comparison DL/RLS-DLA/SolveFISTA.m @ 40:6416fc12f2b8

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+% Copyright ©2010. The Regents of the University of California (Regents). 
+% All Rights Reserved. Contact The Office of Technology Licensing, 
+% UC Berkeley, 2150 Shattuck Avenue, Suite 510, Berkeley, CA 94720-1620, 
+% [phone], for commercial licensing opportunities.
+
+% Authors: Arvind Ganesh, Allen Y. Yang, Zihan Zhou.
+% Contact: Allen Y. Yang, Department of EECS, University of California,
+% Berkeley. <[e-mail]>
+
+% IN NO EVENT SHALL REGENTS BE LIABLE TO ANY PARTY FOR DIRECT, INDIRECT, 
+% SPECIAL, INCIDENTAL, OR CONSEQUENTIAL DAMAGES, INCLUDING LOST PROFITS, 
+% ARISING OUT OF THE USE OF THIS SOFTWARE AND ITS DOCUMENTATION, EVEN IF 
+% REGENTS HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+% REGENTS SPECIFICALLY DISCLAIMS ANY WARRANTIES, INCLUDING, BUT NOT LIMITED
+% TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A 
+% PARTICULAR PURPOSE. THE SOFTWARE AND ACCOMPANYING DOCUMENTATION, IF ANY, 
+% PROVIDED HEREUNDER IS PROVIDED "AS IS". REGENTS HAS NO OBLIGATION TO 
+% PROVIDE MAINTENANCE, SUPPORT, UPDATES, ENHANCEMENTS, OR MODIFICATIONS.
+
+%% This function is modified from Matlab code proximal_gradient_bp
+
+function [x_hat,nIter] = SolveFISTA(A,b, varargin)
+
+% b - m x 1 vector of observations/data (required input)
+% A - m x n measurement matrix (required input)
+%
+% tol - tolerance for stopping criterion.
+%     - DEFAULT 1e-7 if omitted or -1.
+% maxIter - maxilambdam number of iterations
+%         - DEFAULT 10000, if omitted or -1.
+% lineSearchFlag - 1 if line search is to be done every iteration
+%                - DEFAULT 0, if omitted or -1.
+% continuationFlag - 1 if a continuation is to be done on the parameter lambda
+%                  - DEFAULT 1, if omitted or -1.
+% eta - line search parameter, should be in (0,1)
+%     - ignored if lineSearchFlag is 0.
+%     - DEFAULT 0.9, if omitted or -1.
+% lambda - relaxation parameter
+%    - ignored if continuationFlag is 1.
+%    - DEFAULT 1e-3, if omitted or -1.
+% outputFileName - Details of each iteration are dumped here, if provided.
+%
+% x_hat - estimate of coeeficient vector
+% numIter - number of iterations until convergence
+%
+%
+% References
+% "Robust PCA: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization", J. Wright et al., preprint 2009.
+% "An Accelerated Proximal Gradient Algorithm for Nuclear Norm Regularized Least Squares problems", K.-C. Toh and S. Yun, preprint 2009.
+%
+% Arvind Ganesh, Summer 2009. Questions? [e-mail]
+
+DEBUG = 0 ;
+
+STOPPING_GROUND_TRUTH = -1;
+STOPPING_DUALITY_GAP = 1;
+STOPPING_SPARSE_SUPPORT = 2;
+STOPPING_OBJECTIVE_VALUE = 3;
+STOPPING_SUBGRADIENT = 4;
+STOPPING_DEFAULT = STOPPING_SUBGRADIENT;
+
+stoppingCriterion = STOPPING_DEFAULT;
+maxIter = 1000 ;
+tolerance = 1e-3;
+[m,n] = size(A) ;
+x0 = zeros(n,1) ;
+xG = [];
+
+%% Initializing optimization variables
+t_k = 1 ; 
+t_km1 = 1 ;
+L0 = 1 ;
+G = A'*A ;
+nIter = 0 ;
+c = A'*b ;
+lambda0 = 0.99*L0*norm(c,inf) ;
+eta = 0.6 ;
+lambda_bar = 1e-4*lambda0 ;
+xk = zeros(n,1) ;
+lambda = lambda0 ;
+L = L0 ;
+beta = 1.5 ;
+
+% Parse the optional inputs.
