Database Reference
In-Depth Information
Fig. 9.3 Matricization of
a 3-mode tensor after
adding a matrix “to the
right,” i.e., in mode
2(d
¼
2)
1-mode
2-mode
3-mode
9.1.4
...
And How to Compute It Adaptively
The tensor generalization of the SVD updating problem discussed in Sect.
8.2
is as
follows: how can an HOSVD of a tensor
A
n
with entries
∈
R
a
i
, i
n
e
d
b
i
ðÞ
, i
d
¼ n
d
e
a
i
¼
n
ðÞ
, and (e
k
)
z
:
¼ δ
zk
be expressed in terms of an
HOSVD of
A
? We refer to the
d
-1-mode subtensor
B
of
A
as a slice.
Establishing a new tensor by adding a slice to a given one generalizes the previously
discussed situation in which a new matrix is established by adding a column.
To extend the adaptive framework presented in Sect.
8.3.2
, we need to observe
the resulting changes in the mode matricizations, which are graphically illustrated
by Fig.
9.3
. Let us first consider the case where
k 6¼ d
and assume that the multi-
index enumeration chosen for the matricization
A
ðÞ
satisfies
n
e
d
where
A ¼ aðÞ∈
R
B
∈
R
,
i
d
j
d
) υ
ðÞυ
ðÞ8
i, j
∈
n
:
(This may, e.g., be achieved by using a
lexicographic ordering
.) Then the
k
th
mode matricization is of the form
h
i
,
A
ðÞ
¼
A
ðÞ
B
ðÞ
where
A
(
k
)
,
B
(
k
)
are
k
th mode matricizations of the tensors
A, B
with respect to a suitably
chosen enumeration. Hence, if
denotes the number of columns of
B
(
k
)
,wemay
obtain an SVD of
A
ðÞ
, given an SVD of
A
(
k
)
, which is assumed to be available from
previous computations, by applying the procedure presented in Sect.
8.3.2
α
α
times.
