Decision Trees and Recommender Systems - Data Mining with Decision Trees: Theory and Applications - page 264

The SVD CF methods transform users and items into a joint latent
factor space,

f where f indicate the number of latent factors. Both users
u and items i are represented by corresponding vectors p u ,q i ∈ R

R

f .The

cosine similarity
The final rating is created by adding baseline predictors that depend
only on the user or item. Thus, a rating is predicted by the rule:

r ui = µ + b i + b u + q i p u . (16.5)

We distinguish predicted ratings from known ones, by using the hat
notation for the predicted value. In order to estimate the model parameters
( b u ,b i ,p u and q i ) one can solve the regularized least squares error problem
using a stochastic gradient descent procedure:

q i p u ) 2 + λ 4 ( b i + b u + q i 2 + p u 2 ) . min

b ∗ ,q ∗ ,p ∗

( r ui − µ − b i − b u −

( u,i ) ∈K

(16.6)

16.3.4 Pairwise Comparisons

A pairwise comparison is used for eliciting new user profiles. The main idea
is to present the user with two items and ask her which of them is preferred.
Usually, she has to choose the answer from among several discrete pairwise
choices.
It has previously been shown in psychological studies that due to the
human capacity, the best number of discrete choices is 7 2, each of which

is a linguistic phrase such as “I much prefer item A to item B” or “I equally
like item A and item B”.
While we are using pairwise comparisons to generate the initial profile,
we still assume that the feedback from existing users is provided as a
rating of individual items (the rating matrix). As the preferences are
expressed in different ways, they are not directly comparable. Mapping
from one system to another poses a challenge. Satty explains how a rating
scale can be converted into a pairwise comparison and vice versa. We
illustrate the mapping process with the following example: we are given
four items A,B,C,D rated in the scale [1 , 5] as following r uA =5, r uB =1,
r uC

±

=3, r uD = 2. The pairwise comparison value between two items is

set to:

C uij = r ui /r uj . (16.7)