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n
3
n
3
n
3
n
2
n
2
n
1
U
3
n
3
=
U
1
A
n
1
n
1
n
1
n
2
C
n
2
U
2
Fig. 9.2 Illustration of the HOSVD for a 3-mode tensor A. The
dotted lines
indicate the
boundaries of the submatrices or tensors, respectively, corresponding to a truncated HOSVD
where
:¼ A
1
U
1
2
...
d
U
d
C
and
U
i
,
i
d
, is a matrix of left singular vectors corresponding to the t
i
largest
singular values in an SVD of A
(
i
)
and is referred to as truncated higher-order
singular value decomposition (HOSVD) of rank-t.Thei
k
th k-mode singular
value of A is defined as
∈
s
F
¼
X
s
ðÞ
ik
c
i
:
:¼
cðÞ
i
ðÞ
∈
n
ðÞ
i
ðÞ
∈
n
ðÞ
As in the matrix case, we shall refer to the factorization corresponding to the rank-n
truncated HOSVD of an n-dimensional tensor simply as an HOSVD thereof. Similarly
to Fig.
8.2
, a graphical representation of the (truncated) HOSVD is given in Fig.
9.2
.
Definition 9.4 immediately leads to Algorithm 9.1 of a truncated HOSVD.
Algorithm 9.1 Truncated HOSVD
n
, truncation rank t
n
Input: tensor
A
∈
R
ð
n
k
;t
k
Þ
t
Output: factor matrices
U
k
∈
R
k ¼
1
,
...
,
d
, core tensor
C
∈
R
,
1: for
k ¼
1,
...
,
d
ð
n
k
;t
k
Þ
calculate matrix
U
k
∈
R
2:
of principal
left singular vectors of
matricization
A
(
k
)
3: end for
4:
C
:
¼ A
1
U
1
2
...
d
U
d
Example 9.2
Consider the tensor
A
from the previous Example 9.1. As one easily
verifies, it holds that
,
T
20
02
A
ðÞ
A
ðÞ
¼
