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On Combining Trilinear Decomposition and ICA#
GONG Xiaofeng1, HAO Yana1, LIN Qiuhua1, LIU Zhiwen2, XU Yougen2*
(1. Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology,
LiaoNing DaLian 116024; 5
2. School of Information and Electronics, Bejing Institute of Technology, Beijing 100081)
Foundations: This work was supported in part by Doctoral Fund of Ministry of Education of China under grant
(20110041120019), and National Natural Science Foundation of China under grants (60971097, 61072098, and
61105008).
Abstract: We present a new canonical polyadic decomposition (CPD) algorithm to exploit both source
statistical independence and trilinear structure of a three-way tensor based on independent component
analysis (ICA) and joint diagonalization (JD). More exactly, ICA is first performed on the matricized
tensor to exploit the statistical independence in one mode, JD is then carried out upon the initial ICA 10
results to restore the trilinear structure of the original tensor, and rank-1 approximation is finally used
to extract the parallel factors. The proposed algorithm is able to overcome the converging difficulties of
standard CPD in the presence of collinearities. Moreover, it is able to exploit the trilinearity of the
target tensor more thoroughly than existing methods that combine CPD and ICA, and thus is less
sensitive to the incorporated ICA stage than these methods. Simulations are provided to compare the 15
proposed algorithm with both standard CPD algorithms and algorithms that combine CPD and ICA.
Keywords: Independent Component Analysis; Canonical Polyadic Decomposition; Joint Diagonalization;
Tensor
0 Introduction 20
Blind sources separation (BSS) aims at separating sources from their mixtures with no prior
knowledge on the mixing process and sources other than some practical assumptions (. the
source independence). It was widely concerned in the field of signal processing in recent decades,
and numerous BSS algorithms were developed in the open literature[1-10]. Independent component
analysis (ICA) has been playing an instrumental role in solving BSS problems[2]. These works are 25
mostly based on two-way (. matrix) formulization of the observed mixtures and propose to
decompose the mixture matrix into maximally independent components. Most ICA algorithms
feature the optimzation over higher-order statistics based criteria (. kurtosis maximization,
maximization of negentropy) [1-4]. The algebraic algorithms, on the other hand, resort to
higher-order cumulants to build higher-order tensors (or multiple jointly diagonalizable matrices) 30
to allow BSS based on tensor decomposition [5] or joint diagonalization [6, 7].
In contrary to ICA that uses higher-order statistics for matrix formulized signals, some recent
works are devoted to the analysis of multi-way data, where the signals are aquired in a multilinear
manner (. electromagnetic vector-sensor array signals[8], multi-subject fMRI signals[9]). In such
applications, the mixing procedure is in terms of two sets of parallel factors, yielding trilinear 35
structure in the observed three-way signals. Moreover, noting that this trilinear structure could be
uniquely identified under some mild conditions [10], BSS could be done directly in the data domain
via three-way canonical polyadic decompositions (CPD, also known as trilinear decompositions or
three-way parallel factor analysis) [11, 12]. In terms of computing three-way CPD, we note that the
most typical algorithms (. alternating least squares: ALS[13], or joint diagonalization[14] based 40
algorithms) measure the fitting level of the trilinear structure in the least squares sense, and thus
do not incorporates any statistics which may possibly exist at one mode of the tensorial signal. As
a result, a lot of standard CPD algorithms suffer from converging difficulties in ill-conditioned
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problems where collinearities exist at one or two modes [15].
To cope with the above problem, some recent works focused on inserting ICA into three-way 45
CPD [9,16,17]. More exactly, in [16], CPD is performed over much higher-order tensors (5th-8th
order tensors) constructed from the 4th-order cumulants of trilinear signals, to enable the use of
ICA in three-way signal processing. In [9, 17], the three-way signal is unfolded into a matrix to
facilitate the use of ICA for extraction of factors at one mode, and rank-1 approximation is then
performed on the ICA results to calculate the remainder components. These methods, taking 50
advantages of using source statistics in CPD, generally provide improved converging speed as
well as performance. However, these methods rely heavily on the ICA stage and thus is expected
to be sensitive to the choice of ICA algorithms or may fail in scenarioes where higher-order
statistics are absent.
