Following up from an answer by @Royi on adding weights to BPDN problem , I would like to use CVX to test this approach. How can we formulate in CVX the regularized LS L1 norm with weights given by a vector $c$, as follows:
$$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{ {C}^{-1} }^{2} + \lambda {\left\| \boldsymbol{x} \right\|}_{1} $$
Where $ C $ is the covariance matrix $ \operatorname{diag} \left( \boldsymbol{c} \right) $ ?
Here's a minimal example using Matlab:
% problem data A = [1 0 0 0.5;... 0 1 0.2 0.3;... 0 0.1 1 0.2]; x0 = [1 0 1 0]'; % original signal y = A*x0; % measurements with no noise w = randi(1e3,1,numel(y))'; % random weights vector y = y + 1./(sqrt(w)).*randn(numel(y),1); % measurements with weighted noise CVX that does not include the weights info would be:
lambda = 0.01; % regularization parameter cvx_precision high cvx_solver SeDuMi cvx_begin quiet variable x(size(A,2),1); minimize(norm(A*x-y)+lambda*norm(x1,1)) cvx_end x = 0.9864 -0.0281 1.0108 0 
