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The optimizer is given by
T
¼ U þ
T
ðÞ
Θ
Due to the special structure of U , it is possible to represent Θ * in a simple closed
form. To this end, recall that
δ αβ
U T U ¼
G β
α , β
m
Moreover, we obtain
δ αβ
G β \ G
U T U ¼
β
β , β m
which gives rise to
G β \ G
Θ ¼ X
β m
β
Θ :
G β
With some minor effort, this observation may be extended to the case of a
weighted Frobenius norm in the optimization ( 10.12 ).
As regards the procedure outlined in the previous section, the above-derived least
squares framework may be deployed to obtain initial iterates for the individuals in
each new generation of the GA procedure. Here, it appears reasonable to take the
factors U , Θ
to those of the most recent iterate of one of the parents of the considered
individual.
10.6 How It All Fits Together
We are now going to discuss how the approaches described so far in this topic all fit
together. Almost all of them deal with the problem of complexity: hierarchical
methods to speed up convergence, factorization, and tensors as well as special
empirical assumptions to reduce the complexity of the recommendation model.
We consider the most general task, the k -MDP of Sect. 10.1 , and include
multiple recommendations. As we stated in Sect. 4.2 , multiple recommendations
can be interpreted as single actions, and thus all considerations of Sect. 10.1 remain
valid. However, the problem of multiple recommendations is their increased
complexity.
We proceed similar
to Sect. 10.2 and include the space of multiple
recommendations
m
1 S j ,
A
:¼ [
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