Candes.m

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This Matlab demonstration of compressive sampling (a.k.a. compressed sensing) by Emmanuel Cand├Ęs
comes from his June 6 2007 video on the Optimization Videos page.

%Emmanuel Candes, California Institute of Technology, June 6 2007, IMA Summerschool.
%Transcribed by Jon Dattorro.
%Fails using SDP solver SDPT3  on 7th consecutive run after Matlab R2007b startup.  CVX version 1.2 (build 656).
%Fails using SDP solver Sedumi on 4th consecutive run after Matlab R2007b startup.  CVX version 1.2 (build 656).
clear all, close all                
n = 512;                            % Size of signal
m = 64;                             % Number of samples (undersample by a factor 8)

k = 0:n-1;  t = 0:n-1;
F = exp(-i*2*pi*k'*t/n)/sqrt(n);    % Fourier matrix
freq = randsample(n,m);
A = [real(F(freq,:)); 
     imag(F(freq,:))];              % Incomplete Fourier matrix

S = 28;
support = randsample(n,S);
x0 = zeros(n,1);  x0(support) = randn(S,1);
b = A*x0;

% Solve l1 using CVX
cvx_quiet(true);
%cvx_solver('sedumi'); 
cvx_begin
    variable x(n);
    minimize(norm(x,1));
    A*x == b;
cvx_end

norm(x - x0)/norm(x0)
figure, plot(1:n,x0,'b*',1:n,x,'ro'), legend('original','decoded')

Code between cvx_begin and cvx_end requires CVX.

randsample() is from Matlab Statistics Toolbox.

Revision

Failure modes are reparable by Convex Iteration from Convex Optimization & Euclidean Distance Geometry, Ch.4.5:

%Emmanuel Candes, California Institute of Technology, June 6 2007, IMA Summerschool.
%Convex Iteration implementation by Jon Dattorro.
%Failure modes repaired.
clear all, close all
n = 512;                            % Size of signal
m = 64;                             % Number of samples (undersample by a factor 8)

k = 0:n-1;  t = 0:n-1;
F = exp(-i*2*pi*k'*t/n)/sqrt(n);    % Fourier matrix
freq = randsample(n,m);
A = [real(F(freq,:));
     imag(F(freq,:))];              % Incomplete Fourier matrix

S = 28;
support = randsample(n,S);
x0 = zeros(n,1);  x0(support) = randn(S,1);
b = A*x0;

cvx_quiet(true);
%cvx_solver('sedumi');

%convex iteration
y = ones(n,1);
while 1
    % Solve l0 using CVX and Convex Iteration
    cvx_begin
        variable x(n);
        minimize(norm(y.*x,1));
        A*x == b;
    cvx_end

    % update search direction y
    [x_sorted, indices] = sort(abs(x), 'descend');  
    y = ones(n,1);
    y(indices(1:S)) = 0;

    cardx = sum(abs(x) > 1e-6)
    if cardx <= S, break, end
end
norm(x - x0)/norm(x0)
figure, plot(1:n,x0,'b*',1:n,x,'ro'), legend('original','decoded')
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