+if (mod(length(varargin), 2) ~= 0 ),
+    error(['Extra Parameters passed to the function ''' mfilename ''' lambdast be passed in pairs.']);
+end
+parameterCount = length(varargin)/2;
+
+for parameterIndex = 1:parameterCount,
+    parameterName = varargin{parameterIndex*2 - 1};
+    parameterValue = varargin{parameterIndex*2};
+    switch lower(parameterName)
+        case 'stoppingcriterion'
+            stoppingCriterion = parameterValue;
+        case 'groundtruth'
+            xG = parameterValue;
+        case 'tolerance'
+            tolerance = parameterValue;
+        case 'linesearchflag'
+            lineSearchFlag = parameterValue;
+        case 'lambda'
+            lambda_bar = parameterValue;
+        case 'maxiteration'
+            maxIter = parameterValue;
+        case 'isnonnegative'
+            isNonnegative = parameterValue;
+        case 'continuationflag'
+            continuationFlag = parameterValue;
+        case 'initialization'
+            xk = parameterValue;
+            if ~all(size(xk)==[n,1])
+                error('The dimension of the initial xk does not match.');
+            end
+        case 'eta'
+            eta = parameterValue;
+            if ( eta <= 0 || eta >= 1 )
+                disp('Line search parameter out of bounds, switching to default 0.9') ;
+                eta = 0.9 ;
+            end
+        otherwise
+            error(['The parameter ''' parameterName ''' is not recognized by the function ''' mfilename '''.']);
+    end
+end
+clear varargin
+
+if stoppingCriterion==STOPPING_GROUND_TRUTH && isempty(xG)
+    error('The stopping criterion must provide the ground truth value of x.');
+end
+
+keep_going = 1 ;
+nz_x = (abs(xk)> eps*10);
+f = 0.5*norm(b-A*xk)^2 + lambda_bar * norm(xk,1);
+xkm1 = xk;
+while keep_going && (nIter < maxIter)
+    nIter = nIter + 1 ;
+    
+    yk = xk + ((t_km1-1)/t_k)*(xk-xkm1) ;
+    
+    stop_backtrack = 0 ;
+    
+    temp = G*yk - c ; % gradient of f at yk
+    
+    while ~stop_backtrack
+        
+        gk = yk - (1/L)*temp ;
+        
+        xkp1 = soft(gk,lambda/L) ;
+        
+        temp1 = 0.5*norm(b-A*xkp1)^2 ;
+        temp2 = 0.5*norm(b-A*yk)^2 + (xkp1-yk)'*temp + (L/2)*norm(xkp1-yk)^2 ;
+        
+        if temp1 <= temp2
+            stop_backtrack = 1 ;
+        else
+            L = L*beta ;
+        end
+        
+    end
+    
+    switch stoppingCriterion
+        case STOPPING_GROUND_TRUTH
+            keep_going = norm(xG-xkp1)>tolerance;
+        case STOPPING_SUBGRADIENT
+            sk = L*(yk-xkp1) + G*(xkp1-yk) ;
+            keep_going = norm(sk) > tolerance*L*max(1,norm(xkp1));
+        case STOPPING_SPARSE_SUPPORT
+            % compute the stopping criterion based on the change
+            % of the number of non-zero components of the estimate
+            nz_x_prev = nz_x;
+            nz_x = (abs(xkp1)>eps*10);
+            num_nz_x = sum(nz_x(:));
+            num_changes_active = (sum(nz_x(:)~=nz_x_prev(:)));
+            if num_nz_x >= 1
+                criterionActiveSet = num_changes_active / num_nz_x;
+                keep_going = (criterionActiveSet > tolerance);
+            end
+        case STOPPING_OBJECTIVE_VALUE
+            % compute the stopping criterion based on the relative
+            % variation of the objective function.
+            prev_f = f;
+            f = 0.5*norm(b-A*xkp1)^2 + lambda_bar * norm(xk,1);
+            criterionObjective = abs(f-prev_f)/(prev_f);
+            keep_going =  (criterionObjective > tolerance);
+        case STOPPING_DUALITY_GAP
+            error('Duality gap is not a valid stopping criterion for PGBP.');
+        otherwise
+            error('Undefined stopping criterion.');
+    end
+    
+    lambda = max(eta*lambda,lambda_bar) ;
+    
+    
+    t_kp1 = 0.5*(1+sqrt(1+4*t_k*t_k)) ;
+    
+    t_km1 = t_k ;
+    t_k = t_kp1 ;
+    xkm1 = xk ;
+    xk = xkp1 ;
+end
+
+x_hat = xk ;
+
+function y = soft(x,T)
+if sum(abs(T(:)))==0
+    y = x;
+else
+    y = max(abs(x) - T, 0);
+    y = sign(x).*y;
+end