Database Reference
In-Depth Information
1.
Subtensors of order d
-1
are mutually orthogonal,
i.e.
,
D
c
j
j
ðÞ
∈
n
ðÞ
i/δ
i
k
j
k
8k
cðÞ
i
ðÞ
∈
n
ðÞ
;
∈
d
i
k
,
j
k
∈
n
k
:
,
3.
The k-mode singular values are ordered,
i.e.
,
s
ðÞ
i
k
þ
1
s
ðÞ
i
k
8k
∈
d
i
k
<
n
k
:
,
The subsequent bound is an improvement over that in Property 10 in
[DLDMV00] by a factor of
d
through a slight modification of the proof therein.
Proposition 9.1
A rank-
t
truncated HOSVD A
t
of A
n
satisfies
∈
R
2
d
X
k
X
1
s
ðÞ
i
k
2
F
k
A A
t
k
:
∈
d
t
k
<
i
k
n
k
Proof
¼
X
t
<
i
n
2
F
c
i
k
A A
t
k
d
X
k∈
d
X
1
c
i
¼
i
n
t
<
d
X
k
X
X
1
c
i
∈
d
t
k
<
i
k
n
k
i
ðÞ
∈
n
ðÞ
2
d
X
k∈
d
X
1
s
ðÞ
i
k
¼
:
□
t
k
<i
k
n
k
As opposed to the matrix case, the rank-t truncated HOSVD does
not
provide an
optimal rank-t approximation. Nevertheless, it appears to be a promising candidate
for a tensor generalization of PCA-based CF because:
1. Certain properties of the matrix case are preserved.
2. The quality of approximation may be estimated.
3. The computation of the HOSVD reduces to the computation of matrix SVDs.
4. In particular, this permits incremental computation.
5. The projection approach to the prediction of unknown values may be carried
over in a straightforward fashion.
A major drawback, however, is the exponential dependency of computational
complexity on the number of modes. Therefore, it is suitable for applications with a
moderate number of modes only. To conclude this section, we point out that a
mode-scalable generalization of the SVD will be introduced in Sect.
9.3
.
