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Recommender systems and collaborative filtering

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Chapter 12 (Section 12.4):
Recommender Systems
Second edition of the book, coming soon
Road Map
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Introduction
Content-based recommendation
Collaborative filtering based recommendation
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K-nearest neighbor
Association rules
Matrix factorization
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Introduction
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Recommender systems are widely used on the
Web for recommending products and services
to users.
Most e-commerce sites have such systems.
These systems serve two important functions.
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They help users deal with the information overload
by giving them recommendations of products, etc.
They help businesses make more profits, i.e.,
selling more products.
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E.g., movie recommendation
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The most common scenario is the following:
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A set of users has initially rated some subset of
movies (e.g., on the scale of 1 to 5) that they have
already seen.
These ratings serve as the input. The
recommendation system uses these known
ratings to predict the ratings that each user would
give to those not rated movies by him/her.
Recommendations of movies are then made to
each user based on the predicted ratings.
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Different variations
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In some applications, there is no rating
information while in some others there are also
additional attributes
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about each user (e.g., age, gender, income, marital
status, etc), and/or
about each movie (e.g., title, genre, director,
leading actors or actresses, etc).
When no rating information, the system will not
predict ratings but predict the likelihood that a
user will enjoy watching a movie.
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The Recommendation Problem
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We have a set of users U and a set of items S
to be recommended to the users.
Let p be an utility function that measures the
usefulness of item s (пѓЋ S) to user u (пѓЋ U), i.e.,
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p:UГ—S п‚® R, where R is a totally ordered set (e.g.,
non-negative integers or real numbers in a range)
Objective
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Learn p based on the past data
Use p to predict the utility value of each item s (пѓЋ
S) to each user u (пѓЋ U)
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As Prediction
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Rating prediction, i.e., predict the rating score
that a user is likely to give to an item that s/he
has not seen or used before. E.g.,
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rating on an unseen movie. In this case, the utility
of item s to user u is the rating given to s by u.
Item prediction, i.e., predict a ranked list of
items that a user is likely to buy or use.
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Two basic approaches
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Content-based recommendations:
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Collaborative filtering (or collaborative
recommendations):
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The user will be recommended items similar to the
ones the user preferred in the past;
The user will be recommended items that people
with similar tastes and preferences liked in the past.
Hybrids: Combine collaborative and contentbased methods.
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Road Map
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Introduction
Content-based recommendation
Collaborative filtering based recommendation
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K-nearest neighbor
Association rules
Matrix factorization
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Content-Based Recommendation
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Perform item recommendations by predicting
the utility of items for a particular user based
on how “similar” the items are to those that
he/she liked in the past. E.g.,
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In a movie recommendation application, a movie
may be represented by such features as specific
actors, director, genre, subject matter, etc.
The user’s interest or preference is also
represented by the same set of features, called
the user profile.
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Content-based recommendation (contd)
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Recommendations are made by comparing
the user profile with candidate items
expressed in the same set of features.
The top-k best matched or most similar items
are recommended to the user.
The simplest approach to content-based
recommendation is to compute the similarity
of the user profile with each item.
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Road Map
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Introduction
Content-based recommendation
Collaborative filtering based recommendations
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K-nearest neighbor
Association rules
Matrix factorization
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Collaborative filtering
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Collaborative filtering (CF) is perhaps the
most studied and also the most widely-used
recommendation approach in practice.
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k-nearest neighbor,
association rules based prediction, and
matrix factorization
Key characteristic of CF: it predicts the utility
of items for a user based on the items
previously rated by other like-minded users.
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k-nearest neighbor
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kNN (which is also called the memory-based
approach) utilizes the entire user-item
database to generate predictions directly, i.e.,
there is no model building.
This approach includes both
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User-based methods
Item-based methods
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User-based kNN CF
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A user-based kNN collaborative filtering
method consists of two primary phases:
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the neighborhood formation phase and
the recommendation phase.
There are many specific methods for both.
Here we only introduce one for each phase.
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Neighborhood formation phase
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Let the record (or profile) of the target user be
u (represented as a vector), and the record of
another user be v (v пѓЋ T).
The similarity between the target user, u, and
a neighbor, v, can be calculated using the
Pearson’s correlation coefficient:
sim(u, v) пЂЅ


iпѓЋC
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iпѓЋC
(ru,i пЂ­ ru )(rv,i пЂ­ rv )
(ru,i пЂ­ ru )
2

iпѓЋC
(rv,i пЂ­ rv )
2
,
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Recommendation Phase
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Use the following formula to compute the
rating prediction of item i for target user u
p(u, i) пЂЅ ru

