IEEE Geoscience and Remote Sensing Magazine - March 2023 - 35

LEMMA 1 (CIRCULAR DIMENSIONAL PERMUTATION
INVARIANCE [27])
If the TR decomposition of X is (, ,, ),
X RII IInn 11 n 1+- is defined as circular shifting
XG GG() ()
=U
!n
!
!n
## ## #gg
the dimensions of X by n, we obtain the following
relation:
X GG GG GG()
=U ff+nn NN() () () () ()11 2
^h (5)
,, ,, ,, ,.
DEFINITION 8 (MIXED l1,0 PSEUDONORM [124])
Given a vector y Rm
! and index sets ,, ,, (1in1
nm ),# where each ii is a subset of ,, ,m1f il
any il ,! and , 1 i
of y is defined as
<< ff<< << << <<
10, ii i
yy ,, ,yy1 11 10in
=
ii iff #
ii+ Q= for
i = i f=1 ,, ,m the mixed l10, pseudonorm
n
i ^h (6)
where y ii denotes a subvector of y, with its entries specified
by ,ii and << · 0 calculates the number of the nonzero
entries in ().$
DEFINITIONS OF COMMON TENSOR
DECOMPOSITIONS
Before providing a comprehensive survey of five practical
HS applications, we summarize eight types of tensor decompositions
widely used in HS data processing. Practitioners
can obtain a basic theoretical knowledge of the relevant
background in preparation for the following research.
TENSOR TRACE NORM
This is the sum of the nuclear norm (SNN) of the mode-k
unfolding matrix for a three-way HS tensor [125]:
3
X
SNN = a X() )
k=1
| / k
a $
=
k
where weights ka satisfy (, ,) and R a = 1 .
k 01k 23
3
k=1 k
TUCKER DECOMPOSITION
The Tucker decomposition of an N-order tensor X !
RII
12 IN## #g
is defined as [20], [126], [127]
X A BB BNN1122
= ## #g
where A Rrr
B Rn
!
!
12 rN
## #g
nnnN represent factor matrices. The
Tucker ranks are represented by rank X ( )[ ,, ,].
Here, one variation of Tucker decomposition is
Ir ,, ,,
#
= 12 f
Tucker
= frr r
12 N
worth noting. The " Tucker 1 " decomposition of a thirdorder
array sets two of the factor matrices to be the
identity matrix:
X A B11
= #
(9)
which is equivalent to standard 2D principal component
analysis (PCA) since
MARCH 2023 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
xxi
i
2 = /
XX=t
X()k
X k12
Xk
! n
X
N
[ ]GG()n
n
= %
=1
fft( ,[ ], )3
2
(8)
stands for a core tensor and
(7)
f
12 ()N
XA .
() B= #1111 ()
(10)
" Tucker 1 " can find those components that obtain the
variation in mode n, independent of the other modes, later
known as higher-order SVD (HOSVD). Lathauwer et al.
[126] were convinced that HOSVD is a generalization of
the matrix SVD and discussed how to compute the leading
left singular vectors of
X .()n Lathauwer et al. [128] proposed
and named the higher-order orthogonal iteration. This
more efficient technique iteratively calculates factor matrices,
specifically computing only the dominant singular
vectors of X(n) and using an SVD rather than an eigenvalue
decomposition or even just an orthonormal basis of the
dominant subspace.
CANONICAL DECOMPOSITION/PARALLEL FACTOR
ANALYSIS DECOMPOSITION
The CP decomposition of an N-order tensor
is defined as [20], [129], [130]
!
X RII IN12
##$#
TABLE 3. THE NOTATIONS USED IN THIS ARTICLE.
NOTATION
x
x
X
vec(X)
ii
x ii
X Rhv z
##
!
I
X ,,
hv zii i
XX and
X (: ,:,)i
(,:, :),(:, ,:),ii
XX ,,
,,
1 = /
hv zii i
XX,,
,,
F = /
hv zii i
z
XX
k
):(:,:, )k
=1
= / t
v
i()X
XXi
i
xxi
i
1 = /
) = v/ ()
)
Singular values of matrix X
Nuclear norm
l1 norm
l2 norm
Fourier transformation of X along mode 3
Unfold( ,)kX
Second mode-k unfolding
Mode-k permutation
Circular shifting the dimensions of X by n
Multiproduct of G()n
Tensor nuclear norm
2
hv zii i
Frobenius norm
hv zii i
DESCRIPTION
Scalars
Vectors
Matrices
vec(X) stacks the columns of X
Index sets
Subvector of x, with its entries specified by i
Tensors with three modes
i
Identity tensor
(, ,)hv zii i element of X
ith horizontal, lateral, and frontal slices
l1 norm
35
IEEE Geoscience and Remote Sensing Magazine - March 2023

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