introduce some notation. Thus, the set of users in the system will be denoted by U,
and the set of items by I. Moreover, we denote by Rthe set of ratings recorded
in the system, and write Sthe set of possible values for a rating (e.g., S= [1,5]or
S={like,dislike}). Also, we suppose that no more than one rating can be made
by any user u∈ U for a particular item i∈ I and write rui this rating. To identify
the subset of users that have rated an item i, we use the notation Ui. Likewise, Iu
represents the subset of items that have been rated by a user u. Finally, the items
that have been rated by two users uand v, i.e. Iu∩Iv, is an important concept in our
presentation, and we use Iuv to denote this concept. In a similar fashion, Ui j is used
to denote the set of users that have rated both items iand j.
Two of the most important problems associated with recommender systems are
the best item and top-N recommendation problems. The first problem consists in
finding, for a particular user u, the new item i∈ I \ Iufor which uis most likely
to be interested in. When ratings are available, this task is most often defined as a
regression or (multi-class) classification problem where the goal is to learn a func-
tion f:U × I → S that predicts the rating f(u,i)of a user ufor a new item i. This

A comprehensive survey of neighborhood-based recommendation methods 3
function is then used to recommend to the active user uaan item i∗for which the
estimated rating has the highest value:
i∗=argmax
j∈I\Iu
f(ua,j).(1)
Accuracy is commonly used to evaluate the performance of the recommendation
method. Typically, the ratings Rare divided into a training set Rtrain used to learn
f, and a test set Rtest used to evaluate the prediction accuracy. Two popular measures
of accuracy are the Mean Absolute Error (MAE):
MAE(f) = 1
|Rtest|∑
rui∈Rtest
|f(u,i)−rui|,(2)
and the Root Mean Squared Error (RMSE):
RMSE(f) = s1
|Rtest|∑
rui∈Rtest
(f(u,i)−rui)2.(3)
When ratings are not available, for instance, if only the list of items purchased by
each user is known, measuring the rating prediction accuracy is not possible. In
such cases, the problem of finding the best item is usually transformed into the task
recommending to an active user uaa list L(ua)containing Nitems likely to interest
him or her [18, 66]. The quality of such method can be evaluated by splitting the
items of Iinto a set Itrain, used to learn L, and a test set Itest . Let T(u)⊂ Iu∩ Itest
be the subset of test items that a user ufound relevant. If the user responses are
binary, these can be the items that uhas rated positively. Otherwise, if only a list of
purchased or accessed items is given for each user u, then these items can be used
as T(u). The performance of the method is then computed using the measures of
precision and recall:
Precision(L) = 1
|U| ∑
u∈U
|L(u)∩T(u)|/|L(u)|(4)
Recall(L) = 1
|U| ∑
u∈U
|L(u)∩T(u)|/|T(u)|.(5)
A drawback of this task is that all items of a recommendation list L(u)are considered
equally interesting to user u. An alternative setting, described in [18], consists in
learning a function Lthat maps each user uto a list L(u)where items are ordered
by their “interestingness” to u. If the test set is built by randomly selecting, for each
user u, a single item iuof Iu, the performance of Lcan be evaluated with the Average
Reciprocal Hit-Rank (ARHR):
ARHR(L) = 1
|U| ∑
u∈U
1
rank(iu,L(u)),(6)

4 Christian Desrosiers and George Karypis
where rank(iu,L(u)) is the rank of item iuin L(u), equal to ∞if iu6∈ L(u). More
details can be found in [18].
1.2 Overview of recommendation approaches
While the recommendation problem truly emerged as an independent area of re-
search in the mid 1990’s, it has deeper roots in several other fields like cognitive
science [61] and information retrieval [65]. Approaches for this problem are nor-
mally divided in two broad categories: content-based and collaborative filtering ap-
proaches.
1.2.1 Content-based approaches
The general principle of content-based (or cognitive) approaches [3, 8, 44, 58] is to
identify the common characteristics of items that have received a favorable rating
from a user u, and then recommend to unew items that share these characteristics.
In content-based recommender systems, rich information describing the nature of
each item iis assumed to be available in the form of a feature vector xi. For items in
the form of text documents, such as news articles [8, 44] or Web documents [3, 58],
this vector often contains the Term Frequency-Inverse Document Frequency (TF-
IDF) [65] weights of the most informative keywords. Moreover, for each user u, a
preference profile vector xuis usually obtained from the contents of items of Iu.
A technique to compute these profiles, used in several content-based recommender
systems such as Newsweeder [44] and Fab [3], is the Rocchio algorithm [63, 12].
This technique updates the profile xuof user uwhenever this user rates an item iby
adding the weights of xito xu, in proportion to the appreciation of ufor i:
xu=∑
i∈Iu
rui xi.
The user profiles can then be used to recommend new items to a user u, by sug-
gesting the item whose feature vector xiis most similar to the profile vector xu, for
example, using the cosine similarity [3, 8, 44] or the Minimum Description Length
(MDL) [44, 62]. This approach can also be used to predict the rating of user ufor
a new item i[44], by building for each rating value r∈ S a content profile vector
x(r)
uas the average of the feature vectors of items that have received this rating value
from u. The predicted rating ˆrui for item iis the value rfor which x(r)
uis most similar
to xi. Bayesian approaches have also been proposed to predict ratings [8, 53, 58].
In these approaches, the probability of an item ito receive a rating rui from user u,
given this user’s profile vector xu, is computed as
Pr(rui|xu) = Pr(rui )∏
j∈xi
Pr(xi j|rui ).

A comprehensive survey of neighborhood-based recommendation methods 5
The values of Pr(rui)and Pr(xi j |rui)are usually estimated from the underlying data,
and the predicted rating ˆrui is the one for which Pr(rui|xu)is maximized.
Recommender systems based purely on content generally suffer from the prob-
lems of limited content analysis and over-specialization [70]. Limited content anal-
ysis stems from the fact that the system may have only a limited amount of informa-
tion on its users or the content of its items. The reasons for this lack of information
can be numerous. For instance, privacy issues might refrain a user from providing
personal information, or the precise content of items may be difficult or costly to
obtain for some types of items, such as music or images. Finally, the content of an
item is often insufficient to determine its quality. For example, it may be impossi-
ble to distinguish between a well written and a badly written article if both use the
same terms. Over-specialization, on the other hand, is a side effect of the way in
which content-based systems recommend new items, where the predicted rating of
a user for an item is high if this item is similar to the ones liked by this user. For
example, in a movie recommendation application, the system may recommend to a
user a movie of the same genre or having the same actors as movies already seen by
this user. Because of this, the system may fail to recommend items that are different
but still interesting to the user. Solutions proposed for this problem include adding
some randomness [71] or filtering out items that are too similar [8, 77].
1.2.2 Collaborative filtering approaches
Unlike content-based approaches, which use the content of items previously rated
by a user u, collaborative (or social) filtering approaches [18, 31, 41, 47, 60, 66, 70]
rely on the ratings of uas well as those of other users in the system. The key idea is
that the rating of ufor a new item iis likely to be similar to that of another user v,
if uand vhave rated other items in a similar way. Likewise, uis likely to rate two
items iand jin a similar fashion, if other users have given similar ratings to these
two items.
Collaborative approaches overcome some of the limitations of content-based
ones. For instance, items for which the content is not available or difficult to ob-
tain can still be recommended to users through the feedback of other users. Fur-
thermore, collaborative recommendations are based on the quality of items as eval-
uated by peers, instead of relying on content that may be a bad indicator of quality.
Finally, unlike content-based systems, collaborative filtering ones can recommend
items with very different content, as long as other users have already shown interest
for these different items.
Following [1, 5, 10, 18], collaborative filtering methods can be grouped in the
two general classes of neighborhood and model-based methods. In neighborhood-
based (memory-based [10] or heuristic-based [1]) collaborative filtering [17, 18,
31, 41, 47, 54, 60, 66, 70], the user-item ratings stored in the system are directly
used to predict ratings for new items. This can be done in two ways known as user-
based or item-based recommendation. User-based systems, such as GroupLens [41],
Bellcore video [31], and Ringo [70], evaluate the interest of a user ufor an item i