In this paper, we propose a new algorithm to combine ICA and three-way CPD. More 55
precisely, after ICA is performed on the matricization of a tensor, a joint diagonalization (JD)
based strategy is used upon the initial ICA results to recover the original trilinear structure, and
rank-1 approximation is further performed to generate the parallel factors. We note here that this
strategy has been partially used in [14] but was not previously combined with ICA. In the rest of
the paper, problem formulation is given in Section 1, the new algorithm as well as discussion and 60
remarks is presented in Section 2. In Section 3, we use simulations to compare the performance of
the proposed algorithm with other existing ones. Section 4 presents the conclusion.
1 Problem Formulation
We assume R mutually independent non-Gaussian sources 1, , KR ∈s s" ^ are mixed with
two sets of parallel loading factors 1, , IR ∈a a" ^ , and 1, , JR ∈b b" ^ 65
a b
1
, ,a b s A B S
=
= =∑ D DR r r r
r
X (1)
where “ D ” denotes tensor outer product, I J K× ×∈^X , A � 1[ , , ]Ra a" , 1[ , , ]RB b b� " , and
1[ , , ]RS s s� " .
The matricization of the above trilinear models is: 70
T( )X A B S= : (2)
where X is defined as (( 1) , ) ( , , )i J j k i j k− + =X X , and “ : ” denotes Khatri-Rao
(column-wise Kronecker) product.
The above trilinear model could be uniquely identified (up to scaling and permutation 75
ambiguities) under mild conditions [10], and many algorithms have been proposed to this end in the
open literature [13, 14]. However, most of the conventional methods exploit only the trilinear
structure present in the tensorial dataset while ignore the source statistical independence that might
be useful for the identification.
As such, our goal is to identify the trilinear model (2) with independence constraint on the 80
columns of S .
2 The Proposed Algorithm
CDP Based on ICA and Joint Diagonalization
Our algorithm consists of three stages. In the first stage, we note that the matrix formulization
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in (2) is actually an instantaneous mixing model if we denote ( )M A B� : , and therefore ICA 85
could be performed to obtain an initial result:
-1arg max G ( )
M
M M X′ =� (3)
where ′M� is the estimate of M and G ( )⋅ is a measure of statistical independence (.
kurtosis or non-gaussianity [3, 4]). Many ICA algorithms could be used in this stage [1-4] which,
however, mostly ignore the Khatri-Rao structure in M , and thus ′M� would differ from M 90
with an invertible matrix P : -1′ =M MP� . Therefore, the considered problem amounts to
estimating P to restore the Khatri-Rao structure in M :
2
, ,
, , arg min -
A B P
A B P M P A B′=� � � � :
F
(4)
In the second stage, we propose to use the matrix joint diagonalization (JD) based stratedgy
[14] to tackle the problem in(4). The main steps are summarized as follows: 95
Unvectorize the columns of ′M� into matrices I Jr ×∈M ^ , 1, 2,...,r R= , and calculate a set
of 4th-order tensors:
, Φ( , )M M=r u r uT (5)
where operator Φ( )⋅ is defined as:
[ ( , )]X YΦ = + − −ijkl ij kl ij kl il kj il kjx y y x x y y x (6) 100
- Estimate the null subspace of tensors ,{ , 1,..., }r u r u R=T to obtain R target matrices R Rr ×∈C ^ ,
1, 2,...,r R= :
, ,, 1 ( ) 0C= =∑ R r s t s ts t T (7)
- Jointly diagonalize 1, ..., RC C with R R×∈F ^ :
, 1, 2,...,C FΛ F= =Tr r r R (8) 105
where rΛ is diagonal, F is the estimate of -1P , and then we refine ′M� obtained in the
first stage via 1−′=M M F� � .
With the above steps we note that the Khatri-Rao structure of ′M� is restored, and thus in
the third stage we unvectorize the rth column of M� into a matrix, and do rank-1 approximation
(simply by taking the left and right dominant singular vectors of this matrix) to obtain the 110
estimates of the rth columns of A and B . S could be estimated with(2).
In practice, we usually let the third mode be the longest one (if not, we permute the index of
the longest mode of the tensor to the third), so as to reduce the complexity of subsequent JD based
scheme. Similar idea has been adopted in [14] and it is proven quite efficient in decomposing a
tensor with one long mode and two short modes. However, in case that two modes are long, 115
memory problem would occur and solution to this problem will be addressed in the next
subsection.
