# Candes.m

### From Wikimization

This Matlab demonstration of *compressive sampling* (`a.k.a.` *compressed sensing*) by Emmanuel Candès

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')