6 Christian Desrosiers and George Karypis
using the ratings for this item by other users, called neighbors, that have similar
rating patterns. The neighbors of user uare typically the users vwhose ratings on
the items rated by both uand v, i.e. Iuv, are most correlated to those of u. Item-based
approaches [18, 47, 66], on the other hand, predict the rating of a user ufor an item
ibased on the ratings of ufor items similar to i. In such approaches, two items are
similar if several users of the system have rated these items in a similar fashion.
In contrast to neighborhood-based systems, which use the stored ratings directly
in the prediction, model-based approaches use these ratings to learn a predictive
model. The general idea is to model the user-item interactions with factors repre-
senting latent characteristics of the users and items in the system, like the pref-
erence class of users and the category class of items. This model is then trained
using the available data, and later used to predict ratings of users for new items.
Model-based approaches for the task of recommending items are numerous and
include Bayesian Clustering [10], Latent Semantic Analysis [32], Latent Dirichlet
Allocation [9], Maximum Entropy [78], Boltzmann Machines [64], Support Vector
Machines [27], and Singular Value Decomposition [4, 42, 57, 74, 75]. A survey of
state-of-the-art model-based methods can be found in Chapter ?? of this book.
1.3 Advantages of neighborhood approaches
While recent investigations show state-of-the-art model-based approaches superior
to neighborhood ones in the task of predicting ratings [42, 73], there is also an
emerging understanding that good prediction accuracy alone does not guarantee
users an effective and satisfying experience [30]. Another factor that has been iden-
tified as playing an important role in the appreciation of users for the recommender
system is serendipity [30, 66]. Serendipity extends the concept of novelty by helping
a user find an interesting item he might not have otherwise discovered. For exam-
ple, recommending to a user a movie directed by his favorite director constitutes
a novel recommendation if the user was not aware of that movie, but is likely not
serendipitous since the user would have discovered that movie on his own.
Model-based approaches excel at characterizing the preferences of a user with
latent factors. For example, in a movie recommender system, such methods may de-
termine that a given user is a fan of movies that are both funny and romantic, without
having to actually define the notions “funny” and “romantic”. This system would be
able to recommend to the user a romantic comedy that may not have been known
to this user. However, it may be difficult for this system to recommend a movie
that does not quite fit this high-level genre, for instance, a funny parody of horror
movies. Neighborhood approaches, on the other hand, capture local associations in
the data. Consequently, it is possible for a movie recommender system based on this
type of approach to recommend the user a movie very different from his usual taste
or a movie that is not well known (e.g. repertoire film), if one of his closest neigh-
bors has given it a strong rating. This recommendation may not be a guaranteed

A comprehensive survey of neighborhood-based recommendation methods 7
success, as would be a romantic comedy, but it may help the user discover a whole
new genre or a new favorite actor/director.
The main advantages of neighborhood-based methods are:
•Simplicity: Neighborhood-based methods are intuitive and relatively simple to
implement. In their simplest form, only one parameter (the number of neighbors
used in the prediction) requires tuning.
•Justifiability: Such methods also provide a concise and intuitive justification for
the computed predictions. For example, in item-based recommendation, the list
of neighbor items, as well as the ratings given by the user to these items, can be
presented to the user as a justification for the recommendation. This can help the
user better understand the recommendation and its relevance, and could serve as
basis for an interactive system where users can select the neighbors for which a
greater importance should be given in the recommendation [4].
•Efficiency: One of the strong points of neighborhood-based systems are their
efficiency. Unlike most model-based systems, they require no costly training
phases, which need to be carried at frequent intervals in large commercial ap-
plications. While the recommendation phase is usually more expensive than for
model-based methods, the nearest-neighbors can be pre-computed in an offline
step, providing near instantaneous recommendations. Moreover, storing these
nearest neighbors requires very little memory, making such approaches scalable
to applications having millions of users and items.
•Stability: Another useful property of recommender systems based on this ap-
proach is that they are little affected by the constant addition of users, items and
ratings, which are typically observed in large commercial applications. For in-
stance, once item similarities have been computed, an item-based system can
readily make recommendations to new users, without having to re-train the sys-
tem. Moreover, once a few ratings have been entered for a new item, only the
similarities between this item and the ones already in the system need to be com-
puted.
1.4 Objectives and outline
This chapter has two main objectives. It first serves as general guide on neighborhood-
based recommender systems, and presents practical information on how to im-
plement such recommendation approaches. In particular, the main components of
neighborhood-based methods will be described, as well as the benefits of the most
common choices for each of these components. Secondly, it presents more spe-
cialized techniques on the subject that address particular aspects of recommending
items, such as data sparsity. Although such techniques are not required to imple-
ment a simple neighborhood-based system, having a broader view of the various

8 Christian Desrosiers and George Karypis
difficulties and solutions for neighborhood methods may help making appropriate
decisions during the implementation process.
The rest of this document is structured as follows. In Secton 2, the principal
neighborhood approaches, predicting user ratings for new items based on regres-
sion or classification, are introduced, and the main advantages and flaws of these
approaches are described. This section also presents two complementary ways of
implementing such approaches, either based on user or item similarities, and anal-
yses the impact of these two implementations on the accuracy, efficiency, stability,
justfiability ans serendipity of the recommender system. Section 3, on the other
hand, focuses on the three main components of neighborhood-based recommenda-
tion methods: rating normalization, similarity weight computation, and neighbor-
hood selection. For each of these components, the most common approaches are
described, and their respective benefits compared. In Section 4, the problems of
limited coverage and data sparsity are introduced, and several solutions proposed to
overcome these problems are described. In particular, several techniques based on
dimensionality reduction and graphs are presented. Finally, the last section of this
document summarizes the principal characteristics and methods of neighorhood-
based recommendation, and gives a few more pointers on implementing such meth-
ods.
2 Neighborhood-based recommendation
Recommender systems based on neighborhood automate the common principle of
word-of-mouth, where one relies on the opinion of like-minded people or other
trusted sources to evaluate the value of an item (movie, book, articles, album, etc.)
according to his own preferences. To illustrate this, consider the following example
based on the ratings of Figure 1.
Example 1 User Eric has to decide whether or not to rent the movie “Titanic” that
he has not yet seen. He knows that Lucy has very similar tastes when it comes to
movies, as both of them hated “The Matrix” and loved “Forrest Gump”, so he asks
her opinion on this movie. On the other hand, Eric finds out he and Diane have
different tastes, Diane likes action movies while he does not, and he discards her
opinion or considers the opposite in his decision.
2.1 User-based rating prediction
User-based neighborhood recommendation methods predict the rating rui of a user
ufor a new item iusing the ratings given to iby users most similar to u, called
nearest-neighbors. Suppose we have for each user v6=ua value wuv representing the
preference similarity between uand v(how this similarity can be computed will be

A comprehensive survey of neighborhood-based recommendation methods 9
The Titanic Die Forrest Wall-E
Matrix Hard Gump
John 5 1 2 2
Lucy 1 5 2 5 5
Eric 2 ? 3 5 4
Diane 4 3 5 3
Fig. 1 A “toy example” showing the ratings of four users for five movies.
discussed in Section 3.2). The k-nearest-neighbors (k-NN) of u, denoted by N(u),
are the kusers vwith the highest similarity wuv to u. However, only the users who
have rated item ican be used in the prediction of rui, and we instead consider the
kusers most similar to uthat have rated i. We write this set of neighbors as Ni(u).
The rating rui can be estimated as the average rating given to iby these neighbors:
ˆrui =1
|Ni(u)|∑
v∈Ni(u)
rvi.(7)
A problem with (7) is that is does not take into account the fact that the neighbors
can have different levels of similarity. Consider once more the example of Figure
1. If the two nearest-neighbors of Eric are Lucy and Diane, it would be foolish to
consider equally their ratings of the movie “Titanic”, since Lucy’s tastes are much
closer to Eric’s than Diane’s. A common solution to this problem is to weigh the
contribution of each neighbor by its similarity to u. However, if these weights do
not sum to 1, the predicted ratings can be well outside the range of allowed values.
Consequently, it is customary to normalize these weights, such that the predicted
rating becomes
ˆrui =
∑
v∈Ni(u)
wuv rvi
∑
v∈Ni(u)
|wuv|.(8)
In the denominator of (8), |wuv|is used instead of wuv because negative weights can
produce ratings outside the allowed range. Also, wuv can be replaced by wα
uv, where
α>0 is an amplification factor [10]. When α>1, as is it most often employed, an
even greater importance is given to the neighbors that are the closest to u.
Example 2 Suppose we want to use (8) to predict Eric’s rating of the movie “Ti-
tanic” using the ratings of Lucy and Diane for this movie. Moreover, suppose the
similarity weights between these neighbors and Eric are respectively 0.75 and 0.15.
The predicted rating would be
ˆr=0.75×5+0.15×3
0.75 +0.15 '4.67,
which is closer to Lucy’s rating than to Diane’s.