Memory Saving Stratedgy to The Proposed Algorithm
The memory problem generally occurs in the second stage if the target tensor has two long
modes. More exactly, we note that (7) is usually solved with following two substeps: 120
- Calculate a matrix
2 2 ( 1)/ 2CI J R R× +∈T as follows:
1,1 , 1,2 1,3 1,(vec( ),..., vec( ), vec( ), vec( ),..., vec( ))T −= R R R RT T T T T (9)
where ,vec( )r uT denotes the vectorization of ,r uT . Then (7) is converted to the following
problem:
1,1 ,R 1,2 1,3 -1,(( ) ,..., ( ) ,2( ) ,2( ) ,..., 2( ) ) 0T C C C C C =Tr r R r r r R R (10) 125
- Select R right singular vectors rw , r = 1,2,…,R, that are associated with the R smallest
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singular values of T . Stacking back the elements of rw in the way defined by (10) yields the
estimates of rC .
It is important to note that the row dimensionality of T would become extremely large
when the target tensor has two long modes, and thus would take huge memory which prevents the 130
proposed algorithm from practical use. To tackle this, we propose the following memory saving
scheme to replace (9) and (10) (we assume K J I> � without loss of generality):
- Calculate matrix ( , , , )r u j lT for fixed indices ,j l as follows:
( , , , ) ( , ) [ ( , )]T M M= Φr u j l r u ijkli k (11)
and further calculate the following matrix: 135
( , ) (1,1, , ) ( , , , ) (1,2, , ) (1,3, , ) ( 1, , , )[vec( ),..., vec( ), vec( ), vec( ),..., vec( ))]W T T T T T −=j l j l R R j l j l j l R R j l (12)
It can be easily proven that the following equation holds:
( , ) ( , )
1 1
E T T W W
= =
= ∑∑� J JH Hj l j l
j l
(13)
- Select R eigenvectors rw , r = 1,2,…,R, associated with the R smallest eigenvalues of E .
The estimates of rC can then be obtained by stacking back elements of rw in the way defined 140
by (10). The above procedure is based on the fact that eigenvectors of E are exactly the same as
the right singular vectors of T .
We note from (13) that E could be calculated by sequential summation of
2 ( 1) /2
( , ) CI R Rj l × +∈W over indices (j,l), and this procedure takes much less memory than T , at the
cost that the exhaustion of indices (j,l) in sequential manner might be time consuming if J is large. 145
To further tackle the above problem, we note that the columns of ( , )j lW are in fact columns
selected from T , and thus a small number of sequential summations in the manner defined in (13)
would yield a matrix with theoretically equivalent column subspace as T . Therefore, when
handling large tensors, a more practical way is let the upperbound of indices (j,l) in (13) be small
so as to accelerate the algorithm. 150
Comparison with Existing Algorithms
We note that the proposed algorithm, termed as TRIlinear canonical polyadic decomposition
via ICA and JD (TRICA-JD), distincts itself from existing algorithms that combine CPD and ICA.
More exactly, TRICA-JD does not involve decomposition of higher-order tensors constructed via
4th order cumulant of three way data, as is in ICA-CPA [16], and thus is computationally more 155
efficient than ICA-CPA. When compared with Tensor probablistic ICA (TICA) [9] and CPD based
on Single Mode BSS (CPD-SMBSS) [17], we note that TRICA-JD involves a stage of Khatri-Rao
structure restoration via JD, which provides precise exploitation of the trilinear structure of the
tensorial data, and this stage is not incorporated in TICA and CPD-SMBSS. Therefore, TRICA-JD
is able to exploit the trilinearity of the tensorial data more thoroughly than TICA and 160
CPD-SMBSS, and this may result in improved performance.
Moreover, we note that ICA-CPA, TICA, CPD-SMBSS rely heavily on the source statistics
or the incorporated ICA stage. More exactly, ICA-CPA would be invalid for gaussian sources, as
the fourth order cumulants would vanish. CPD-SMBSS and TICA would be sensitive to the ICA
stage as the parallel factors are obtained directly via sequential rank-1 approximation of the ICA 165
results. The proposed TRICA-JD, on the other hand, uses a JD based stratedgy to exploit the
trlinear structure of the tensor data, which is independent of the source statistics as well as the ICA
stage. Therefore, TRICA-JD is able to generate robust performances for sources of various
statistics and for a variety of ICA choices.
