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Let us now turn to the “troublemaker mode,” i.e., the frontal mode,
k ¼ d
. In this
case, a matricization is of the form
:
A
ðÞ
B
ðÞ
A
ðÞ
¼
T
Here, applying the procedure to
A
ðÞ
and assigning the appropriate matrix
of right singular values to
U
d
may do the trick. Please note, however, that avoiding
any explicit representation of a matrix of right singular vectors of a matrix to
which columns are appended is critical as regards realtime scalability of the
procedure. Therefore, we shall develop a method to compute the projection of
the added slide onto the range of the considered matrix of singular vectors in the
following. Let
T
¼:
A
¼ U
Sz
0
T
A
ðÞ
V
T
c
|
{z
}
¼:S
be a decomposition according to (
8.15
).
Moreover, let
U S V
T
be a truncated SVD of
S
,
U
:¼ VV
. Then
U S V
T
is a truncated SVD of
A
. We shall seek after
A V
V
T
e
n
, i.e., the projection of
the last column of
A
onto the principal subspace spanned by
V
, where
e
n
denotes
the vector of all zeros except for the last entry, which is 1. Since
:¼ UU
, and
V
z
c
S V
T
e
n
¼ U
T
Se
n
¼ U
T
,
z
c
A
V V
T
e
n
¼ UU U
T
ð
9
:
3
Þ
which avoids the “column scaling.”
The entire update procedure is summarized in Algorithm 9.2.
Algorithm 9.2 HOSVD update
ð
n
k
;
t
k
Þ
of principal left singular vectors of
A
(
k
)
,
k ¼
1,
Input: matrices
U
k
∈
R
...
,
d
1, matrix
U
of principal left singular vectors of (
A
(
d
)
)
T
, new slice
B
R
n
ðÞ
∈
(continued)
