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ALGORITHMS FOR NONORTHOGONAL APPROXIMATE JOINT Institute of Information Theory and Automation Pod vod´arenskou vˇeˇz´ı 4, 18208 Prague 8, Czech Republic Approximate joint block diagonalization (AJBD) of a set of matrices has applications in blind source separation, e.g., nWT ,
when the signal mixtures contain mutually independent sub- are all approximately block diagonal, having the blocks on the spaces of dimension higher than one. In this paper we present main diagonal of the same size as the original matrices M
three novel AJBD algorithms which differ in the final target Ideally, one may wish to estimate W = A1 and get RW
criterion to be minimized in the optimization process. Two Bdiag(RW, . . . , RW ), where RW M
of the algorithms extend the principle of Jacobi elementary task is reversed and consists in fitting the given data matrices rotations to the more general concept of matrix elementary rotations. The proposed algorithms are compared to existing n by the model (1) with parameters A, {Mn}.
In general, however, it is impossible to recover the orig- inal blocks Mnk (even in the “noiseless” case), because of
Index Terms— Source separation, independent sub- inherent ambiguities of the problem (e.g., [6]), but it is possi- ble to recover “independent subspaces”, as explained below.
Let W0 = A1, also called demixing matrix, be parti-
tioned in K blocks Wk of size Ik×d, W0 = [WT , . . . , WT ]T
Each block Wk represents a linear space of all linear com-
Consider a set of square symmetric matrices Mn, n =
1, . . . , N , that are all block diagonal, with K blocks along its uniquely identifiable [6, 2]. Let W be an estimated demixing
main diagonal, Mn = Bdiag(Mn1, . . . , MnK ), where Mnk
matrix. We say that W is “essentially equivalent” to W0 (and
is the k−th block of Mn and the Bdiag(·) operator constructs
therefore represents an ideal joint block diagonalization), if a block-diagonal matrix from its argument matrices. The size there exists a suitable d × d permutation matrix Π such that
of the blocks might not be identical. We assume that the size for each k = 1, . . . , K the subspaces spanned by Wk and by
of the block Mnk is Ik × Ik. The dimension of the matrices
the respective k-th block of ΠW coincide, and their mutual
Mn is d = I1 + . . . IK .
Next, assume that (possibly perturbed) congruence trans- Some existing AJBD algorithms are restricted to the case formations of these matrices are given as where A (and therefore also W) are orthogonal [3], some
other algorithms consider a general matrix A [4, 6]. In this
Rn = AMnAT + Nn,
where the superscript T denotes a matrix transposition, A is
It was shown in [8] that reasonable solutions to AJBD can an unknown square “mixing matrix”, and N
be obtained using a two step approach, by first applying an tion (or “noise”) matrix. We shall refer to the case where all ordinary approximate joint diagonalization (AJD) algorithm, and then clustering the separated components (rows of the n = 0, n = 1, . . . , N as the “unperturbed” (or “noiseless”)
case. The choice of symbol R reflects the fact that the matri-
demixing matrix). More specifically, it was shown that un- ces in the set often play a role of (sample-) covariance matri- like several popular AJD approaches, one recently proposed ces of a partitioned data, or time-lagged (sample-) covariance AJD method (U-WEDGE, Uniformly Weighted Exhaustive Diagonalization with Gauss itErations [9]) features a unique The goal in Approximate Joint Block Diagonalization ability to attain ideal separation in the unperturbed (“noise- (AJBD) is to find a “demixing” matrix W, such that the
less”) case, for general (not necessarily orthogonal) matrices Supported by Grant Agency of the Czech Republic through the project 1The mutual angle between two subspaces can be obtained in Matlab us- A. The above mentioned methods are computationally ap-
Success of the joint block diagonalization can be measured by pealing, but it appeared that in the presence of the noise, ded- these three criteria or, if the ideal mixing/demixing matrix is icated block algorithms may perform better. This paper is known, by the angle between the true and estimated subspaces therefore devoted to finding AJBD solutions that minimize mentioned in the previous subsection.
some a priori chosen criteria.