10 Christian Desrosiers and George Karypis
Equation (8) also has an important flaw: it does not consider the fact that users
may use different rating values to quantify the same level of appreciation for an item.
For example, one user may give the highest rating value to only a few outstanding
items, while a less difficult one may give this value to most of the items he likes. This
problem is usually addressed by converting the neighbors’ ratings rvi to normalized
ones h(rvi)[10, 60], giving the following prediction:
ˆrui =h−1


∑
v∈Ni(u)
wuv h(rvi)
∑
v∈Ni(u)
|wuv|

.(9)
Note that the predicted rating must be converted back to the original scale, hence
the h−1in the equation. The most common approaches to normalize ratings will be
presented in Section 3.1.
2.2 User-based classification
The prediction approach just described, where the predicted ratings are computed as
a weighted average of the neighbors’ ratings, essentially solves a regression prob-
lem. Neighborhood-based classification, on the other hand, finds the most likely
rating given by a user uto an item i, by having the nearest-neighbors of uvote on
this value. The vote vir given by the k-NN of ufor the rating r∈ S can be obtained
as the sum of the similarity weights of neighbors that have given this rating to i:
vir =∑
v∈Ni(u)
δ(rvi =r)wuv,(10)
where δ(rvi =r)is 1 if rvi =r, and 0 otherwise. Once this has been computed for
every possible rating value, the predicted rating is simply the value rfor which vir
is the greatest.
Example 3 Suppose once again that the two nearest-neighbors of Eric are Lucy
and Diane with respective similarity weights 0.75 and 0.15. In this case, ratings 5
and 3each have one vote. However, since Lucy’s vote has a greater weight than
Diane’s, the predicted rating will be ˆr=5.
A classification method that considers normalized ratings can also be defined.
Let S0be the set of possible normalized values (that may require discretization), the
predicted rating is obtained as:
ˆrui =h−1 argmax
r∈S0∑
v∈Ni(u)
δ(h(rvi) = r)wuv !.(11)

A comprehensive survey of neighborhood-based recommendation methods 11
2.3 Regression VS classification
The choice between implementing a neighborhood-based regression or classifica-
tion method largely depends on the system’s rating scale. Thus, if the rating scale
is continuous, e.g. ratings in the Jester joke recommender system [23] can take any
value between −10 and 10, then a regression method is more appropriate. On the
contrary, if the rating scale has only a few discrete values, e.g. “good” or “bad”, or
if the values cannot be ordered in an obvious fashion, then a classification method
might be preferable. Furthermore, since normalization tends to map ratings to a
continuous scale, it may be harder to handle in a classification approach.
Another way to compare these two approaches is by considering the situation
where all neighbors have the same similarity weight. As the number of neighbors
used in the prediction increases, the rating rui predicted by the regression approach
will tend toward the mean rating of item i. Suppose item ihas only ratings at either
end of the rating range, i.e. it is either loved or hated, then the regression approach
will make the safe decision that the item’s worth is average. This is also justified
from a statistical point of view since the expected rating (estimated in this case) is
the one that minimizes the RMSE. On the other hand, the classification approach
will predict the rating as the most frequent one given to i. This is more risky as
the item will be labeled as either “good” or “bad”. However, as mentioned before,
taking risk may be be desirable if it leads to serendipitous recommendations.
2.4 Item-based recommendation
While user-based methods rely on the opinion of like-minded users to predict a
rating, item-based approaches [18, 47, 66] look at rating given to similar items. Let
us illustrate this approach with our toy example.
Example 4 Instead of consulting with his peers, Eric instead determines whether
the movie “Titanic” is right for him by considering the movies that he has already
seen. He notices that people that have rated this movie have given similar ratings
to the movies “Forrest Gump” and “Wall-E”. Since Eric liked these two movies he
concludes that he will also like the movie “Titanic”.
This idea can be formalized as follows. Denote by Nu(i)the items rated by user
umost similar to item i. The predicted rating of ufor iis obtained as a weighted
average of the ratings given by uto the items of Nu(i):
ˆrui =
∑
j∈Nu(i)
wi j ru j
∑
j∈Nu(i)
|wi j|.(12)
Example 5 Suppose our prediction is again made using two nearest-neighbors, and
that the items most similar to “Titanic” are “Forrest Gump” and “Wall-E”, with

12 Christian Desrosiers and George Karypis
respective similarity weights 0.85 and 0.75. Since ratings of 5and 4were given by
Eric to these two movies, the predicted rating is computed as
ˆr=0.85×5+0.75×4
0.85 +0.75 '4.53.
Again, the differences in the users’ individual rating scales can be considered by
normalizing ratings with a h:
ˆrui =h−1


∑
j∈Nu(i)
wi j h(ru j )
∑
j∈Nu(i)
|wi j|

.(13)
Moreover, we can also define an item-based classification approach. In this case,
the items jrated by user uvote for the rating to be given to a new item i, and these
votes are weighted by the similarity between iand j. The normalized version of this
approach can be expressed as follows:
ˆrui =h−1 argmax
r∈S0∑
j∈Nu(i)
δ(h(ru j) = r)wi j !.(14)
2.5 User-based VS item-based recommendation
When choosing between the implementation of a user-based and an item-based
neighborhood recommender system, five criteria should be considered:
•Accuracy: The accuracy of neighborhood recommendation methods depends
mostly on the ratio between the number of users and items in the system. As
will be presented in the Section 3.2, the similarity between two users in user-
based methods, which determines the neighbors of a user, is normally obtained
by comparing the ratings made by these users on the same items. Consider a
system that has 10,000 ratings made by 1,000 users on 100 items, and suppose,
for the purpose of this analysis, that the ratings are distributed uniformly over
the items1. Following Table 1, the average number of users available as potential
neighbors is roughly 650. However, the average number of common ratings used
to compute the similarities is only 1. On the other hand, an item-based method
usually computes the similarity between two items by comparing ratings made
by the same user on these items. Assuming once more a uniform distribution of
ratings, we find an average number of potential neighbors of 99 and an average
number of ratings used to compute the similarities of 10.
1The distribution of ratings in real-life data is normally skewed, i.e. most ratings are given to a
small proportion of items.

A comprehensive survey of neighborhood-based recommendation methods 13
In general, a small number of high-confidence neighbors is by far preferable to
a large number of neighbors for which the similarity weights are not trustable. In
cases where the number of users is much greater than the number of items, such
as large commercial systems like Amazon.com, item-based methods can therefore
produce more accurate recommendations [19, 66]. Likewise, systems that have
less users than items, e.g., a research paper recommender with thousands of users
but hundreds of thousands of articles to recommend, may benefit more from user-
based neighborhood methods [30].
Table 1 The average number of neighbors and average number of ratings used in the computation
of similarities for user-based and item-based neighborhood methods. A uniform distribution of
ratings is assumed with average number of ratings per user p=|R|/|U|, and average number of
ratings per item q=|R|/|I|
Avg. neighbors Avg. ratings
User-based (|U| − 1)1−|I|−p
|I|pp2
|I|
Item-based (|I| − 1)1−|U|−q
|U|qq2
|U|
•Efficiency: As shown in Table 2, the memory and computational efficiency of
recommender systems also depends on the ratio between the number of users
and items. Thus, when the number of users exceeds the number of items, as is it
most often the case, item-based recommendation approaches require much less
memory and time to compute the similarity weights (training phase) than user-
based ones, making them more scalable. However, the time complexity of the
online recommendation phase, which depends only on the number of available
items and the maximum number of neighbors, is the same for user-based and
item-based methods.
In practice, computing the similarity weights is much less expensive than the
worst-case complexity reported in Table 2, due to the fact that users rate only
a few of the available items. Accordingly, only the non-zero similarity weights
need to be stored, which is often much less than the number of user pairs. This
number can be further reduced by storing for each user only the top Nweights,
where Nis a parameter [66]. In the same manner, the non-zero weights can be
computed efficiently without having to test each pair of users or items, which
makes neighborhood methods scalable to very large systems.
•Stability: The choice between a user-based and an item-based approach also
depends on the frequency and amount of change in the users and items of the
system. If the list of available items is fairly static in comparison to the users
of the system, an item-based method may be preferable since the item similarity
weights could then be computed at infrequent time intervals while still being able
to recommend items to new users. On the contrary, in applications where the list
of available items is constantly changing, e.g., an online article recommender,
user-based methods could prove to be more stable.

