Database Reference
In-Depth Information
Take the derivative of the above with respect to
x
, and set it equal to 0, in order to find the
value of
x
that minimizes the RMSE. That is,
As in the previous examples, the common factor −2 can be dropped. We solve the above
equation for
x
, and get
There is an analogous formula for the optimum value of an element of
V
. If we want to
vary
v
rs
=
y
, then the value of
y
that minimizes the RMSE is
Here, ∑
i
is shorthand for the sum over all
i
such that
m
is
is nonblank, and ∑
k
≠
r
is the sum
over all values of
k
between 1 and
d
, except for
k
=
r
.
9.4.5
Building a Complete UV-Decomposition Algorithm
Now, we have the tools to search for the global optimum decomposition of a utility matrix
M
. There are four areas where we shall discuss the options.
(1) Preprocessing of the matrix
M
.
(2) Initializing
U
and
V
.
(3) Ordering the optimization of the elements of
U
and
V
.
(4) Ending the attempt at optimization.
Preprocessing
Because the differences in the quality of items and the rating scales of users are such im-
portant factors in determining the missing elements of the matrix
M
, it is often useful to
remove these influences before doing anything else. The idea was introduced in
Section
the resulting matrix can be modified by subtracting the average rating (in the modified mat-
rix) of item
j
. It is also possible to first subtract the average rating of item
j
and then sub-
tract the average rating of user
i
in the modified matrix. The results one obtains from doing
things in these two different orders need not be the same, but will tend to be close. A third
option is to normalize by subtracting from
m
ij
the average of the average rating of user
i
and item
j
, that is, subtracting one half the sum of the user average and the item average.