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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
,