14 Christian Desrosiers and George Karypis
Table 2 The space and time complexity of user-based and item-based neighborhood methods,
as a function of the maximum number of ratings per user p=maxu|Iu|, the maximum number of
ratings per item q=maxi|Ui|, and the maximum number of neighbors used in the rating predictions
k.
Space Time
Training Online
User-based O(|U|2)O(|U|2p)O(|I|k)
Item-based O(|I|2)O(|I|2q)O(|I|k)
•Justifiability: An advantage of item-based methods is that they can easily be
used to justify a recommendation. Hence, the list of neighbor items used in the
prediction, as well as their similarity weights, can be presented to the user as an
explanation of the recommendation. By modifying the list of neighbors and/or
their weights, it then becomes possible for the user to participate interactively in
the recommendation process. User-based methods, however, are less amenable
to this process because the active user does not know the other users serving as
neighbors in the recommendation.
•Serendipity: In item-based methods, the rating predicted for an item is based on
the ratings given to similar items. Consequently, recommender systems using this
approach will tend to recommend to a user items that are related to those usually
appreciated by this user. For instance, in a movie recommendation application,
movies having the same genre, actors or director as those highly rated by the user
are likely to be recommended. While this may lead to safe recommendations, it
does less to help the user discover different types of items that he might like as
much.
Because they work with user similarity, on the other hand, user-based ap-
proaches are more likely to make serendipitous recommendations. This is par-
ticularly true if the recommendation is made with a small number of nearest-
neighbors. For example, a user Athat has watched only comedies may be very
similar to a user Bonly by the ratings made on such movies. However, if Bis fond
of a movie in a different genre, this movie may be recommended to Athrough
his similarity with B.
3 Components of neighborhood methods
In the previous section, we have seen that deciding between a regression and a clas-
sification rating prediction method, as well as choosing between a user-based or
item-based recommendation approach, can have a significant impact on the accu-
racy, efficiency and overall quality of the recommender system. In addition to these
crucial attributes, three very important considerations in the implementation of a
neighborhood-based recommender system are 1) the normalization of ratings, 2)

A comprehensive survey of neighborhood-based recommendation methods 15
the computation of the similarity weights, and 3) the selection of neighbors. This
section reviews some of the most common approaches for these three components,
describes the main advantages and disadvantages of using each one of them, and
gives indications on how to implement them.
3.1 Rating normalization
When it comes to assigning a rating to an item, each user has its own personal
scale. Even if an explicit definition of each of the possible ratings is supplied (e.g.,
1=“strongly disagree”, 2=“disagree”, 3=“neutral”, etc.), some users might be reluc-
tant to give high/low scores to items they liked/disliked. Two of the most popular
rating normalization schemes that have been proposed to convert individual ratings
to a more universal scale are mean-centering and Z-score.
3.1.1 Mean-centering
The idea of mean-centering [10, 60] is to determine whether a rating is positive or
negative by comparing it to the mean rating. In user-based recommendation, a raw
rating rui is transformation to a mean-centered one h(rui)by subtracting to rui the
average ruof the ratings given by user uto the items in Iu:
h(rui) = rui −ru.
Using this approach the user-based prediction of a rating rui is obtained as
ˆrui =ru+
∑
v∈Ni(u)
wuv (rvi −rv)
∑
v∈Ni(u)
|wuv|.(15)
In the same way, the item-mean-centered normalization of rui is given by
h(rui) = rui −ri,
where ricorresponds to the mean rating given to item iby user in Ui. This normal-
ization technique is most often used in item-based recommendation, where a rating
rui is predicted as:
ˆrui =ri+
∑
j∈Nu(i)
wi j (ru j −rj)
∑
j∈Nu(i)
|wi j|.(16)
An interesting property of mean-centering is that one can see right-away if the ap-
preciation of a user for an item is positive or negative by looking at the sign of the

16 Christian Desrosiers and George Karypis
normalized rating. Moreover, the module of this rating gives the level at which the
user likes or dislikes the item.
Example 6 As shown in Figure 2, although Diane gave an average rating of 3 to the
movies “Titanic” and “Forrest Gump”, the user-mean-centered ratings show that
her appreciation of these movies is in fact negative. This is because her ratings are
high on average, and so, an average rating correspond to a low degree of apprecia-
tion. Differences are also visible while comparing the two types of mean-centering.
For instance, the item-mean-centered rating of the movie “Titanic” is neutral, in-
stead of negative, due to the fact that much lower ratings were given to that movie.
Likewise, Diane’s appreciation for “The Matrix” and John’s distaste for “Forrest
Gump” are more pronounced in the item-mean-centered ratings.
User mean-centering:
The Titanic Die Forrest Wall-E
Matrix Hard Gump
John 2.50 -1.50 -0.50 -0.50
Lucy -2.60 1.40 -1.60 1.40 1.40
Eric -1.50 -0.50 1.50 0.50
Diane 0.25 -0.75 1.25 -0.75
Item mean-centering:
The Titanic Die Forrest Wall-E
Matrix Hard Gump
John 2.00 -2.00 -1.75 -1.67
Lucy -2.00 2.00 -1.33 1.25 1.33
Eric -1.00 -0.33 1.25 0.33
Diane 1.00 0.00 1.67 -0.75
Fig. 2 The user and item mean-centered ratings of Figure 1.
3.1.2 Z-score normalization
Consider, two users Aand Bthat both have an average rating of 3. Moreover, sup-
pose that the ratings of Aalternate between 1 and 5, while those of Bare always 3.
A rating of 5 given to an item by Bis more exceptional than the same rating given
by A, and, thus, reflects a greater appreciation for this item. While mean-centering
removes the offsets caused by the different perceptions of an average rating, Z-
score normalization [29] also considers the spread in the individual rating scales.
Once again, this is usually done differently in user-based than in item-based rec-
ommendation. In user-based methods, the normalization of a rating rui divides the
user-mean-centered rating by the standard deviation σuof the ratings given by user

A comprehensive survey of neighborhood-based recommendation methods 17
u:
h(rui) = rui −ru
σu
.
A user-based prediction of rating rui using this normalization approach would there-
fore be obtained as
ˆrui =ru+σu
∑
v∈Ni(u)
wuv (rvi −rv)/σv
∑
v∈Ni(u)
|wuv|.(17)
Likewise, the z-score normalization of rui in item-based methods divides the item-
mean-centered rating by the standard deviation of ratings given to item i:
h(rui) = rui −ri
σi
.
The item-based prediction of rating rui would then be
ˆrui =ri+σi
∑
j∈Nu(i)
wi j (ru j −rj)/σj
∑
j∈Nu(i)
|wi j|.(18)
3.1.3 Choosing a normalization scheme
In some cases, rating normalization can have undesirable effects. For instance, imag-
ine the case of a user that gave only the highest ratings to the items he has purchased.
Mean-centering would consider this user as “easy to please” and any rating below
this highest rating (whether it is a positive or negative rating) would be considered
as negative. However, it is possible that this user is in fact “hard to please” and care-
fully selects only items that he will like for sure. Furthermore, normalizing on a few
ratings can produce unexpected results. For example, if a user has entered a single
rating or a few identical ratings, his rating standard deviation will be 0, leading to
undefined prediction values. Nevertheless, if the rating data is not overly sparse,
normalizing ratings has been found to consistently improve the predictions [29, 33].
Comparing mean-centering with Z-score, as mentioned, the second one has the
additional benefit of considering the variance in the ratings of individual users or
items. This is particularly useful if the rating scale has a wide range of discrete
values or if it is continuous. On the other hand, because the ratings are divided and
multiplied by possibly very different standard deviation values, Z-score can be more
sensitive than mean-centering and, more often, predict ratings that are outside the
rating scale. Lastly, while an initial investigation found mean-centering and Z-score
to give comparable results [29], a more recent one showed Z-score to have more
significant benefits [33].

