IEEE Geoscience and Remote Sensing Magazine - March 2023 - 36

R
X bb b N
12
= x %%% f
=
/ r rr
1
() ()
r
where r
f % b()
r
x are nonzero weight parameters and bb
() ()
rr
12
N denotes a rank 1 tensor, with b R()
n
r !
rank denoted by rank ( ) RXCP
In
()
r
(11)
%%
. The CP
= is the sum of the rank 1
tensors. The core consistency diagnostic (CORCONDIA)
has gained popularity as a useful tool for calculating the
number of components in parallel factor analysis and Tucker
decomposition models. Several modifications for CORCONDIA
have been proposed to improve efficiency [131],
[132]. To address this problem of rank estimation, Shi et
al. [133] regarded the weight vector of the orthogonal CP
decomposition as the vector of singular values. Das et al.
[134] proposed a noise-invariant tensor-based rank estimation
approach and verified the proficiency of our proposed
estimation in noise cases.
BLOCK TERM DECOMPOSITION
The BT decomposition of a three-order tensor
is defined as [135]
R
XG AB C
1
= ## #
=
/
r
r 12 3rr r
(12)
z
XS (, ,)kk 1
k
)|= /
=1
TABLE 4. THE MAIN ABBREVIATIONS USED IN THIS ARTICLE.
ABBREVIATION FULL NAME
AD
ADMM
AVIRIS
CP
CS
BT
HOSVD
HYDICE
HS
HR
LMM
LR
MS
MODIS
NMF
PCA
ROSIS
RS
SNN
SU
SVD
TNN
t-SVD
TV
TT
TTN
TR
Anomaly detection
Alternating direction method of multipliers
Airborne Visible/Infrared Imaging Spectrometer
Canonical decomposition/parallel factor analysis
Compressive sensing
Block term
Higher-order SVD
Hyperspectral Digital Imagery Collection Experiment
Hyperspectral
High resolution
Linear mixing model
Low rank
Multispectral
Moderate-Resolution Imaging Spectroradiometer
Nonnegative matrix factorization
Principal component analysis
Reflective Optics System Imaging Spectrometer
Remote sensing
Sum of the nuclear norm
Spectral unmixing
Singular value decomposition
Tensor nuclear norm
Tensor SVD
Total variation
Tensor train
Tensor trace norm
Tensor ring
36
and it can also be expressed as the SNN of all the frontal
slices of :Xt
z
X
)|= / t
k=1
X (: ,:,) .k
)
(16)
For a more intuitive understanding of the preceding
tensor decompositions, examples of third-order tensors
are provided in Figure 3, which benefits the consequent
tensor decomposition-based research of thirdorder
HS data.
TENSOR SINGULAR VALUE DECOMPOSITION
Here,
X Rhv z
##
!
= )) )
can be factorized by t-SVD as [54]
XU SV
where U RR ,Vhh zv vz
##
!!
!
(15)
X Rhv z
##
!
where ,, ,ABhv zh vvL and C Rr
## ##
G RR R
LL L
!! !
hL
r
r
!
zLz
#
.
Each of R component tensors can be expressed by rank
(, ,)LL Lhv z
Tucker decomposition. BT decomposition
can be regarded as the combination of Tucker and CP
decomposition. On the one hand, (12) becomes Tucker decomposition
when
component is represented by a rank (L, L, 1) tensor, (12) is
written as
R
X AB cr
1
= $%
=
/ r
r
where matrix A Rr
rank L. If rank L E Rr
be rewritten as
!
!
hLr
#
and matrix B Rr
!
R
X Ec .
r=1
= / %
r
TENSOR NUCLEAR NORM
Let XU SV
= ))
r
) be the t-SVD of X R . The tensor
!
hv z
##
nuclear norm (TNN) is the sum of singular values of X ,
that is,
(14)
vLr
#
r $
r
are also
hv is factorized as AB ,T (13) can
#
T
r
(13)
R .1= On the other hand, when each
(17)
## are orthogonal tensors and
S Rhv z## is an f-diagonal tensor. The details of t-SVD are
described in Algorithm 1.
TENSOR TRAIN DECOMPOSITION
The TT decomposition [26] of an N-order
GG G()
RrI rnn n
composition is defined as rank X ( )[ ,, ,]. Each
- ##1 ,, ,, ,.nN rrand
TT
==f
12
= f
entry of the tensor, X is formulated as
(, ,)
ii (:,,:).ii(:,,:)
1ff=
N
1 (:,,:)
1
()
2
()
N
N
01 N
XG1 GG()
2
()N
X RII
!
12 IN
## #g
is represented by cores {, ,}, where G()n
1
!
N =1 The rank of TT de=
frr r
(18)
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE MARCH 2023
IEEE Geoscience and Remote Sensing Magazine - March 2023

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