The paper is organized as follows. Section 2 presents a survey of three main cost functions used for AJBD in the lit- erature. In Section 3, a general AJBD algorithm based ongeneralized Jacobi rotations is proposed. Specific details of Each of the criterion mentioned in the former section (and the general procedure for the two criteria are derived in Sec- namely the first two criteria) can be optimized by applying tion 4. Section 5 presents an algorithm for minimizing the set of elementary rotations that follow the idea of well known third criterion. In Section 6, a novel initialization method is Jacobi rotations. The elementary rotations follow the block proposed. A simulation study in Section 7 verifies effective- structure of the assumed block matrices Mn. The rotations
ij (B, C) = 
In the literature, we can see three main criteria used for joint where the diagonal is as in the identity matrix and the only CLS(W) =
Boff(RW)2
nonzero off-diagonal blocks B and C of the sizes Ii × Ij
and Ij × Ii lie at positions (i, j) and (j, i), respectively, for where the operator “Boff” nullifies the elements of a matrix that lie in the diagonal blocks. This is used in [3] to build a The blocks B and C are selected so that the updated
unitary block diagonalization algorithm. In the case of non- demixing matrix W= Eij(B, C)W approximates the low-
orthogonal diagonalization, the criterion has to be completed est attainable criterion function C(W), where C(W) is one
by adding a constraint that prevents the algorithm from con- of the criteria (3), (4), (5). The idea of the minimization is verging to the trivial solution W = 0. The constraint is that
that the elementary rotations Eij(B, C) should be small, i.e.
all rows of W have unit Euclidean norm. An example is [4].
the matrices B and C should be small. The true criterion
A different criterion was proposed by Lahat et al [5].
function C(W) is replaced by the second order approxi-
mation in terms of B and C, and is minimized in a closed
det Bdiag(RW)
We will show in the next section that the approximation of C(W) might have the form C(W) = φ(B, C),
where Bdiag is the opposite of the operator “Boff”, nullifies the elements of a matrix that lie out of the diagonal blocks.
φ(B, C) = φ
The criterion in motivated by maximum likelihood estima- n1B) + tr(Jn2C)
tion. It is a generalization of the criterion proposed for ap-proximate joint diagonalization in [7]. Minimization of this + tr(Jn3BJn4B) + tr(Jn5BJn6BT )
criterion leads to the maximum likelihood estimator, if Rn
+ tr(Jn7CJn8C) + tr(Jn9CJn10CT )
are sample covariance matrices from Gaussian distributed iid +tr(Jn11BJn12C) + tr(Jn13BJn14CT )
random vectors that have covariance matrices admitting theexactly block diagonal structure.
where the matrices Jnm, m = 1, . . . , 14, depend on RW
The third possible criterion was suggested by Nion [6].
φ0 = φ(0, 0), and “tr” denotes the trace of a matrix (sum
The latest method is based on fitting the block diagonal model of diagonal elements). Explicit expressions for the matrices to the given set of matrices, seeking the mixing matrix A =
Jnm, m = 1, . . . , 14, for the criteria, (3) and (4) will be de-
W1,
Rn − AMAAT ∥2
φ(B, C) = φ0 + gT v(B, C) + v(B, C)T Hv(B, C) (9)
R
n − AMnAT ∥2
v(B, C) =
n =Bdiag(Mn )
g is a suitable vector of the same length, and H is a symmetric
Let RW and ˜
R = ERWET be partitioned to blocks
R11, . . . , R33 and ˜
R11, . . . , ˜
R33, respectively. Since
vec(JT )
vec(JT )
CT
BT
HBCn
HT
HCCn
R12 = R12 +R11CT +BR22, ˜
R13 = R13 +BR23
R23 = R23 + CR13. Then
HBBn
Jn6 JT + JT ⊗ J
LS (B, C) = ˜
R122F
+(Jn3 JT + J
n4 JT
A straightforward computation gives J1 = 2R22RT12, J2 =
HBCn
Jn14 JT
n12 JT
2R11R12, J5 = I, J6 = R222, J9 = I, J10 = R211, J11 =
HCCn
Jn10 JT + JT
J
R11, J12 = 2R22, J3 = J4 = J7 = J8 = J13 = J14 = 0.
+(Jn7 JT + J
)P,
n8 JT
is the Kronecker product, and where P stands for a permu-
Consider the following Taylor series expansion of the matrix tation matrix such that Pvec(M) = vec(MT ) for any matrix
function log det R in a neighborhood of a positive definite
M of a suitable dimension (Ii × Ij).
Once g and H are computed, the optimum v(B, C) that
0. Let ∆R = R R0. It holds
minimizes φ(B, C) can be found as
tr(R1∆R) 1tr(R1∆RR1∆R) .
v(B, C) = H1g .