18 Christian Desrosiers and George Karypis
Finally, if rating normalization is not possible or does not improve the results, an-
other possible approach to remove the problems caused by the individual rating scale
is preference-based filtering. The particularity of this approach is that it focuses on
predicting the relative preferences of users instead of absolute rating values. Since,
an item preferred to another one remains so regardless of the rating scale, predicting
relative preferences removes the need to normalize the ratings. More information on
this approach can be found in [13, 21, 37, 36].
3.2 Similarity weight computation
The similarity weights play a double in neighborhood-based recommendation meth-
ods: 1) they allow to select trusted neighbors whose ratings are used in the predic-
tion, and 2) they provide the means to give more or less importance to these neigh-
bors in the prediction. The computation of the similarity weights is one of the most
critical aspects of building a neighborhood-based recommender system, as it can
have a significant impact on both its accuracy and its performance.
3.2.1 Correlation-based similarity
A measure of the similarity between two objects aand b, often used in information
retrieval, consists in representing these objects in the form of a vector xaand xband
computing the Cosine Vector (CV) (or Vector Space) similarity [3, 8, 44] between
these vectors:
cos(xa,xb) = x>
axb
||xa||||xb||.
In the context of item recommendation, this measure can be employed to compute
user similarities by considering a user uas a vector xu∈R|I|, where xui =rui if user
uhas rated item i, and 0 otherwise. The similarity between two users uand vwould
then be computed as
CV (u,v) = cos(xu,xv) =
∑
i∈Iuv
rui rvi
r∑
i∈Iu
r2
ui ∑
j∈Iv
r2
v j
,(19)
where Iuv once more denotes the items rated by both uand v. A problem with this
measure is that is does not consider the differences in the mean and variance of the
ratings made by users uand v.
A popular measure that compares ratings where the effects of mean and variance
have been removed is the Pearson Correlation (PC) similarity:

A comprehensive survey of neighborhood-based recommendation methods 19
PC(u,v) =
∑
i∈Iuv
(rui −ru)(rvi −rv)
r∑
i∈Iuv
(rui −ru)2∑
i∈Iuv
(rvi −rv)2.(20)
Note that this is different from computing the CV similarity on the Z-score nor-
malized ratings, since the standard deviation of the ratings in evaluated only on the
common items Iuv, not on the entire set of items rated by uand v, i.e. Iuand Iv. The
same idea can be used to obtain similarities between two items iand j[18, 66], this
time by comparing the ratings made by users that have rated both these items:
PC(i,j) =
∑
u∈Ui j
(rui −ri)(ru j −rj)
r∑
u∈Ui j
(rui −ri)2∑
u∈Ui j
(ru j −rj)2.(21)
While the sign of a similarity weight indicates whether the correlation is direct or
inverse, its magnitude (ranging from 0 to 1) represents the strength of the correla-
tion.
Example 7 The similarities between the pairs of users and items of our toy exam-
ple, as computed using PC similarity, are shown in Figure 3. We can see that Lucy’s
taste in movies is very close to Eric’s (similarity of 0.922) but very different from
John’s (similarity of −0.938). This means that Eric’s ratings can be trusted to pre-
dict Lucy’s, and that Lucy should discard John’s opinion on movies or consider the
opposite. We also find that the people that like “The Matrix” also like “Die Hard”
but hate “Wall-E”. Note that these relations were discovered without having any
knowledge of the genre, director or actors of these movies.
User-based Pearson correlation
John Lucy Eric Diane
John 1.000 -0.938 -0.839 0.659
Lucy -0.938 1.000 0.922 -0.787
Eric -0.839 0.922 1.000 -0.659
Diane 0.659 -0.787 -0.659 1.000
Item-based Pearson correlation
The Titanic Die Forrest Wall-E
Matrix Hard Gump
Matrix 1.000 -0.943 0.882 -0.974 -0.977
Titanic -0.943 1.000 -0.625 0.931 0.994
Die Hard 0.882 -0.625 1.000 -0.804 -1.000
Forrest Gump -0.974 0.931 -0.804 1.000 0.930
Wall-E -0.977 0.994 -1.000 0.930 1.000
Fig. 3 The user and item PC similarity for the ratings of Figure 1.

20 Christian Desrosiers and George Karypis
The differences in the rating scales of individual users are often more pronounced
than the differences in ratings given to individual items. Therefore, while computing
the item similarities, it may be more appropriate to compare ratings that are centered
on their user mean, instead of their item mean. The Adjusted Cosine (AC) similarity
[66], is a modification of the PC item similarity which compares user-mean-centered
ratings:
AC(i,j) =
∑
u∈Ui j
(rui −ru)(ru j −ru)
r∑
u∈Ui j
(rui −ru)2∑
u∈Ui j
(ru j −ru)2.
In some cases, AC similarity has been found to outperform PC similarity on the
prediction of ratings using an item-based method [66].
3.2.2 Other similarity measures
Several other measures have been proposed to compute similarities between users or
items. One of them is the Mean Squared Difference (MSD) [70], which evaluate the
similarity between two users uand vas the inverse of the average squared difference
between the ratings given by uand von the same items:
MSD(u,v) = |Iuv|
∑
i∈Iuv
(rui −rvi)2.(22)
While it could be modified to compute the differences on normalized ratings, the
MSD similarity is limited compared to PC similarity because it does not allows to
capture negative correlations between user preferences or the appreciation of dif-
ferent items. Having such negative correlations may improve the rating prediction
accuracy [28].
Another well-known similarity measure is the Spearman Rank Correlation (SRC)
[39]. While PC uses the rating values directly, SRC instead considers the ranking of
these ratings. Denote by kui the rating rank of item iin user u’s list of rated items
(tied ratings get the average rank of their spot). The SRC similarity between two
users uand vis evaluated as:
SRC(u,v) =
∑
i∈Iuv
(kui −ku)(kvi −kv)
r∑
i∈Iuv
(kui −ku)2∑
i∈Iuv
(kvi −kv)2
,(23)
where kuis the average rank of items rated by u(which can differ from |Iu|+1 if
there are tied ratings).
The principal advantage of SRC is that it avoids the problem of rating normaliza-
tion, described in the last section, by using rankings. On the other hand, this measure
may not be the best one when the rating range has only a few possible values, since

A comprehensive survey of neighborhood-based recommendation methods 21
that would create a large number of tied ratings. Moreover, this measure is typically
more expensive than PC as ratings need to be sorted in order to compute their rank.
Table 3 shows the user-based prediction accuracy (MAE) obtained with MSD,
SRC and PC similarity measures, on the MovieLens2dataset [28]. Results are given
for different values of k, which represents the maximum number of neighbors used
in the predictions. For this data, we notice that MSD leads to the least accurate
predictions, possibly due to the fact that it does not take into account negative corre-
lations. Also, these results show PC to be slightly more accurate than SRC. Finally,
although PC has been generally recognized as the best similarity measure, see e.g.
[28], a more recent investigation has shown that the performance of such measure
depended greatly on the data [33].
Table 3 The rating prediction accuracy (MAE) obtained using the Mean Squared Difference
(MSD), the Spearman Rank Correlation and the Pearson Correaltion (PC) similarity. Results are
shown for predictions using an increasing number of neighbors k.
kMSD SRC PC
5 0.7898 0.7855 0.7829
10 0.7718 0.7636 0.7618
20 0.7634 0.7558 0.7545
60 0.7602 0.7529 0.7518
80 0.7605 0.7531 0.7523
100 0.7610 0.7533 0.7528
3.2.3 Accounting for significance
Because the rating data is frequently sparse in comparison to the number of users
and items of a system, it is often the case that similarity weights are computed using
only a few ratings given to common items or made by the same users. For example,
if the system has 10,000 ratings made by 1,000 users on 100 items (assuming a
uniform distribution of ratings), Table 1 shows us that the similarity between two
users is computed, on average, by comparing the ratings given by these users to
asingle item. If these few ratings are equal, then the users will be considered as
“fully similar” and will likely play an important role in each other’s recommenda-
tions. However, if the users’ preferences are in fact different, this may lead to poor
recommendations.
Several strategies have been proposed to take into account the significance of a
similarity weight. The principle of these strategies is essentially the same: reduce
the magnitude of a similarity weight when this weight is computed using only a few
ratings. For instance, in Significance Weighting [29, 49], a user similarity weight
wuv is penalized by a factor proportional to the number of commonly rated item, if
this number is less than a given parameter γ>0:
2http://www.grouplens.org/

