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+% % Non-negative Matrix Factorization via Nesterov's Optimal Gradient Method: Improved via Randomized Subspace Iterations NeNMF (RSI)
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+
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+% Reference
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+% F. Yahaya, M. Puigt, G. Delmaire, G. Roussel, Faster-than-fast NMF using
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+% random projections and Nesterov iterations, to appear in the Proceedings
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+% of iTWIST: international Traveling Workshop on Interactions between
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+% low-complexity data models and Sensing Techniques, Marseille, France,
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+% November 21-23, 2018
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+
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+
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+% <Inputs>
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+% X : Input data matrix (m x n)
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+% r : Target low-rank
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+%
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+% MAX_ITER : Maximum number of iterations. Default is 1,000.
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+% MIN_ITER : Minimum number of iterations. Default is 10.
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+
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+% TOL : Stopping tolerance. Default is 1e-5. If you want to obtain a more accurate solution, decrease TOL and increase MAX_ITER at the same time.
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+
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+% <Outputs>
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+% W : Obtained basis matrix (m x r).
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+% H : Obtained coefficients matrix (r x n).
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+% T : CPU TIME.
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+% RRE: Relative reconstruction error in each iteration
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+
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+
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+% Tmax : CPU time in seconds.
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+% Note: another file 'stop_rule.m' should be included under the same
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+% directory as this code.
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+
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+function [W,H,RRE,T]=RSI_NeNMF( X,W,H,r,Tmax)
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+MinIter=10;
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+tol=1e-5;
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+T=zeros(1,301);
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+RRE=zeros(1,301);
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+
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+ITER_MAX=500; % maximum inner iteration number (Default)
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+ITER_MIN=10; % minimum inner iteration number (Default)
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+
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+[L,R]=RSI_compression(X,r);
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+
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+
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+% Compress left and right
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+X_L = L * X;
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+X_R = X * R;
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+
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+
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+H_comp= H* R;
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+W_comp = L*W;
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+
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+HVt=H_comp*X_R';
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+HHt=H_comp*H_comp';
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+
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+WtV=W_comp'*X_L;
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+WtW=W_comp'*W_comp;
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+
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+GradW=W*HHt-HVt';
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+GradH=WtW*H-WtV;
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+
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+init_delta=stop_rule([W',H],[GradW',GradH]);
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+tolH=max(tol,1e-3)*init_delta;
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+tolW=tolH; % Stopping tolerance
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+
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+
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+% Iterative updating
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+W=W';
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+k=1;
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+RRE(k) = nmf_norm_fro( X, W', H);
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+T(k) =0;
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+tic
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+% main loop
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+while(toc<= Tmax+0.05)
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+
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+ % Optimize H with W fixed
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+ [H,iterH]=NNLS(H,WtW,WtV,ITER_MIN,ITER_MAX,tolH);
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+
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+ if iterH<=ITER_MIN
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+ tolH=tolH/10;
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+ end
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+ H_comp=H*R;
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+ HHt=H_comp*H_comp'; HVt=H_comp*X_R';
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+ % Optimize W with H fixed
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+ [W,iterW,GradW]=NNLS(W,HHt,HVt,ITER_MIN,ITER_MAX,tolW);
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+
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+ if iterW<=ITER_MIN
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+ tolW=tolW/10;
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+ end
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+ W_comp=W * L';
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+ WtW=W_comp*W_comp';
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+ WtV=W_comp*X_L;
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+ GradH=WtW*H-WtV;
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+
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+ % HIS.niter=niter+iterH+iterW;
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+ delta=stop_rule([W,H],[GradW,GradH]);
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+
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+ % Stopping condition
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+ if (delta<=tol*init_delta && k>=MinIter)
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+ break;
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+ end
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+
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+
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+ if toc-(k-1)*0.05>=0.05
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+ k = k+1;
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+ RRE(k) = nmf_norm_fro( X, W', H);
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+ T(k) = toc;
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+ end
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+
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+end %end of loop
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+W=W';
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+
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+
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+return;
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+
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+
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+function [H,iter,Grad]=NNLS(Z,WtW,WtV,iterMin,iterMax,tol)
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+
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+if ~issparse(WtW)
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+ L=norm(WtW); % Lipschitz constant
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+else
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+ L=norm(full(WtW));
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+end
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+H=Z; % Initialization
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+Grad=WtW*Z-WtV; % Gradient
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+alpha1=1;
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+
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+for iter=1:iterMax
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+ H0=H;
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+ H=max(Z-Grad/L,0); % Calculate sequence 'Y'
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+ alpha2=0.5*(1+sqrt(1+4*alpha1^2));
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+ Z=H+((alpha1-1)/alpha2)*(H-H0);
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+ alpha1=alpha2;
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+ Grad=WtW*Z-WtV;
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+
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+ % Stopping criteria
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+ if iter>=iterMin
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+ % Lin's stopping condition
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+ pgn=stop_rule(Z,Grad);
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+ if pgn<=tol
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+ break;
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+ end
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+ end
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+end
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+
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+Grad=WtW*H-WtV;
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+
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+return;
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+function f = nmf_norm_fro(X, W, H)
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+% Author : F. Yahaya
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+% Date: 13/04/2018
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+% Contact: farouk.yahaya@univ-littoral.fr
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+% Goal: compute a normalized error reconstruction of the mixing matrix V
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+% "Normalized" means that we divide the squared Frobenius norm of the error
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+% by the squared Frobenius norm of the matrix V
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+% Note: To express the error in dB, you have to compute 10*log10(f)
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+%
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+
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+f = norm(X - W * H,'fro')^2/norm(X,'fro')^2;
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+
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+return;
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+
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+function [ L,R ] = RSI_compression(X,r)
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+% Tepper, M., & Sapiro, G. (2016). Compressed nonnegative
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+% matrix factorization is fast and accurate. IEEE Transactions
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+% on Signal Processing, 64(9), 2269-2283.
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+% see: https://arxiv.org/pdf/1505.04650
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+% The corresponding code is originally created by the authors
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+% Then, it is modified by F. Yahaya.
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+% Date: 13/04/2018
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+%
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+
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+compressionLevel=20;
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+[m,n]=size(X);
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+
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+l = min(n, max(compressionLevel, r + 10));
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+
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+OmegaL = randn(n,l);
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+
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+Y = X * OmegaL;
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+
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+for i=1:4
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+
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+ [Y,~]=qr(Y,0);
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+ S=X'*Y;
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+ [Z,~]=qr(S,0);
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+ Y=X* Z;
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+end
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+[L,~]=qr(Y,0);
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+L=L';
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+
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+OmegaR = randn(l, m);
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+Y = OmegaR * X;
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+
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+for i=1:4
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+ [Y,~]=qr(Y',0);
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+ S=X*Y;
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+ [Z,~]=qr(S,0);
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+
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+ Y=Z'*X;
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+end
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+Y=Y';
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+[R,~] = qr(Y,0);
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+
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+
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+return
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