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We summarize the proposed TRICA-JD in Table 1: 170
Summerization of TRICA-JD
Input: A third-order tensor I J K× ×∈^X and number of sources R
1: Matricize X via (( 1) , ) ( , , )i J j k i j k− + =X X ;
2: Perform ICA on X to initially estimate ( )M A B� : ;
3: Calculate 4th-order tensors ,r uT , , 1, 2,...,r u R= via (5);
4: Obtain matrices rC , 1, 2,...,r R= via (9) and (10), or via (11) - (13) as a memory saving alternative;
5: Calculate the joint diagonalisor F of rC , 1, 2,...,r R= ;
6: 1−←M MF , estimate A , B via SVD based stratedgy, and S by (2);
Output: The source matrix S and two parallel loading matrices A , B
3 Simulation Results
In this section, we illustrate with numerical experiments the comparison of the proposed
TRICA-JD algorithm with two standard trilinear CPD algorithms: COMFAC [13], and trilinear 175
CPD via joint diagonalization (Tri-JD) [14], as well as two algorithms that combine CPD and ICA:
TICA [9], and CPD-SMBSS [17], with particular concern on the performance in difficult scenarioes,
. in the presence of collinearities.
Simulation 1: We construct a trilinear tensor following (1) with the same setting as the first
simulation in [17] to examine the performance of the compared algorithms in case that two 180
loading matrices contain nearly collinear structures. More exactly, the columns of A are sine
waves with shifted phases and equal frequencies: sin(2 / 50)j ft jπ π= +a , sampled from 0 to 990
ms with the interval of 10 ms. The jth and (j-1)th columns of B are generated as:
1 j j−= +b b v , j = 2,3,…, 10, and 1 1=b v , with the entries of jv drawn from independent
standard normal distributions,. The columns of S are independent sine waves with different 185
frequencies that are included in ICALAB benchmark acsin10d[18]. We note here that the
neighboring columns of A and B are highly correlated[17] and S contains independent
components. Signal-to-interference ratio (SIR) is used as the performance measure defined as:
2
10 2SIR( , ) 10 log ( )
a a = −
∑
∑� �
tt
t tt
a
a a
(14)
where a and a� are normalized random variables with zero mean and unit variance, and 190
a� is the estimate of a . In addition, we carry out 50 independent runs and denote by SIR A ,
SIRB , SIR S the mean SIR values for all the columns of , ,A B S , respectively, averaged over all
the 50 independent runs. The ICA algorithm for TRICA-JD, TICA, CP-SMBSS is the second
order blind identification (SOBI) [7] and the results are listed in Table 2.
195
Performance of compared methods in decomposing trilinear tensor with collinearities in two modes and
independence in the other
SIR_A(dB) SIR_B(dB) SIR_C(dB) Runtime(s)
TRICA-JD
COMFAC
Tri-JD
TICA
CPD-SMBSS
From this table we note the proposed TRICA-JD algorithm provides the best estimation
accuracy, as is indicated by the SIR values, and quite nice convergence speed (only slightly 200
inferior to Tri-JD and TICA). In addition, we note that the two standard CPD algorithms behave
quite diversely, with Tri-JD providing nice performance while COMFAC suffering seriously from
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the double bottlenecks in the target tensor. The reason might be that COMFAC is based on
alternating least squares (ALS), and slow convergence is likely to occur for ALS if collinearities
are present. Tri-JD, on the other hand, uses only algebraic tools (SVD and JD) in the 205
decomposition, and thus does not have such problem. Moreover, we note that TICA and
CPD-SMBSS converge faster than COMFAC and yield improved estimation accuracy, thanks to
the ICA stage which is capable to exploit the independence in S . However, TICA and
CPD-SMBSS use rank-1 approximation techniques in the second stage and thus the exploitation of
trilinear structure in the target tensor is rough, resulting in the fact that the SIR improvement of 210
TICA and CPD-SMBSS over COMFAC is slight.
Simulation 2: We consider the application of trilinear CPD in direction-of-arrival (DOA)
estimation with electromagnetic (EM) vector sensor array [8]. We assume 4 far-field narrow-band
BPSK signals impinge upon an L-shaped array of 8 EM vector sensors (see Fig1) with
interspacing of twice the impinging wavelength. The DOA of the mth impinging source is 215
represented with the azimuth mθ and elevation mφ , and the polarization is represented with the
polarization amplitude mγ and polarization phase difference mη .
