init4acdc.m

联合对角化方法

function A0=init4acdc(M,wix)
%this function finds an initial guess for
%the acdc algorithm by performing
%pre-whitening on one of the target
%matrices, and then finding the orthogonal
%diagonalizer of the transformed matrices
%using Cardoso's joint_diag algorithm.
%
%inputs:
%
%M(N,N,K) - a set of the K MxM target
%           matrices;
%
%wix - (optional) index (1 to K) of the 
%      matrix M(:,:,wix) by which to 
%      determine the "whitening" factor.
%      default: wix=1.
%
%output:
% 
%A0 - the initial guess for acdc.

if ~exist('wix','var'), wix=1; end  
[N N1 K]=size(M);
if N~=N1
   error('input matrices must be square')
end

[V D]=eig(M(:,:,wix));
sD=sqrt(D);
W=inv(sD)*V';
Mw=zeros(N,N,K);
for k=1:K
   Mw(:,:,k)=W*M(:,:,k)*W';
end
U=joint_diag(Mw,1e-10);
A0=V*sD*U;

function [ V , D , N] =  joint_diag(A,jthresh)
% Joint approximate diagonalization
% 
% Joint approximate of n (complex) matrices of size m*m stored in the
% m*mn matrix A by minimization of a joint diagonality criterion
%
% Usage:  [ V , D , N] =  joint_diag(A,jthresh)
%
% Input :
% * the m*nm matrix A is the concatenation of n matrices with size m
%   by m. We denote A = [ A1 A2 .... An ]
% * threshold is an optional small number (typically = 1.0e-8 see the M-file).
%
% Output :
% * V is an m*m unitary matrix.
% * D = V'*A1*V , ... , V'*An*V has the same size as A and is a
%   collection of diagonal matrices if A1, ..., An are exactly jointly
%   unitarily diagonalizable.
% * N is the required number of iterations
%

% The algorithm finds a unitary matrix V such that the matrices
% V'*A1*V , ... , V'*An*V are as diagonal as possible, providing a
% kind of `average eigen-structure' shared by the matrices A1 ,...,An.
% If the matrices A1,...,An do have an exact common eigen-structure ie
% a common orthonormal set eigenvectors, then the algorithm finds it.
% The eigenvectors THEN are the column vectors of V and D1, ...,Dn are
% diagonal matrices.
% 
% The algorithm implements a properly extended Jacobi algorithm.  The
% algorithm stops when all the Givens rotations in a sweep have sines
% smaller than 'threshold'.
%
% In many applications, the notion of approximate joint
% diagonalization is ad hoc and very small values of threshold do not
% make sense because the diagonality criterion itself is ad hoc.
% Hence, it is often not necessary in applications to push the
% accuracy of the rotation matrix V to the machine precision.
%
% PS: If a numrical analyst knows `the right way' to determine jthresh
%     in terms of 1) machine precision and 2) size of the problem,
%     I will be glad to hear about it.
% 
%
% This version of the code is for complex matrices, but it also works
% with real matrices.  However, simpler implementations are possible
% in the real case.
%
% See more info, references and version history at the bottom of this
% m-file

%
%----------------------------------------------------------------
% Version 1.2
%
% Copyright     : Jean-Francois Cardoso. 
% Author        : Jean-Francois Cardoso. [e-mail]
% Comments, bug reports, etc are welcome.
%----------------------------------------------------------------


[m,nm] = size(A);

%% Better declare the variables used in the loop :
B       = [ 1 0 0 ; 0 1 1 ; 0 -i i ] ;
Bt      = B' ;
Ip      = zeros(1,nm) ;
Iq      = zeros(1,nm) ;
g       = zeros(3,nm) ;
g       = zeros(3,m);
G       = zeros(2,2) ;
vcp     = zeros(3,3);
D       = zeros(3,3);
la      = zeros(3,1);
K       = zeros(3,3);
angles  = zeros(3,1);
pair    = zeros(1,2);
G       = zeros(3);
c       = 0 ;
s       = 0 ;


%% Init
V       = eye(m);
encore  = 1; 
N=0;

while encore, encore=0;
 N=N+1;
 for p=1:m-1, Ip = p:m:nm ;
 for q=p+1:m, Iq = q:m:nm ;

        %% Computing the Givens angles
        g       = [ A(p,Ip)-A(q,Iq)  ; A(p,Iq) ; A(q,Ip) ] ; 
        [vcp,D] = eig(real(B*(g*g')*Bt));
        [la, K] = sort(diag(D));
        angles  = vcp(:,K(3));
        if angles(1)<0 , angles= -angles ; end ;
        c       = sqrt(0.5+angles(1)/2);
        s       = 0.5*(angles(2)-j*angles(3))/c; 

        if abs(s)>jthresh, %%% updates matrices A and V by a Givens rotation
                encore          = 1 ;
                pair            = [p;q] ;
                G               = [ c -conj(s) ; s c ] ;
                V(:,pair)       = V(:,pair)*G ;
                A(pair,:)       = G' * A(pair,:) ;
                A(:,[Ip Iq])    = [ c*A(:,Ip)+s*A(:,Iq) -conj(s)*A(:,Ip)+c*A(:,Iq) ] ;

   end%% if
  end%% q loop
 end%% p loop
end%% while

D = A ;

return

% Revision history
%
% Version 1.2.  Nov. 2, 1997.
%   o some Matlab tricks to have a cleaner code.
%   o Changed (angles=sign(angles(1))*angles) to (if angles(1)<0 ,
%   angles= -angles ; end ;) as kindly suggested by Iain Collings
%   <[e-mail]>.  This is safer (with probability 0 in
%   the case of sample statistics)
%
% Version 1.1.  Jun. 97.
%       Made the code available on the WEB




%----------------------------------------------------------------
% References:
%
% The 1st paper below presents the Jacobi trick.
% The second paper is a tech. report the first order perturbation
% of joint diagonalizers
%
%
%@article{SC-siam,
%  HTML        = "ftp://sig.enst.fr/pub/jfc/Papers/siam_note.ps.gz",
%  author       = "Jean-Fran\c{c}ois Cardoso and Antoine Souloumiac",
%  journal      = "{SIAM} J. Mat. Anal. Appl.",
%  title        = "Jacobi angles for simultaneous diagonalization",
%  pages        = "161--164",
%  volume       = "17",
%  number       = "1",
%  month        = jan,
%  year         = {1996}
%  }
%
%
%
%@techreport{PertDJ,
%  author       = "Jean-Fran\c{c}ois Cardoso",
%  HTML         = "ftp://sig.enst.fr/pub/jfc/Papers/joint_diag_pert_an.ps",
%  institution  = "T\'{e}l\'{e}com {P}aris",
%  title        = "Perturbation of joint diagonalizers. Ref\# 94D027",
%  year         = "1994"
%}