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sim(u, v) п‚ґ (rv,i пЂ­r v )
vV sim(u, v)
vпѓЋV
where V is the set of k similar users, rv,i is the
rating of user v given to item i,
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Issue with the user-based kNN CF
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The problem with the user-based formulation
of collaborative filtering is the lack of
scalability:
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it requires the real-time comparison of the target
user to all user records in order to generate
predictions.
A variation of this approach that remedies
this problem is called item-based CF.
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Item-based CF
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The item-based approach works by
comparing items based on their pattern of
ratings across users. The similarity of items i
and j is computed as follows:
sim(i, j) пЂЅ
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
uпѓЋU
(ru,i пЂ­ ru )(ru, j пЂ­ ru )
2
(
r
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r
)
uU u,i u
2
(
r
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r
)
uU u, j u
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Recommendation phase
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After computing the similarity between items
we select a set of k most similar items to the
target item and generate a predicted value of
user u’s rating
r

p(u, i) пЂЅ

jпѓЋJ u, j
п‚ґ sim(i, j)
sim
(
i
,
j
)
jпѓЋJ
where J is the set of k similar items
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Road Map
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Introduction
Content-based recommendation
Collaborative filtering based recommendation
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K-nearest neighbor
Association rules
Matrix factorization
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Association rule-based CF
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Association rules obviously can be used for
recommendation.
Each transaction for association rule mining
is the set of items bought by a particular user.
We can find item association rules, e.g.,
buy_X, buy_Y -> buy_Z
Rank items based on measures such as
confidence, etc.
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See Chapter 3 for details
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Road Map
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Introduction
Content-based recommendation
Collaborative filtering based recommendation
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K-nearest neighbor
Association rules
Matrix factorization
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Matrix factorization
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The idea of matrix factorization is to
decompose a matrix M into the product of
several factor matrices, i.e.,
M пЂЅ F1F2 ...Fn
where n can be any number, but it is usually
2 or 3.
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CF using matrix factorization
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Matrix factorization has gained popularity for
CF in recent years due to its superior
performance both in terms of recommendation
quality and scalability.
Part of its success is due to the Netflix Prize
contest for movie recommendation, which
popularized a Singular Value Decomposition
(SVD) based matrix factorization algorithm.
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The prize winning method of the Netflix Prize
Contest employed an adapted version of SVD
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The abstract idea
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Matrix factorization a latent factor model.
Latent variables (also called features,
aspects, or factors) are introduced to account
for the underlying reasons of a user
purchasing or using a product.
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When the connections between the latent
variables and observed variables (user, product,
rating, etc.) are estimated during the training
recommendations can be made to users by
computing their possible interactions with each
product through the latent variables.
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Netflix Prize Contest
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Netflix Prize Task
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Training data: Quadruples of the form
(user, movie, rating, time)
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For our purpose here, we only use triplets, i.e.,
(user, movie, rating)
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For example, (132456, 13546, 4) means that the
user with ID 132456 gave the movie with ID 13546
a rating of 4 (out of 5).
Testing: predict the rating of each triplet:
(user, movie, ?)
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SVD factorization
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The technique discussed here is based on
the SVD method given by
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Simon Funk at his blog site,
the derivation of Funk’s method described by
Wagman in the Netflix forums.
the paper by Takacs et al.
The method was later improved by Koren et
al., Paterek and several other researchers.
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Intuitive Idea
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Simon Funk’s SVD method
where U = [u1, u2, …, uI] and M = [m1, m2, …, mJ]
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SVD method (contd)
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Let us use K = 90 latent aspects (K needs to
be set experimentally).
Then, each movie will be described by only
ninety aspect values indicating how much
that movie exemplifies each aspect.
Correspondingly, each user is also described
by ninety aspect values indicating how much
he/she prefers each aspect.
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SVD method (contd)
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To combine these together into a rating, we
multiply each user preference by the
corresponding movie aspect, and then sum
them up to give a rating to indicate how much
that user likes that movie:
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U = [u1, u2, …, uI] and M = [m1, m2, …, mJ]
Using SVD, we can perform the task
K
rij  ui m j  uki  mkj
T
k пЂЅ1
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SVD method (contd)
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SVD is a mathematical way to find these two
smaller matrices which minimizes the
resulting approximation error, the mean
square error (MSE).
We can use the resulting matrices U and M to
predict the ratings in the test set.
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SVD method (contd)
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SVD method (contd)
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To minimize the error, the gradient descent
approach is used.
For gradient descent, we take the partial
derivative of the square error with respect to
each parameter, i.e. with respect to each uki
and mkj.
(eij )2
eij
пЂЅ 2eij
uki
uki
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SVD method (contd)
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SVD method (contd)
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The final update rules
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By the same reasoning, we can also compute
the update rule for mkj.
Finally, we have both rules
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The final prediction uses Eq. (11)
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Further improvements
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The two basic rules need some
improvements to make them work well.
There are also some pre-processing.
Time was also added later.
Etc
Note:
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Funk used stochastic gradient descent
Not the batch (global) gradient descent.
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