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

ALPS toolbox - Algebraic Pursuit added to smallbox

+function [x_hat, numiter, x_path] = infty_ALPS(y, Phi, K, params)
+% =========================================================================
+%               infty-ALPS(#) algorithm - Beta Version
+% =========================================================================
+% Algebraic Pursuit (ALPS) algorithm with infty-memory 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. 
+%    solveNewtonb,...       Value: solveNewtonb = 0. Not used in infty
+%                           methods.
+%    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
+%    tau,...                Variable that controls the momentum in
+%                           non-memoryless case. Ignored in memoryless
+%                           case. Default value: tau = 1/2.
+%                           Special cases:
+%                               - tau = -1: momentum step size is automatically
+%                               optimized in every step.
+%                               - tau as a function handle: user defined
+%                               behavior of tau momentum term.
+%    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.
+% =========================================================================
+
+[~,N] = size(Phi);
+
+%% Initialize transpose of measurement matrix
+
+Phi_t = Phi';
+
+%% Initialize to zero vector
+x_cur = zeros(N,1);
+y_cur = zeros(N,1);
+X_i = [];
+
+x_path = zeros(N, params.ALPSiters);
+
+%% CG params
+if (params.solveNewtonx == 1 || params.solveNewtonb == 1)
+    cg_verbose = 0;
+    cg_A = Phi_t*Phi;
+    cg_b = Phi_t*y;
+end;
+
+%% Determine momentum step size selection strategy
+optimizeTau = 0;
+function_tau = 0;
+
+if (isa(params.tau,'float'))
+    if (params.tau == -1)
+        optimizeTau = 1;
+    end;
+elseif (isa(params.tau, 'function_handle'))
+    function_tau = 1;
+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);
+setXi = zeros(N,1);
+setYi = zeros(N,1);
+
+i = 1;
+%% infty-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;    
+    [~, 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);    
+    
+    setder = zeros(N,1);
+    setder(S_i) = 1;
+    if (adaptive_mu)
+        % Step size selection via eq. (12) and eq. (13)        
+        Pder = Phi(:,S_i)*ider;
+        mu_bar = ider'*ider/(Pder'*Pder);        
+    end;
+    
+    iy_cur = y_cur.*setYi;    
+    if (~function_tau)      % If tau is not a function handle...
+        if (optimizeTau)    % Compute optimized tau
+            
+            % tau = argmin || u - Phi(x_i + y_i) ||
+            %     = <Phi*y_i, u - Phi(x_i - mu/2 * grad_Si f(xi))>/||Phi*y_i||^2
+            
+            if (i == 1)
+                params.tau = 0;
+            else
+                % u - Phi*(x_i - mu/2 grad_Si f(xi)) = u - Phi*b
+                if (adaptive_mu)
+                    b = x_cur(S_i) + mu_bar*ider;        % Non-zero elems: S_i
+                elseif (function_mu)
+                    b = x_cur(S_i) + params.mu(i)*ider;
+                else b = x_cur(S_i) + params.mu*ider;
+                end;
+                
+                y_Phi_b = y - Phi(:,S_i)*b;               
+                Phi_y_prev = Phi(:,Y_i)*y_cur(Y_i);     % Phi * y_i
+                params.tau = y_Phi_b'*Phi_y_prev/(Phi_y_prev'*Phi_y_prev);
+            end;            
+            
+            if (adaptive_mu)
+                y_cur = params.tau*iy_cur + mu_bar*der.*setder;
+            elseif (function_mu)
+                y_cur = params.tau*iy_cur + params.mu(i)*der.*setder;
+            else y_cur = params.tau*iy_cur + params.mu*der.*setder;
+            end;
+            
+            Y_i = ne(y_cur,0);
+            setYi = zeros(N,1);
+            setYi(Y_i) = 1;
+        else    % Tau fixed and user-defined
+            if (adaptive_mu)
+                y_cur = params.tau*iy_cur + mu_bar*der.*setder;
+            elseif (function_mu)
+                y_cur = params.tau*iy_cur + params.mu(i)*der.*setder;
+            else y_cur = params.tau*iy_cur + params.mu*der.*setder;
+            end;
+            
+            Y_i = ne(y_cur,0);
+            setYi = zeros(N,1);
+            setYi(Y_i) = 1;
+        end;
+    else
+        if (adaptive_mu)
+            y_cur = params.tau(i)*iy_cur + mu_bar*der.*setder;
+        elseif (function_mu)
+            y_cur = params.tau(i)*iy_cur + params.mu(i)*der.*setder;
+        else y_cur = params.tau(i)*iy_cur + params.mu*der.*setder;
+        end;
+        
+        Y_i = ne(y_cur,0);
+        setYi = zeros(N,1);
+        setYi(Y_i) = 1;
+    end;
+       
+    % Hard-threshold b and compute X_{i+1}
+    set_Xi_Yi = setXi + setYi;
+    ind_Xi_Yi = find(set_Xi_Yi > 0);
+    z = x_cur(ind_Xi_Yi) + y_cur(ind_Xi_Yi);
+    [~, ind_z] = sort(abs(z), 'descend');
+    x_cur = zeros(N,1);
+    x_cur(ind_Xi_Yi(ind_z(1:K))) = z(ind_z(1:K));
+    X_i = ind_Xi_Yi(ind_z(1:K));
+    
+    setXi = zeros(N,1);
+    setXi(X_i) = 1;
+    
+    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_cur(X_i) + params.mu(i)*ider;
+        else x_cur = x_cur(X_i) + params.mu*ider;
+        end;
+    elseif (params.solveNewtonx == 1)                
+        % Similar to HTP
+        if (params.useCG == 1)
+            [v, ~, ~] = 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;