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meaningful way. This entails a considerable amount of mathematical as well as
engineering-related issues that will be tackled in the following.
Furthermore, we shall be interested in applying tensorial factorization models as
a means of regularized estimation of transition probability functions of Markov
decision processes so as to tackle large state space arising from state augmentation.
9.1.3 PCA for Tensorial Data: Tucker Tensor
and Higher-Order SVD
Definition 9.3 A d-mode n-dimensional Tucker tensor of (Tucker) rank t n is a
tensor
A ¼ C 1 U 1 2 ... d U d ,
ð 9
:
1 Þ
t denotes the core tensor and U k R
ð
n k ;t k
Þ
where C
R
k
d the mode factors.
,
Remark 9.1 A Tucker tensor is completely determined by its core tensor and the
mode factors. Therefore, with a slight abuse of language, we shall henceforth
identify a Tucker tensor ( 9.1 ) with the tuple ( U 1 ,
...
, U d , C ).
Thus, for each component a i 1 , ...
, i d of A decomposition, ( 9.1 ) reads as
, i d ¼ X
X
t 1
j 1 ¼ 1 ...
t d
, j d u i 1 , j 1 ...
u i 1 d , j d ,
a i 1 , ...
c j 1 , ...
j d ¼ 1
where c j 1 , ... , j d are the components of the core tensor C and u i;j are the elements of
the factor U k .
In utmost generality, tensor factorization problems in terms of the Tucker format
may be stated as instances of the framework
min
fA , C 1 U 1 2 ... d U d
ð
Þ ,
ð 9
:
2 Þ
, ... ,
,
U 1 ∈C 1
U d ∈C d
C∈C 0
where the cost function f stipulates a notion of approximation quality and con-
straints, e.g., nonnegativity or orthonormality, are encoded in the sets C 0 ,
, C d .
Please note that the matrix factorization framework ( 8.1 ) coincides with the special
case of ( 9.2 ) where d ¼ 2 .
...
n und t n. Then a Tucker tensor
Definition 9.4 Let A
R
A t ¼ C 1 U 1 2 ... d U d ,
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