comparison toolboxes/alps/ALPS/zero_ALPS.m @ 154:0de08f68256b ivand_dev

ALPS toolbox - Algebraic Pursuit added to smallbox

+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;