We apply this approximation to the last line of the expression After each elementary transformation, i.e. update W=
E
ij (B, C)W, it is recommended to check if the original op-
φLL(B, C) =
timization criterion has really decreased its value. If this is not the case, it means that the quadratic approximation was not R) 1
det Bdiag(RW )
accurate enough. One can proceed by replacing the step (16) det RW
by another one as it is done in the so called damped Gauss- Newton algorithm, also known as Levenberg-Marquardt, det Bdiag(RW )
det RW
v(B, C) = (H + µI)1g
After some computation we get (8) with J1 = J3 =
R21R1, J
R1,
2 = J7 = R12R1
where µ is a suitable positive parameter. Discussion of this J6 = R22 R21R1R
R1,
= 1 7, J9 = 1 22
modification, however, exceeds the scope of this paper.
J10 = R11 R12R1R21, J11 = J12 = I, J13 = J14 = 0.
The main algorithm consists after a suitable initialization The resultant algorithm is similar to the one proposed in [5].
in cyclic minimization of the elementary rotations with re-
spect to all pairs (i, j), 1 ≤ i < j ≤ K, until norms of the
matrices B and C are smaller than a threshold for all the pairs.
The generalized Jacobi rotations presented in Section 3, ap- pear not to be so effective in the case of the FIT criterion (5).
The reason is that while in the case of the former two crite- In this section we analyze the criteria (3), (4) in order to de- ria, the Hessian matrix H can be shown to be always positive
rive matrices J
(semi-)definite, it is no longer true in the last case. There- nm that are needed for computation of the el- ementary rotations considered in the previous section. For fore, we propose another simple iterative procedure for this simplicity of presentation we assume here that there are only cost function. Starting from an initial estimate of the mixing three blocks in the matrices M
matrix A, its update Ais obtained as
is sought for i = 1 and j = 2. We can assume without any loss in generality that the third block includes all remaining A= arg min
Rn − ˜
AMAAT ∥2 .
blocks. We skip the running index n.
global minimum, it is convenient to find a suitable initializa-tion. We propose to use the method advocated in [8], apply Rn − ˜
AMAAT ∥2 =
I)
vec(Rn) (AMA
A)2
the AJD algorithm U-WEDGE, followed by a suitable clus-tering of rows of the mixing matrix, which would reveal the A) can be found by solving the overdeter-
block structure of the demixed matrices.
First, we propose a modified method of clustering the rows, compared to the one proposed in [8]. It is a greedy al- AMA I
gorithm again. Given the AJD demixing matrix W, compute
A)
an auxiliary matrix D as
AMA I
|WRnWT |
vec(A) =
(AMA I)T (AMA I)
where the absolute value is taken elementwise. If the demix- ing is perfect, D should have, after arranging columns and
rows, the same block structure as the original matrices Mn.
(AMA I)T
Let the block sizes be ordered increasingly, I1 ≤ . . . ≤ IK.
vec(Rn)
Elements of all columns of D are sorted decreasingly to form
a matrix D. Then, sort the (I1 +1)-th row of Dand denote it
r1 for easy reference. The first block of the demixing matrix
will be composed of the indices that correspond to the small- A=
1 elements in r1. It means that W1 is built of the rows
nAMA
MAAT AMA
of W with these indices. The rows and columns of D at the
positions i1, . . . , iI are set to zero, and the procedure iterated It remains to explain computation of MA
until K − 1 subspaces (blocks) W
k , k = 1, . . . , K − 1, have ing matrix A be vertically partitioned as A = [A
been determined. The K-th block W
1, . . . , AK ],
where the size of blocks A
the rows that have not been selected before.
k is d × Ik, and let Mnk be the kth
diagonal block of MA
Rn − AMAAT ∥2 = R
k Mnk AT
We have compared performance of five AJBD algorithms: (1) U-WEDGE completed by clustering of rows of a demix- ing matrix, this algorithm is used to initialize all subsequent = vec(Rn)
(Ak ⊗ Ak)vec(Mnk)2
ones, (2) algorithm JBD NCG [6], (3) the ad hoc algorithm of Ghennioui et al [4], and two algorithms proposed in this pa- The desired elements of MA
per: (4) LSJBD and (5) FITJBD, minimizing the criteria (3) and (5), respectively. The LLAJD algorithm is not included, mn = [vec(Mn1)T , . . . , vec(MnK )T ]T .
because it would need a different setup with all target matricespositive (semi-)definite.