22 Christian Desrosiers and George Karypis
w0
uv =min{|Iuv|,γ}
γ×wuv.(24)
Likewise, an item similarity wi j, obtained from a few ratings, can be adjusted as
w0
i j =min{|Ui j |,γ}
γ×wi j.(25)
In [29, 28], it was found that using γ≥25 could significantly improve the accuracy
of the predicted ratings, and that a value of 50 for γgave the best results. However,
the optimal value for this parameter is data dependent and should be determined
using a cross-validation approach.
A characteristic of significance weighting is its use of a threshold γdetermin-
ing when a weight should be adjusted. A more continuous approach, described in
[4], is based on the concept of shrinkage where a weak or biased estimator can be
improved if it is “shrunk” toward a null-value. This approach can be justified us-
ing a Bayesian perspective, where the best estimator of a parameter is the posterior
mean, corresponding to a linear combination of the prior mean of the parameter
(null-value) and an empirical estimator based fully on the data. In this case, the pa-
rameters to estimate are the similarity weights and the null value is zero. Thus, a
user similarity wuv estimated on a few ratings is shrunk as
w0
uv =|Iuv|
|Iuv|+β×wuv ,(26)
where β>0 is a parameter whose value should also be selected using cross-
validation. In this approach, wuv is shrunk proportionally to β/|Iuv|, such that almost
no adjustment is made when |Iuv|  β. Item similarities can be shrunk in the same
way:
w0
i j =|Ui j |
|Ui j|+β×wi j ,(27)
As reported in [4], a typical value for βis 100.
3.2.4 Accounting for variance
Ratings made by two users on universally liked/disliked items may not be as infor-
mative as those made for items with a greater rating variance. For instance, most
people like classic movies such as “The Godfather” so basing the weight compu-
tation on such movies would produce artificially high values. Likewise, a user that
always rates items in the same way may provide less predictive information than
one whose preferences vary from one item to another.
A recommendation approach that addresses this problem is the Inverse User Fre-
quency [10]. Based on the information retrieval notion of Inverse Document Fre-
quency (IDF), a weight λiis given to each item i, in proportion to the log-ratio of
users that have rated i:

A comprehensive survey of neighborhood-based recommendation methods 23
λi=log |U|
|Ui|.
While computing the Frequency-weighted Pearson Correlation (FPC) between
users uand v, the correlation between the ratings given to an item iis weighted
by λi:
FPC(u,v) =
∑
i∈Iuv
λi(rui −ru)(rvi −rv)
r∑
i∈Iuv
λi(rui −ru)2∑
i∈Iuv
λi(rvi −rv)2.(28)
This approach, which was found to improve the prediction accuracy of a user-based
recommendation method [10], could also be adapted to the computation of item
similarities.
A more advanced approach computes the factors λiby maximizing the average
similarity between users. In this approach, the similarity between two users uand v,
given an item weight vector λ= (λ1,...,λ|I|), is evaluated as the likelihood of uto
have the same rating behavior as user v:
Pr(u|v,λ) = 1
Zv
exp ∑
i∈Iuv
λirui rvi!,
where Zvis a normalization constant. The optimal item weight vector is the one
maximizing the average similarity between users:
f(λ) = ∏
u∈U
1
|U|(|U | − 1)∑
v6=u
Pr(u|v,λ).
More details can be found in [35].
3.3 Neighborhood selection
The number of nearest-neighbors to select and the criteria used for this selection can
also have a serious impact on the quality of the recommender system. The selection
of the neighbors used in the recommendation of items is normally done in two steps:
1) a global filtering step where only the most likely candidates are kept, and 2) a per
prediction step which chooses the best candidates for this prediction.
3.3.1 Pre-filtering of neighbors
In large recommender systems that can have millions of users and items, it is usually
not possible to store the (non-zero) similarities between each pair of users or items,
due to memory limitations. Moreover, doing so would be extremely wasteful as only
the most significant of these values are used in the predictions. The pre-filtering of

24 Christian Desrosiers and George Karypis
neighbors is an essential step that makes neighborhood-based approaches practica-
ble by reducing the amount of similarity weights to store, and limiting the number of
candidate neighbors to consider in the predictions. There are several ways in which
this can be accomplished:
•Top-Nfiltering: For each user or item, only a list of the Nnearest-neighbors
and their respective similarity weight is kept. To avoid problems with efficiency
or accuracy, Nshould be chosen carefully. Thus, if Nis too large, an excessive
amount of memory will be required to store the neighborhood lists and predicting
ratings will be slow. On the other hand, selecting a too small value for Nmay
reduce the coverage of the recommendation method, which causes some items to
be never recommended.
•Threshold filtering: Instead of keeping a fixed number of nearest-neighbors, this
approach keeps all the neighbors whose similarity weight’s magnitude is greater
than a given threshold wmin. While this is more flexible than the previous filtering
technique, as only the most significant neighbors are kept, the right value of wmin
may be difficult to determine.
•Negative filtering: In general, negative rating correlations are less reliable than
positive ones. Intuitively, this is because strong positive correlation between two
users is a good indicator of their belonging to a common group (e.g., teenagers,
science-fiction fans, etc.). However, although negative correlation may indicate
membership to different groups, it does not tell how different are these groups, or
whether these groups are compatible for some other categories of items. While
experimental investigations [29, 30] have found negative correlations to provide
no significant improvement in the prediction accuracy, whether such correlations
can be discarded depends on the data.
Note that these three filtering approaches are not exclusive and can be combined
to fit the needs of the recommender system. For instance, one could discard all
negative similarities as well as those with a magnitude lower than a given threshold.
3.3.2 Neighbors in the predictions
Once a list of candidate neighbors has been computed for each user or item, the
prediction of new ratings is normally made with the k-nearest-neighbors, that is,
the kneighbors whose similarity weight has the greatest magnitude. The important
question is which value to use for k.
As shown in Table 3, the prediction accuracy observed for increasing values of
ktypically follows a concave function. Thus, when the number of neighbors is re-
stricted by using a small k(e.g., k<20), the prediction accuracy is normally low.
As kincreases, more neighbors contribute to the prediction and the variance intro-
duced by individual neighbors is averaged out. As a result, the prediction accuracy
improves. Finally, the accuracy usually drops when too many neighbors are used in
the prediction (e.g., k>50), due to the fact that the few strong local relations are

A comprehensive survey of neighborhood-based recommendation methods 25
“diluted” by the many weak ones. Although a number of neighbors between 20 to
50 is most often described in the literature, see e.g. [28, 30], the optimal value of k
should be determined by cross-validation.
On a final note, more serendipitous recommendations may be obtained at the cost
of a decrease in accuracy, by basing these recommendations on a few very similar
users. For example, the system could find the user most similar to the active one and
recommend the new item that has received the highest rated from this user.
4 Advanced techniques
The neighborhood approaches based on rating correlation, such as the ones pre-
sented in the previous sections, have two important flaws:
•Limited coverage: Because rating correlation measures the similarity between
two users by comparing their ratings for the same items, users can be neighbors
only if they have rated common items. This assumption is very limiting, as users
having rated a few or no common items may still have similar preferences. More-
over, since only items rated by neighbors can be recommended, the coverage of
such methods can also be limited.
•Sensitivity to sparse data: Another consequence of rating correlation, addressed
briefly in Section 2.5, is the fact that the accuracy of neighborhood-based rec-
ommendation methods suffers from the lack of available ratings. Sparsity is a
problem common to most recommender systems due to the fact that users typi-
cally rate only a small proportion of the available items [7, 25, 68, 67]. This is
aggravated by the fact that users or items newly added to the system may have
no ratings at all, a problem known as cold-start [69]. When the rating data is
sparse, two users or items are unlikely to have common ratings, and consequently,
neighborhood-based approaches will predict ratings using a very limited number
of neighbors. Moreover, similarity weights may be computed using only a small
number of ratings, resulting in biased recommendations (see Section 3.2.3 for
this problem).
A common solution for these problems is to fill the missing ratings with default
values [10, 18], such as the middle value of the rating range, and the average user
or item rating. A more reliable approach is to use content information to fill out the
missing ratings [16, 25, 41, 50]. For instance, the missing ratings can be provided
by autonomous agents called filterbots [25, 41], that act as ordinary users of the
system and rate items based on some specific characteristics of their content. The
missing ratings can instead be predicted by a content-based approach [50], such as
those described in Section 1.2.1. Finally, content similarity can also be used “instead
of” or “in addition to” rating correlation similarity to find the nearest-neighbors
employed in the predictions [3, 46, 59, 72].
These solutions, however, also have their own drawbacks. For instance, giving
a default values to missing ratings may induce bias in the recommendations. Also,