2λ
y
x
Array configuration used in simulation 2 220
Each EM vector sensor generates 6-component measurement for the complete received EM
fields [8], and the observed array signal could be written in the following trilinear manner:
4
1
d p s
=
+∑ m m m
m
� D DX N (15)
where N is the additive noise (assumed to be gaussian white), ms is the complex envelop 225
of the mth source, m ∈d CN and 6Cm ∈p denote the mth spatial steering vector and
polarization steering vector, respectively. The formulization of md and mp with DOA and
polarization parameters could be found in [8]. We note here that [ , ]T T Tm m m=p e h where
3, Cm m ∈e h are the complex measurements of the electric and magnetic fields, respectively, and
the cross-product of them would yield the DOA of the mth signal [8]. As a result, DOA estimation 230
with EM vector-sensors amounts to identifying the trilinear signal model in (15), and this could be
done with CPD methods. We evaluate the estimation accuracy with overall root mean squared
angular errors (RMSAE) defined as:
1 ,1 E(arccos( ))v v=∑ m m m m- TM θ φ θ φmχ M � �� (16)
where T, [cos cos ,sin cos ,sin ]v m mθ φ m m m m mθ φ θ φ φ� denotes the propagation vector associated 235
with DOA of the mth source, and ,vθ ϕ� �m m is its estimate.
We let signal-to-noise ratio (SNR) vary between 0dB ~ 10dB, fix the number of temporal
samples to 1000, and calculate the overall RMSAE values averaged over 1000 Monte-Carlo runs,
for TRICA-JD, TICA, COMFAC, and Tri-JD. CPD-SMBSS is not included in the comparison as
it mainly concerns real-valued CPD while the problem in this simulation is complex-valued. Two 240
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ICA algorithms are used for TRICA-JD and TICA: entropy bound minimization (EBM) [1] and
complex FastICA [2]. The results for the following two scenarioes are plotted in Fig2.
- The sources are with distant DOA’s and polarizations:
1 1 1 1( , , , ) (30 ,20 ,35 ,62 )θ φ γ η = D D D D , 2 2 2 2( , , , ) (68 , 40 , 47 , 25 )θ φ γ η = D D D D ,
3 3 3 3( , , , ) (43 ,84 ,12 ,53 )θ φ γ η = D D D D , 4 4 4 4( , , , ) (12 ,60 ,65 ,70 )θ φ γ η = D D D D . 245
In this scenario, no collinearity exists.
- Two sources are with close DOA’s and close polarizations:
1 1 1 1( , , , ) (30 ,20 ,35 ,62 )θ φ γ η = D D D D , 2 2 2 2( , , , ) (28 , 21 ,36 ,61 )θ φ γ η = D D D D ,
3 3 3 3( , , , ) (43 ,84 ,12 ,53 )θ φ γ η = D D D D , 4 4 4 4( , , , ) (12 ,60 ,65 ,70 )θ φ γ η = D D D D .
In this case, double collinearities exist for the first and second modes of the observed signals. 250
0 1 2 3 4 5 6 7 8 9 10
0
1
2
SNR(dB)
O
ve
ra
ll
R
M
S
A
E
(d
eg
re
e)
COMFAC
TRICA−JD(EBM)
TRICA−JD(FASTICA)
TRI−JD
TICA(EBM)
TICA(FASTICA)
(a) The sources are with distant DOA's and polarizations
0 1 2 3 4 5 6 7 8 9 10
10
−1
10
0
10
1
10
2
SNR(dB)
O
ve
ra
ll
R
M
S
A
E
(d
eg
re
e)
COMFAC
TRICA−JD(EBM)
TRICA−JD(FASTICA)
TRI−JD
TICA(EBM)
TICA(FASTICA)
(b) Two sources are with close DOA's and close polarizations
Overall RMSAE vs. SNR 255
We see from .(a) that TRICA-JD (with both FastICA and EBM), COMFAC, and Tri-JD
yield almost equal performance, while TICA provides inferior DOA estimation accuracy than
these methods. Moreover, we note that the overall RMSAE curves of both TICA-FastICA and
TICA-EBM fluctuate dramatically with the increase of SNR, while those of TRICA-JD, 260
COMFAC and Tri-JD drop in a nice monotonical manner. The reason for the above observation is
that TICA uses rank-1 approximation of the ICA results for the estimation of polarization steering
vector pm , which relies heavily on the ICA stage, and thus is sensitive to the choice of ICA
algorithms as well as noise. TRICA-JD, on the other hand, features a Khatri-Rao restoration step
before rank-1 approximation and thus is less dependent on the previous ICA stage. Moreover, the 265
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observation that the performances of TRICA-JD, COMFAC, and Tri-JD are almost equal infers
that the use of source independence via ICA does not contribute much to the overall performance
in the absence of collinearity.
