TWF.m

%% Implementation of the truncated Wirtinger Flow (TWF) algorithm proposed in the paper
%  ``Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems'' by Y. Chen and E. J. Cand�s.
%  The code below is adapted from implementation of the Wirtinger Flow algorithm designed and implemented by E. Candes, X. Li, and M. Soltanolkotabi

function [x_hat, Relerrs, TWFtime] = TWF(y, Params, A, At, x)    
%% Initialization
tic;
    npower_iter = Params.npower_iter;           % Number of power iterations 
    z0 = randn(Params.n,1); z0 = z0/norm(z0,'fro');    % Initial guess 
    normest = sqrt(sum(y(:))/numel(y(:)));    % Estimate norm to scale eigenvector  
    
    for tt = 1:npower_iter,                     % Truncated power iterations
        ytr = y.* (abs(y) <= Params.alpha_y^2 * normest^2 );
        z0 = At( ytr.* (A(z0)) ); z0 = z0/norm(z0,'fro');
    end
    
    z = normest * z0;                   % Apply scaling 
%     Relerrs = norm(x - exp(-1i*angle(trace(x'*z))) * z, 'fro')/norm(x,'fro'); % Initial rel. error
%TWFtime(1)=toc;
%tic;
    %% Loop
    grad_type = Params.grad_type;
    if strcmp(grad_type, 'TWF_Poiss') == 1
        mu = @(t) Params.mu; % Schedule for step size 
    elseif strcmp(grad_type, 'WF_Poiss') == 1
        tau0 = 330;                         % Time constant for step size
        mu = @(t) min(1-exp(-t/tau0), 0.2); % Schedule for step size  
    end

    for t = 1: Params.T,
        grad = compute_grad2(z, y, Params, A, At);
        z = z - mu(t) * grad;             % Gradient update 
%         Relerrs = [Relerrs, norm(x - exp(-1i*angle(trace(x'*z))) * z, 'fro')/norm(x,'fro')];  
TWFtime(t)=toc;
Relerrs(t)=norm(x - exp(-1i*angle(trace(x'*z))) * z, 'fro')/norm(x,'fro');
    end
x_hat   =   z;
%TWFtime(2)=toc;