26 Christian Desrosiers and George Karypis
as discussed in Section 1.2.1, item content may not be available to compute ratings
or similarities. This section presents two approaches proposed for the problems of
limited coverage and sparsity: dimensionality reduction and graph-based methods.
4.1 Dimensionality reduction methods
Dimensionality reduction methods [4, 7, 23, 42, 67, 74, 75] address the problems
of limited coverage and sparsity by projecting users and items into a reduced latent
space that captures their most salient features. Because users and items are com-
pared in this dense subspace of high-level features, instead of the “rating space”,
more meaningful relations can be discovered. In particular, a relation between two
users can be found, even though these users have rated different items. As a result,
such methods are generally less sensitive to sparse data [4, 7, 67].
There are essentially two ways in which dimensionality reduction can be used to
improve recommender systems: 1) decomposition of a user-item rating matrix, and
2) decomposition of a sparse similarity matrix.
4.1.1 Decomposing the rating matrix
A popular dimensionality reduction approach to item recommendation is Latent Se-
mantic Indexing (LSI) [15]. In this approach, the |U|×|I| user-item rating matrix R
of rank nis approximated by a matrix ˆ
R=PQ>of rank k<n, where Pis a |U|×k
matrix of users factors and Qa|I|×kmatrix of item factors. Intuitively, the u-th row
of P,pu∈Rk, represents the coordinates of user uprojected in the k-dimensional
latent space. Likewise, the i-th row of Q,qi∈Rk, can be seen as the coordinates of
item iin this latent space. Matrices Pand Qare normally found by minimizing the
reconstruction error defined with the squared Frobenius norm:
err(P,Q) = ||R−PQ>||2
F
=∑
u,irui −puq>
i2.
Minimizing this error is equivalent to finding the Singular Value Decomposition
(SVD) of R[24]:
R=UΣV>,
where Uis the |U|×nmatrix of left singular vectors, Vis the |I|×nmatrix of right
singular vectors, and Σis the n×ndiagonal matrix of singular values. Denote by
Σk,Ukand Vkthe matrices obtained by selecting the subset containing the khighest
singular values and their corresponding singular vectors, the user and item factor
matrices correspond to P=UkΣ1/2
kand Q=VkΣ1/2
k.

A comprehensive survey of neighborhood-based recommendation methods 27
Once Pand Qhave been obtained, the typical model-based prediction of a rating
rui is:
rui =puq>
i.
There is, however, a major problem with applying SVD to the rating matrix R: most
values rui of Rare undefined, since there may not be a rating given to iby u. Al-
though it is possible to assign a default value to rui, as mentioned above, this would
introduce a bias in the data. More importantly, this would make the large matrix
Rdense and, consequently, render impractical the SVD decomposition of R. The
common solution to this problem is to learn Pand Qusing only the known ratings
[4, 42, 73, 75]:
err(P,Q) = ∑
rui∈R
(rui −puq>
i)2+λ||pu||2+||qi||2,(29)
where λis a parameter that controls the level of regularization. A more compre-
hensive description of this recommendation approach can be found in Chapter ?? of
this book.
In neighborhood-based recommendation, the same principle can be used to com-
pute the similarity between users or items in the latent-space [7]. This can be done
by solving the following problem:
err(P,Q) = ∑
rui∈R
zui −puq>
i2(30)
subject to:
||pu|| =1,∀u∈ U,||qi|| =1,∀i∈ I,
where zui is the mean-centered rating rui normalized to the [−1,1]range. For exam-
ple, if rmin and rmax are the lowest and highest values in the original rating range,
zui =rui −ru
rmax −rmin
.
This problem corresponds to finding, for each user uand item i, coordinates on
the surface of the k-dimensional unit sphere such that uwill give a high rating to i
if their coordinates are near on the surface. If two users uand vare nearby on the
surface, then they will give similar ratings to the same items, and, thus, the similarity
between these users can be computed as
wuv =pup>
v.
Likewise, the similarity between two items iand jcan be obtained as
wi j =qiq>
j.

28 Christian Desrosiers and George Karypis
4.1.2 Decomposing the similarity matrix
The principle of this second dimensionality reduction approach is the same as the
previous one: decompose a matrix into its principal factors representing projection
of users or items in the latent space. However, instead of decomposing the rating
matrix, a sparse similarity matrix is decomposed. Let Wbe a symmetric matrix of
rank nrepresenting either user or item similarities. To simplify the presentation, we
will suppose the former case. Once again, we want to approximate Wwith a matrix
ˆ
W=PP>of lower rank k<nby minimizing the following objective:
err(P) = ||R−PP>||2
F
=∑
u,vwuv −pup>
v2.
Matrix ˆ
Wcan be seen as a “compressed” version of Wwhich is less sparse than
W. As before, finding the factor matrix Pis equivalent to computing the eigenvalue
decomposition of W:
W=VΛV>,
where Λis a diagonal matrix containing the |U| eigenvalues of W, and Vis a |U |×
|U| orthogonal matrix containing the corresponding eigenvectors. Let Vkbe a matrix
formed by the kprincipal (normalized) eigenvectors of W, which correspond to the
axes of the k-dimensional latent subspace. The coordinates pu∈Rkof a user uin
this subspace is given by the u-th row of matrix P=VkΛ1/2
k. Furthermore, the user
similarities computed in this latent subspace are given by matrix
W0=PP>
=VkΛkV>
k.(31)
This approach was used to recommend jokes in the Eigentaste system [23]. In
Eigentaste, a matrix Wcontaining the PC similarities between pairs of items is de-
composed to obtain the latent subspace defined by the two principal eigenvectors of
W. Denote V2the matrix containing these eigenvectors. A user u, represented by the
u-th row ruof the rating matrix R, is projected in the plane defined by V2:
r0
u=ruV2.
In an offline step, the users of the system are clustered in the plane using a recursive
subdivision technique. Then, the rating of user ufor an item iis evaluated as the
mean rating for imade by users in the same cluster as u.

A comprehensive survey of neighborhood-based recommendation methods 29
4.2 Graph-based methods
In graph-based approaches, the data is represented in the form of a graph where
nodes are users, items or both, and edges encode the interactions or similarities
between the users and items. For example, in Figure 4, the data is modeled as a
bipartite graph where the two sets of nodes represent users and items, and an edge
connects user uto item iif there is a rating given to iby uin the system. A weight
can also be given to this edge, such as the value of its corresponding rating. In
another model, the nodes can represent either users or items, and an edge connects
two nodes if the ratings corresponding two these nodes are sufficiently correlated.
The weight of this edge can be the corresponding correlation value.
Fig. 4 A bipartite graph representation of the ratings of Figure 1 (only ratings with value in
{2,3,4}are shown).
In these models, standard approaches based on correlation predict the rating of
a user ufor an item iusing only the nodes directly connected to uor i. Graph-
based approaches, on the other hand, allow nodes that are not directly connected to
influence each other by propagating information along the edges of the graph. The
greater the weight of an edge, the more information is allowed to pass through it.
Also, the influence of a node on another should be less if the two nodes are further
away in the graph. These two properties, known as propagation and attenuation
[26, 34], are often observed in graph-based similarity measures.
The transitive associations captured by graph-based methods can be used to rec-
ommend items in two different ways. In the first approach, the proximity of a user u
to an item iin the graph is used directly to evaluate the rating of ufor i[19, 26, 34].
Following this idea, the items recommended to uby the system are those that are
the “closest” to uin the graph. On the other hand, the second approach considers the
proximity of two users or item nodes in the graph as a measure of similarity, and uses
this similarity as the weights wuv or wi j of a neighborhood-based recommendation
method [19, 48].