Moreover, we see from .(b) that with 2 out of 4 sources having both close DOA’s and
close polarizations, TRICA-JD (with EBM and FastICA) outperforms all the other competitors 270
with regards to the overall RMSAE. In addition, the standard CPD methods such as COMFAC and
Tri-JD, suffer from the double collinearities and provide much less estimation accuracy. The
behavior of TICA falls in between TRICA-JD and standard CPD (especially when SNR > 4dB),
yet is still observed to be sensitive to the ICA choice, noting the heavy fluctuations of the overall
RMSAE curves. We note as well that TRICA-JD with FastICA and EBM provide almost 275
equivalent performance, inferring again that TRICA-JD is insensitive to the incorporated ICA
algorithms.
Simulation 3: In this simulation, we examine if the proposed algorithm could be used in
scenarioes where ICA fails. We construct a tensor with the same setting as in Simulation 1, except
that the columns of S are generated as gaussian white vectors. As a result, the ICA stage used in 280
TRICA-JD, TICA, and CPD-SMBSS would make no contribution to the final results due to its
inability to separate gaussian white signals. We conduct 50 independent runs and calculate the
averaged SIR in all the three modes. The results are listed in Table 3.
Performance of compared methods in decomposing a trilinear tensor with collinearities in two modes and 285
white gaussianity in the other
SIR_A(dB) SIR_B(dB) SIR_C(dB) Runtime(s)
TRICA-JD
COMFAC
Tri-JD
TICA
CPD-SMBSS
From Table 3 we note the performances of both TICA and CPD-SMBSS deteriorate
dramatically when compared with TABLE II. This is because these two algorithms rely heavily on
the ICA stage, and thus would fail to obtain correct results if ICA is invalid. COMFAC, in 290
addition, still suffers from the double collinearities, as is suggested by the runtime and SIR.
Furthermore, we note that the proposed TRICA-JD is able to perform nicely with regards to
both SIR and runtime, even if ICA used in the first stage makes no contribution. The reason for
this is that the Khatri-Rao restoration step in the second stage enables a complete exploitation of
the trilinear structure of the three-way signal, and as is well-known, that merely exploiting the 295
trilinear structure is enough to generate unique identification [10]. Moreover, comparing TRICA-JD
and Tri-JD we note that they provide almost equal performance, and this suggest that TRICA-JD
would degrade to a standard trlinear CPD algorithm when ICA fails.
4 Conclusion
This study presents a new algorithm to perform trilinear canoncal polyadic decomposition 300
(CPD) with independent component analysis (ICA) and joint diagonalization (JD) (termed as
TRICA-JD for abbreviation). This algorithm is able to make use of the source statistics at one
mode of the tensorial data via ICA, when compared with standard CPD, and is able to provide
more thorough exploitation of the trilinear structure via a JD based stratedgy, when compared with
other CPD methods that are combined with ICA. As such, TRICA-JD could provide improved 305
estimation accuracy of parallel loading factors and sources over existing standard CPD and CPD
combined with ICA, as well as fast convergence, with particular interests in scenarioes that the
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target tensor contains collinear structures.
Simulations demonstrated that TRICA-JD could generate superior performance as well as
competent convergence speed with double collinearities and source independence present, in 310
comparison with standard CPD and CPD combined with ICA. Moreover, in scenarioes that the
incorporated ICA stage fails (. when the sources are gaussian white), TRICA-JD is still able to
generate superior performance (almost equivalent to standard CPD) over other ICA based CPD
algorithms.
315
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结合独立成分分析的三线性分解
龚晓峰1，郝亚娜1，林秋华1，刘志文2，徐友根2
（1. 大连理工大学电子信息与电气工程学部，辽宁 大连 116024；
2. 北京理工大学信息与电子学院，北京 100081） 355
摘要：本文提出了一种新的典范分解算法，该算法基于独立分量分析和联合对角化 ，既利
用了信号源的统计独立性，又结合了三维张量的三线性结构。更确切的说，首先将张量矩阵
化后，应用 ICA 获得某一模上的统计独立特性，之后应用联合对角化恢复原张量的三线性
结构，最后应用秩-1 逼近提取平行因子。所提算法可以克服共线性情况下标准典范分解存
在的收敛困难问题，另外，比起现有的结合典范分解和 ICA 的方法，此算法能更加彻底利360
用目标张量的三线性结构信息，且对于所结合的 ICA 算法不敏感。仿真实验中将所提算法
和其他标准典范分解以及结合典范分解和 ICA 的算法进行了比较。
关键词：独立分量分析；典范分解；联合对角化；张量
中图分类号：TN911