The optimum mn is given as
We consider 10 target matrices, each having four diagonal blocks of the size 5 × 5. The blocks were taken at random, mn = (GT G)1GT vec(Rn)
different at each simulation trial, normally distributed with a zero mean, and were symmetrized by adding their transposi- tion. The mixing matrix A was taken at random, also new
1 A1, . . . , AK ⊗ AK ] .
in each simulation trial. Finally, an additive noise was added The convergence of the iterative process (19) appears to be as in (1); the noise matrices were symmetrized as well, and very good, as it is shown in the simulation section.
scaled to achieve desired signal-to-noise ratio (SNR), whichis defined as 10 log10 of Frobenius norm of the mixtures di- vided by Frobenius norm of the noise.
The noisy mixtures were processed by each of the five The algorithms proposed in the previous session are iterative algorithms in 100 independent trials. Each algorithm has its and convergence to the global minimum of the respective cri- own cost function, which is optimized, therefore we have cho- teria (3), (4), and (5) is not guaranteed. In order to minimize sen to measure the performance by the angular difference of probability of getting the algorithm stuck in a local but not the subspaces spanned by corresponding groups of rows of the true and estimated demixing matrix. The resultant mean and median average angular differences, as a function of SNR are plotted in Figure 1. We note namely excellent performance of GHENNIO
JBD_NCG

FITJBD
LSJBD
GHENNIO

Fig. 2. Average and median number of iterations versus SNR.
Fig. 1. Performance of the five AJBD techniques.
the FITJBD algorithm, which appears to work at lower SNR’s than its competitors. Convergence of the algorithm is nearly linear, as in gradient methods or as in JBD NCG. In this par- GHENNIO
JBD_NCG

ticular example, the latter algorithm does not work well, but it is not bad in general, namely in lower dimensions. Both algorithms need thousands of iterations to converge.
An advantage of the methods based on generalized Jacobi Fig. 3. Average and median CPU time versus SNR.
rotations (here LSJBD) is that they have nearly quadratic con-vergence. In general they require only 10-100 iterations toconverge. The algorithm of Ghennioui is something between.
[3] C. F´evotte and F. J. Theis, “Pivot Selection Strategies in Sometimes it converges quickly, as it had a quadratic conver- Jacobi Joint Block-Diagonalization”, Proc. ICA’2007, gence, namely at high SNR scenarios, in other cases it is slow.
Numbers of iterations and the CPU time of the algorithms are [4] H. Ghennioui, F. El Mostafa, N. Thirion-Moreau, A.
shown in Figures 2 and 3, respectively.
Adib, and E. Moreau, “A Nonunitary Joint Block Diag-onalization algorithm for Blind Separation of Convolu-tive Mixtures of Sources”, IEEE Signal Processing Let- ters, vol. 14, pp. 860–863, 2007.
We have presented novel algorithms for approximate joint [5] D. Lahat, J.-F. Cardoso, and H. Messer, “Joint Block block–diagonalization. They differ in the cost function that Diagonalization Algorithms for Optimal Separation of is optimized. Matlab codes of the proposed algorithms are Multidimensional Components”, F. Theis et al. (Eds.): posted on the Internet at http://si.utia.cas.cz/downloadPT.htm .
LVA/ICA 2012, LNCS 7191, pp. 155-162, 2012.
[6] D. Nion, “A Tensor Framework for Nonunitary Joint Block Diagonalization”, IEEE Trans. Signal Process- ing, vol. 59, no. 10, pp. 4585-4594, Oct. 2011.
The authors wish to give thanks to Dr. Dimitri Nion for send- [7] D.-T. Pham , “Joint Approximate Diagonalization of ing them a matlab code of his algorithm JBD NCG.
Positive Definite Hermitian Matrices”, SIAM J. MatrixAnal. and Appl. vol. 22, pp. 1136-1152, 2001.
[8] P. Tichavsk´y, A. Yeredor, and Z. Koldovsk´y, “On Com- putation of Approximate Joint Block-Diagonalization [1] K. Abed-Meraim and A. Belouchrani, “Algorithms for using Ordinary AJD”, F. Theis et al. (Eds.): LVA/ICA Joint Block Diagonalization”, Proc. of EUSIPCO 2004, 2012, LNCS 7191, pp. 163-171, 2012.
Vienna, Austria, pp. 209 - 212, 2004.
[9] P. Tichavsk´y and A. Yeredor , “Fast Approximate Joint Diagonalization Incorporating Weight Matrices”, IEEE [2] L. de Lathauwer, “Decomposition of Higher-Order Ten- Tr. Signal Processing, vol. 57, pp. 878-891, March 2009.
sor in Block Terms-Part II: Definitions and Unique-ness”, SIAM J. Matrix Anal. and Appl. vol. 30, pp. 1033-1066, 2008.

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