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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
9.3.1 . We can subtract from each nonblank element m ij the average rating of user i . Then,
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.
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