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We have not yet addressed computation of the CP-decomposition. Obviously,
we can no longer apply the hitherto used adaptive HOSVD, since it is based on
the Tucker decomposition ( 9.1 ). We shall address the topic in more detail in
Sect. 9.2.3 .
9.2.2 RE-Specific Factorizations
Besides CANDECOMP/PARAFAC, of course, numerous factorizations are possi-
ble. With regard to recommendation engines, methods of nonnegative tensor
factorization, corresponding to the NMF from Sect. 8.4.3 , are of special interest.
In the context of reinforcement learning, we are especially interested in factorizing
the transition probabilities.
As for this, an interesting approach may be found in [RFST10]. Therein, sequences
of baskets belonging to different users are analyzed with the goal of recommending
products that are most likely to be purchased to an identified user. To this end, the
transition probabilities are factorized.
We now retrofit the approach in such a way that sequences of products instead of
sequences of baskets and sessions instead of users be considered. This is consistent
with our Example 9.5. Hence, we seek after a factorization of the transition
probability tensor P u , ss 0 , which corresponds to our previously considered matrix
P ss 0 of transition probabilities from s to s' for the session u. The proposed factor-
ization is of the form
t s , s 0
t u , s 0
t u , s
A ¼ X
þ X
þ X
v s , s 0
k
v s 0 , s
k
v u , s 0
k
v s 0 , u
k
v u , s
k
v s , u
k
:
ð 9
:
8 Þ
1
1
1
The factorization models the pair-wise interaction between the single tensor modes
u, s, s 0 . Therefore, we are dealing with a special case of the CP-decomposition ( 9.7 ),
where the rank-1 tensors are now established from 2 rather than 3 vectors.
Writing ( 9.8 ) element-wise,
t s , s 0
t u , s 0
t u , s
a u , ss 0 ¼ X
s , k þ X
s , k þ X
v s , s 0
s , k v s , s
v u , s 0
u , k v s , u
v u , s
u , k v s , u
s , k
1
1
1
and considering the difference between two probabilities with respect to s' ,i.e.,
a u , ss 0 a u , ss 00 , we notice that the latter is invariant with respect to the first term.
Hence, if one is interested only in the ordering of the values a u , ss 0 , s 0
S ,itsufficesto
consider
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