30 Christian Desrosiers and George Karypis
4.2.1 Path-based similarity
In path-based similarity, the distance between two nodes of the graph is evaluated
as a function of the number of paths connecting the two nodes, as well as the length
of these paths.
Shortest path
A recommendation approach that computes the similarity between two users based
on their shortest distance in a graph is the one described in [2]. In this method,
the data is modeled as a directed graph whose nodes are users, and in which edges
are determined based on the notions of horting and predictability. Horting is an
asymmetric relation between two users that is satisfied if these users have rated
similar items. Formally, a user uhorts another user vprovided either |Iuv | ≥ αor
|Iuv|/|Iu| ≥ βis satisfied, where α,βare predetermined thresholds. Predictability,
on the other hand, is a stronger property additionally requiring the ratings of uto
be similar to those of v, under a mapping representing the difference in the rating
scales of uand v. Thus, vpredicts u, provided uhorts vand there exists a linear
transformation l:S → S such that
1
|Iuv|∑
i∈Iuv
|rui −l(rvi)| ≤ γ,
where γis another given threshold.
The relations of predictability are represented as directed edges in the graph,
such that there is a directed edge from uto vif vpredicts u. Accordingly, a directed
path connecting two users uand vrepresents the transitive predictability of vfor the
ratings of u, under a sequence of transformations. Following this idea, the rating of
user ufor a new item iis predicted using the shortest directed paths from uto other
users that have rated i. Let P={u,v1,v2,...,vm}be such a path, where vm∈ Ui.
The rating of user vmfor item iis transformed in the rating scale of uusing the
composition of the linear mappings along the path:
ˆr(P)
ui = (lm◦. . . ◦l2◦l1)(rvi).
The final prediction of rating rui is computed as the average of the predictions ˆr(P)
ui
obtained for all shortest paths P.
Number of paths
The number of paths between a user and an item in a bipartite graph can also be used
to evaluate their compatibility [34]. Let Rbe once again the |U|×|I|rating matrix
where rui equals 1 if user uhas rated item i, and 0 otherwise. The adjacency matrix

A comprehensive survey of neighborhood-based recommendation methods 31
Aof the bipartite graph can be defined from Ras
A=0R>
R0.
In this approach, the association between a user uand an item iis defined as the sum
of the weights of all distinctive paths connecting uto v(allowing nodes to appear
more than once in the path), whose length is no more than a given maximum length
K. Note that, since the graph is bipartite, Kshould be an odd number. In order to
attenuate the contribution of longer paths, the weight given to a path of length kis
defined as αk, where α∈[0,1]. Using the fact that the number of klength paths
between pairs of nodes is given by Ak, the user-item association matrix SKis
SK=
K
∑
k=1
αkAk
= (I−αA)−1(αA−αKAK).(32)
This method of computing distances between nodes in a graph is known as the Katz
measure [38]. Note that this measure is closely related to the Von Neumann Diffusion
kernel [20, 40, 43]
KVND =
∞
∑
k=0
αkAk
= (I−αA)−1(33)
and the Exponential Diffusion kernel
KED =
∞
∑
k=0
1
k!αkAk
=exp(αA),(34)
where A0=I.
In recommender systems that have a large number of users and items, computing
these association values may require extensive computational resources. To over-
come these limitations, spreading activation techniques [14] have been used in [34].
Essentially, such techniques work by first activating a selected subset of nodes as
starting nodes, and then iteratively activating the nodes that can be reached directly
from the nodes that are already active, until a convergence criterion is met.
4.2.2 Random walk similarity
Transitive associations in graph-based methods can also be defined within a prob-
abilistic framework. In this framework, the similarity or affinity between users or
items is evaluated as a probability of reaching these nodes in a random walk. For-

32 Christian Desrosiers and George Karypis
mally, this can be described with a first-order Markov process defined by a set of n
states and a n×ntransition probability matrix Psuch that the probability of jumping
from state ito jat any time-step tis
pi j =Prs(t+1) = j|s(t) = i.
Denote π(t)the vector containing the state probability distribution of step t, such
that πi(t) = Pr(s(t) = i), the evolution of the Markov chain is characterized by
π(t+1) = P>π(t).
Moreover, under the condition that Pis row-stochastic, i.e. ∑jpi j =1 for all i, the
process converges to a stable distribution vector π(∞)corresponding to the positive
eigenvector of P>with an eigenvalue of 1. This process is often described in the
form of a weighted graph having a node for each state, and where the probability
of jumping from a node to an adjacent node is given by the weight of the edge
connecting these nodes.
Itemrank
A recommendation approach, based on the PageRank algorithm for ranking Web
pages [11], is ItemRank [26]. This approach ranks the preferences of a user ufor new
items ias the probability of uto visit iin a random walk of a graph in which nodes
correspond to the items of the system, and edges connects items that have been rated
by common users. The edge weights are given by the |I|×|I| transition probability
matrix Pfor which pi j =|Ui j |/|Ui|is the estimated conditional probability of a user
to rate and item jif it has rated an item i.
As in PageRank, the random walk can, at any step t, either jump using Pto an
adjacent node with fixed probability α, or “teleport” to any node with probability
(1−α). Let rube the u-th row of the rating matrix R, the probability distribution of
user uto teleport to other nodes is given by vector du=ru/||ru||. Following these
definitions, the state probability distribution vector of user uat step t+1 can be
expressed recursively as
πu(t+1) = αP>πu(t)+(1−α)du.(35)
For practical reasons, πu(∞)is usually obtained with a procedure that first initializes
the distribution as uniform, i.e. πu(0) = 1
n1n, and then iteratively updates πu, using
(35), until convergence. Once πu(∞)has been computed, the system recommends
to uthe item ifor which πui is the highest.

A comprehensive survey of neighborhood-based recommendation methods 33
Average first-passage/commute time
Other distance measures based on random walks have been proposed for the recom-
mendation problem. Among these are the average first-passage time and the aver-
age commute time [19, 20]. The average first-passage time m(j|i)[56] is the average
number of steps needed by a random walker to reach a node jfor the first time, when
starting from a node i6=j. Let Pbe the n×ntransition probability matrix, m(j|i)
can be obtained expressed recursively as
m(j|i) = 


0,if i=j
1+n
∑
k=1
pik m(j|k),otherwise
A problem with the average first-passage time is that it is not symmetric. A related
measure that does not have this problem is the average commute time n(i,j) =
m(j|i) + m(i|j)[22], corresponding to the average number of steps required by a
random walker starting at node i6=jto reach node jfor the first time and go back
to i. This measure has several interesting properties. Namely, it is a true distance
measure in some Euclidean space [22], and is closely related to the well-known
property of resistance in electrical networks and to the pseudo-inverse of the graph
Laplacian matrix [19].
In [19], the average commute time is used to compute the distance between the
nodes of a bipartite graph representing the interactions of users and items in a rec-
ommender system. For each user uthere is a directed edge from uto every item
i∈ Iu, and the weight of this edge is simply 1/|Iu|. Likewise, there is a directed
edge from each item ito every user u∈ Ui, with weight 1/|Ui|. Average commute
times can be used in two different ways: 1) recommending to uthe item ifor which
n(u,i)is the smallest, or 2) finding the users nearest to u, according to the commute
time distance, and then suggest to uthe item most liked by these users.
5 Conclusion
One of the earliest approaches proposed for the task item recommendation, neighbor-
hood-based recommendation still ranks among the most popular methods for this
problem. Although quite simple to describe and implement, this recommendation
approach has several important advantages, including its ability to explain a recom-
mendation with the list of the neighbors used, its computational and space efficiency
which allows it to scale to large recommender systems, and its marked stability in
an online setting where new users and items are constantly added. Another of its
strengths is its potential to make serendipitous recommendations that can lead users
to the discovery of unexpected, yet very interesting items.
In the implementation of a neighborhood-based approach, one has to make sev-
eral important decisions. Perhaps the one having the greatest impact on the accuracy

34 Christian Desrosiers and George Karypis
and efficiency of the recommender system is choosing between a user-based and
an item-based neighborhood method. In typical commercial recommender systems,
where the number of users far exceeds the number of available items, item-based ap-
proaches are typically preferred since they provide more accurate recommendations,
while being more computationally efficient and requiring less frequent updates. On
the other hand, user-based methods usually provide more original recommenda-
tions, which may lead users to a more satisfying experience. Moreover, the different
components of a neighborhood-based method, which include the normalization of
ratings, the computation of the similarity weights and the selection of the nearest-
neighbors, can also have a significant influence on the quality of the recommender
system. For each of these components, several different alternatives are available.
Although the merit of each of these has been described in this document and in
the literature, it is important to remember that the “best” approach may differ from
one recommendation setting to the next. Thus, it is important to evaluate them on
data collected from the actual system, and in light of the particular needs of the
application.
Finally, when the performance of a neighborhood-based approach suffers from
the problems of limited coverage and sparsity, one may explore techniques based on
dimensionality reduction or graphs. Dimensionality reduction provides a compact
representation of users and items that captures their most significant features. An
advantage of such approach is that it allows to obtain meaningful relations between
pairs of users or items, even though these users have rated different items, or these
items were rated by different users. On the other hand, graph-based techniques ex-
ploit the transitive relations in the data. These techniques also avoid the problems of
sparsity and limited coverage by evaluating the relationship between users or items
that are not “directly connected”. However, unlike dimensionality reduction, graph-
based methods also preserve some of the “local” relations in the data, which are
useful in making serendipitous recommendations.
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