Zoom out Search Issue

C
A
(I
1
× I
2
× I
3
)
(R
1
× R
2
× R
3
)
(R
2
× l
2
)
(l
3
× R
3
)
(I × R
1
)
B
T
=
∼
[FIG4] The Tucker decompostion of a third-order tensor. The
column spaces of ,A ,B and C represent the signal subspaces
for the three modes. The core tensor G is nondiagonal,
accounting for the possibly complex interactions among tensor
components.
IEEE SIGNAL PROCESSING MAGAZINE [151] MARCH 2015
TUCKER DECOMPOSITION
Figure 4 illustrates the principle of TKD, which treats a tensor
RX
II I
N12
!
## #g
as a multilinear transformation of a (typically
dense but small) core tensor RG
RR R
N12
!
###g
by the factor
matrices [, ,, ] ,B bb b R
()
() ()
()
n
nn
R
n
IR
12
n
nn
f !=
#
,, ,nN12f= [3],
[4], given by
,bb bgX
() () ( )
rr r
rr r
N
r
R
r
R
r
R
12
111
N
N
N
N
12
12
2
2
1
1
%%%gg=
g
===
^h
///
(7)
or equivalently
BB BXG
() () ( )
N
N
1
1
2
2
## #g=
;,,, .BB BG
() () ()N12
f=
" ,
(8)
Via the Kronecker products (see Table 2), TKD can be expressed
in a matrix/vector form as
()XBGB B B B
()
()
()
() () () ()
n
n
n
NnnT11 1
77777gg=
+-
.() [ ] ()BB Bvec vecXG
() ( ) ()NN11
777g=
-
Although Tucker initially used the orthogonality and ordering
constraints on the core tensor and factor matrices [3], [4], we
can also employ other meaningful constraints.
MULTILINEAR RANK
For a core tensor of minimal size, R
1
is the column rank (the
dimension of the subspace spanned by mode-1 fibers), R
2
is the
row rank (the dimension of the subspace spanned by mode-2
fibers), and so on. A remarkable difference from matrices is that
the values of
,,,RR R
N12
f can be different for .N 3$ The
N-tuple ( , , , )RR R
N12
f is consequently called the multilinear
rank of the tensor .X
LINKS BETWEEN CPD AND TUCKER DECOMPOSTION
TKD can be considered an expansion in rank-1 terms (polyadic but
not necessary canonical), as shown in (7), while (4) represents
CPD as a multilinear product of a core tensor and factor matrices
(but the core is not necessary minimal); Table 3 shows various
other connections. However, despite the obvious interchangeabil-
ity of notation, the CPD and TKD serve different purposes. In gen-
eral, the Tucker core cannot be diagonalized, while the number of
CPD terms may not be bounded by the multilinear rank. Conse-
quently, in signal processing and data analysis, CPD is typically
used for factorizing data into easy to interpret components (i.e.,
the rank-1 terms), while the goal of unconstrained TKD is most
often to compress data into a tensor of smaller size (i.e., the core
tensor) or to find the subspaces spanned by the fibers (i.e., the col-
umn spaces of the factor matrices).
UNIQUENESS
The unconstrained TKD is in general not unique, i.e., factor matri-
ces
B
()n
are rotation invariant. However, physically, the subspaces
defined by the factor matrices in TKD are unique, while the bases
in these subspaces may be chosen arbitrarily—their choice is
compensated for within the core tensor. This becomes clear upon
realizing that any factor matrix in (8) can be postmultiplied by any
nonsingular (rotation) matrix; in turn, this multiplies the core
tensor by its inverse, i.e.,
;,,,BB BXG
() () ( )N12
f=
" ,
,;,,,BR BR B RH
() () () () ( ) ( )
NN
11 22
f=
" ,
, ;,,,RR RHG
() () ( )N12
11 1
f=
-- -
" ,
(9)
where the matrices R
()
n
are invertible.
MULTILINEAR SVD
Orthonormal bases in a constrained Tucker representation can
be obtained via the SVD of the mode- n matricized tensor
XUV
()nnn
n
T
R= (i.e., ,BU
()n
n
= ,, , ).nN12f= Because of the
orthonormality, the corresponding core tensor becomes
.UU USX
TT
N
N
T
1
1
2
2
## #g= (10)
[TABLE 3] DIFFERENT FORMS OF CPD AND TUCKER
REPRESENTATIONS OF A THIRD-ORDER TENSOR
.RX
IJK
!
##
CPD TKD
TENSOR REPRESENTATION, OUTER PRODUCTS
abcX
rr r r
r
R
1
%%m=
=
/
abcgX
rrr r r r
r
R
r
R
r
R
111
123 1 2 3
3
3
2
2
1
1
%%=
===
///
TENSOR REPRESENTATION, MULTILINEAR PRODUCTS
ABCXD
123
###= ABCXG
123
###=
MATRIX REPRESENTATIONS
()XADCB
()
T
1
9=
()
XAGCB
() ()
T
11
7=
()XBDCA
()
T
2
9= ()XBGCA
() ()
T
22
7=
()XCDBA
()
T
3
9= ()XCGBA
() ()
T
33
7=
VECTOR REPRESENTATION
() (CBA)dvec X 99= () ( ) ()vec vecCBA GX 77=
SCALAR REPRESENTATION
xabc
ijk ir jr krr
r
R
1
m=
=
/
xgabc
ijk ir jr krrrr
r
R
r
R
r
R
111
123 1 2 3
3
3
2
2
1
1
=
===
///
MATRIX SLICES (:,: , )kXX
k
=
(, ,,)cc cdiagXA B
kk
kk
R
T
12
f=
(:,: , )crXA G B
kkr
r
R
T
1
3
3
3
3
=
=
/
Previous Page | Contents | Zoom in | Zoom out | Front Cover | Search Issue | Next Page
q
q
M
M
q
q
M
M
q
M
THE WORLD’S NEWSSTAND
®
Previous Page | Contents | Zoom in | Zoom out | Front Cover | Search Issue | Next Page
q
q
M
M
q
q
M
M
q
M
THE WORLD’S NEWSSTAND
®