Graphics Reference
In-Depth Information
I
3
I
2
I
1
I
3
I
2
I
2
A
(1)
I
1
I
3
I
2
I
1
I
3
I
1
I
1
A
(2)
I
2
I
3
I
2
I
1
I
2
I
1
I
1
I
1
I
1
A
(3)
I
3
Figure 9.21
Matricizing a third-order tensor. The tensor can be matricized in three ways, to obtain
matrices having its mode-1, mode-2, and mode-3 vectors as columns. (From [Vasilescu
and Terzopoulos 04]
c
2004 ACM, Inc. Included here by permission.)
the ordinary matrix product of
B
and the mode-
m
matricized tensor
C
]
=
BA
]
.
(9.9)
[
m
[
m
The mode-
m
product is denoted by
×
m
.
To understand the
M
-mode SVD , it is helpful to see how the SVD works on a
matrix regarded as a tensor. As described in Section 9.3.2, the SVD decomposes
amatrix
D
into a product of a diagonal matrix and two orthonormal matrices
associated with the column and row spaces of
D
:
U
2
T
D
=
U
1
Σ
(9.10)
(here
U
1
and
U
2
are used in place of
U
and
V
). Because a matrix is a second-
order tensor, this can be written in tensor form by using the mode-1 and mode-2
