annotate toolboxes/alps/ALPS/zero_ALPS.m @ 239:71128ec3e532 ver_2.0_beta

added documentation file/folder

function [x_hat, numiter, x_path] = zero_ALPS(y, Phi, K, params)
% =========================================================================
%                0-ALPS(#) algorithm - Beta Version
% =========================================================================
% Algebraic Pursuit (ALPS) algorithm with memoryless acceleration. 
% 
% Detailed discussion on the algorithm can be found in 
% [1] "On Accelerated Hard Thresholding Methods for Sparse Approximation", written
% by Volkan Cevher, Technical Report, 2011.
% =========================================================================
% INPUT ARGUMENTS:
% y                         M x 1 undersampled measurement vector.
% Phi                       M x N regression matrix.
% K                         Sparsity of underlying vector x* or desired
%                           sparsity of solution.
% params                    Structure of parameters. These are:
%
%    tol,...                Early stopping tolerance. Default value: tol =
%                           1-e5
%    ALPSiters,...          Maximum number of algorithm iterations. Default
%                           value: 300. 
%    mode, ...              According to [1], possible values are
%                           [0,1,2,4,5,6]. This value comes from the binary 
%                           representation of the parameters:
%                           (solveNewtob, gradientDescentx, solveNewtonx), 
%                           which are explained next. Default value = 0.
%    solveNewtonb,...       If solveNewtonb == 1: Corresponds to solving a 
%                           Newton system restricted to a sparse support. 
%                           It is implemented via conjugate gradients.  
%                           If solveNewtonb == 0: Step size selection as described
%                           in eqs. (12) and (13) in [1]. 
%                           Default value: solveNewtonb = 0.
%    gradientDescentx,...   If gradientDescentx == 1: single gradient
%                           update of x_{i+1} restricted ot its support with 
%                           line search. Default value: gradientDescentx =
%                           1.
%    solveNewtonx,...       If solveNewtonx == 1: Akin to Hard Thresholding Pursuit
%                           (c.f. Simon Foucart, "Hard Thresholding Pursuit," 
%                           preprint, 2010). Default vale: solveNewtonx = 0
%    mu,...                 Variable that controls the step size selection. 
%                           When mu = 0, step size is computed adaptively 
%                           per iteration. Default value: mu = 0. 
%    cg_maxiter,...         Maximum iterations for Conjugate-Gradients method.
%    cg_tol                 Tolerance variable for Conjugate-Gradients method.                                
% =========================================================================
% OUTPUT ARGUMENTS:
% x_hat                     N x 1 recovered K-sparse vector.
% numiter                   Number of iterations executed.
% x_path                    Keeps a series of computed N x 1 K-sparse vectors 
%                           until the end of the iterative process.
% =========================================================================
% 01/04/2011, by Anastasios Kyrillidis. [e-mail], EPFL.
% =========================================================================
% cgsolve.m is written by Justin Romberg, Caltech, Oct. 2005.
%                         Email: [e-mail]
% =========================================================================
% This work was supported in part by the European Commission under Grant 
% MIRG-268398 and DARPA KeCoM program #11-DARPA-1055. VC also would like 
% to acknowledge Rice University for his Faculty Fellowship.
% =========================================================================

[tmpArg, N] = size(Phi);

%% Initialize transpose of measurement matrix

Phi_t = Phi';

%% Initialize to zero vector
x_cur = zeros(N,1);
X_i = [];

x_path = zeros(N, params.ALPSiters);

%% CG params
if (params.mode == 1 || params.mode == 4 || params.mode == 5 || params.mode == 6)
    cg_verbose = 0;
    cg_A = Phi_t*Phi;
    cg_b = Phi_t*y;
end;

%% Determine step size selection strategy
function_mu = 0;
adaptive_mu = 0;

if (isa(params.mu,'float'))
    function_mu = 0;
    if (params.mu == 0)
        adaptive_mu = 1;
    else
        adaptive_mu = 0;
    end;
elseif (isa(params.mu,'function_handle'))
    function_mu = 1;
end;

%% Help variables
complementary_Xi = ones(N,1);

i = 1;
%% 0-ALPS(#)
while (i <= params.ALPSiters)
    x_path(:,i) = x_cur;
    x_prev = x_cur;

    % Compute the residual
    if (i == 1)
        res = y;
    else         
        Phi_x_cur = Phi(:,X_i)*x_cur(X_i);
        res = y - Phi_x_cur;
    end;
    
    % Compute the derivative
    der = Phi_t*res;         
    
    % Determine S_i set via eq. (11)
    complementary_Xi(X_i) = 0;
    [tmpArg, ind_der] = sort(abs(der).*complementary_Xi, 'descend');
    complementary_Xi(X_i) = 1;
    S_i = [X_i; ind_der(1:K)];
           
    ider = der(S_i);
    if (params.solveNewtonb == 1)
        % Compute least squares solution of the system A*y = (A*A)x using CG                        
        if (params.useCG == 1)
            [b, tmpArg, tmpArg] = cgsolve(cg_A(S_i, S_i), cg_b(S_i), params.cg_tol, params.cg_maxiter, cg_verbose);
        else
            b = cg_A(S_i,S_i)\cg_b(S_i);
        end;        
    else
        % Step size selection via eq. (12) and eq. (13) 
        if (adaptive_mu)
            Pder = Phi(:,S_i)*ider;
            mu_bar = ider'*ider/(Pder'*Pder);
            b = x_cur(S_i) + (mu_bar)*ider;    
        elseif (function_mu)            
            b = x_cur(S_i) + params.mu(i)*ider;
        else
            b = x_cur(S_i) + params.mu*ider;
        end;
    end;

    % Hard-threshold b and compute X_{i+1}
    [tmpArg, ind_b] = sort(abs(b), 'descend');
    x_cur = zeros(N,1);
    x_cur(S_i(ind_b(1:K))) = b(ind_b(1:K));
    X_i = S_i(ind_b(1:K));

    if (params.gradientDescentx == 1)
        % Calculate gradient of estimated vector x_cur
        Phi_x_cur = Phi(:,X_i)*x_cur(X_i);
        res = y - Phi_x_cur;
        der = Phi_t*res;
        ider = der(X_i);
        
        if (adaptive_mu)                
            Pder = Phi(:,X_i)*ider;
            mu_bar = ider'*ider/(Pder'*Pder);
            x_cur(X_i) = x_cur(X_i) + mu_bar*ider;
        elseif (function_mu)
            x_cur(X_i) = x_cur(X_i) + params.mu(i)*ider;
        else x_cur(X_i) = x_cur(X_i) + params.mu*ider;
        end;
    elseif (params.solveNewtonx == 1)                
        % Similar to HTP
        if (params.useCG == 1)
            [v, tmpArg, tmpArg] = cgsolve(cg_A(X_i, X_i), cg_b(X_i), params.cg_tol, params.cg_maxiter, cg_verbose);
        else
            v = cg_A(X_i,X_i)\cg_b(X_i);
        end;
        x_cur(X_i) = v;
    end;
     
    % Test stopping criterion
    if (i > 1) && (norm(x_cur - x_prev) < params.tol*norm(x_cur))
        break;
    end;
    i = i + 1;  
      
end;

x_hat = x_cur;
numiter = i;

if (i > params.ALPSiters)
    x_path = x_path(:,1:numiter-1);
else
    x_path = x_path(:,1:numiter);
end;