greed_gp.m

Sparse Estimation Algorithms by Blumensath and Davies min ||x||_0 subject to ||y - Ax||_2<e

function [s, err_mse, iter_time]=greed_gp(x,A,m,varargin)
% greed_gp: Gradient Pursuit algorithm from [1]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Usage
% [s, err_mse, iter_time]=greed_gp(x,P,m,'option_name','option_value')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Input
%   Mandatory:
%               x   Observation vector to be decomposed
%               P   Either:
%                       1) An nxm matrix (n must be dimension of x)
%                       2) A function handle (type "help function_format" 
%                          for more information)
%                          Also requires specification of P_trans option.
%                       3) An object handle (type "help object_format" for 
%                          more information)
%               m   length of s 
%
%   Possible additional options:
%   (specify as many as you want using 'option_name','option_value' pairs)
%   See below for explanation of options:
%__________________________________________________________________________
%   option_name    |     available option_values                | default
%--------------------------------------------------------------------------
%   stopCrit       | M, corr, mse, mse_change                   | M
%   stopTol        | number (see below)                         | n/4
%   P_trans        | function_handle (see below)                | 
%   maxIter        | positive integer (see below)               | n
%   verbose        | true, false                                | false
%   start_val      | vector of length m                         | zeros
%   GradSteps      | 'auto' or integer                          | 'auto'
%
%   Available stopping criteria :
%               M           -   Extracts exactly M = stopTol elements.
%               corr        -   Stops when maximum correlation between
%                               residual and atoms is below stopTol value.
%               mse         -   Stops when mean squared error of residual 
%                               is below stopTol value.
%               mse_change  -   Stops when the change in the mean squared 
%                               error falls below stopTol value.
%
%   stopTol: Value for stopping criterion.
%
%   P_trans: If P is a function handle, then P_trans has to be specified and 
%            must be a function handle. 
%
%   maxIter: Maximum of allowed iterations.
%
%   verbose: Logical value to allow algorithm progress to be displayed.
%
%   start_val: Allows algorithms to start from partial solution.
%
%   GradSteps: Number of gradient optimisation steps per iteration.
%              'auto' uses inner products to decide if more gradient steps 
%              are required. 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Outputs
%    s              Solution vector 
%    err_mse        Vector containing mse of approximation error for each 
%                   iteration
%    iter_time      Vector containing times for each iteration
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Description
%   greed_gp performs a greedy signal decomposition. 
%   In each iteration a new element is selected depending on the inner
%   product between the current residual and columns in P.
%   Gradient optimisation is used to update all selected non-zero elements.
%   
%   THIS ALGORITHM IS AN ALTERNATIVE TO OMP IF OMP IS NOT FEASIBLE DUE TO
%   COMPUTATION TIME OR STORAGE REQUIREMENTS!
%   
% References
%   [1] T. Blumensath and M.E. Davies, "Gradient Pursuits", submitted, 2007
%
% See Also
%   greed_omp, greed_ols, greed_mp, greed_nomp, greed_pcgp
%
% Copyright (c) 2007 Thomas Blumensath
%
% The University of Edinburgh
% Email: [e-mail]
% Comments and bug reports welcome
%
% This file is part of sparsity Version 0.1
% Created: April 2007
%
% Part of this toolbox was developed with the support of EPSRC Grant
% D000246/1
%
% Please read COPYRIGHT.m for terms and conditions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                    Default values and initialisation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


[n1 n2]=size(x);
if n2 == 1
    n=n1;
elseif n1 == 1
    x=x';
    n=n2;
else
   error('x must be a vector.');
end
    
sigsize     = x'*x/n;
initial_given=0;   
err_mse     = [];
iter_time   = [];
STOPCRIT    = 'M';
STOPTOL     = ceil(n/4);
MAXITER     = n;
verbose     = false;
s_initial   = zeros(m,1);
GradSteps   = 'auto';

if verbose
   display('Initialising...') 
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                           Output variables
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

switch nargout 
    case 3
        comp_err=true;
        comp_time=true;
    case 2 
        comp_err=true;
        comp_time=false;
    case 1
        comp_err=false;
        comp_time=false;
    case 0
        error('Please assign output variable.')        
    otherwise
        error('Too many output arguments specified')
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                       Look through options
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Put option into nice format
Options={};
OS=nargin-3;
c=1;
for i=1:OS
    if isa(varargin{i},'cell')
        CellSize=length(varargin{i});
        ThisCell=varargin{i};
        for j=1:CellSize
            Options{c}=ThisCell{j};
            c=c+1;
        end
    else
        Options{c}=varargin{i};
        c=c+1;
    end
end
OS=length(Options);
if rem(OS,2)
   error('Something is wrong with argument name and argument value pairs.') 
end

for i=1:2:OS
   switch Options{i}
        case {'stopCrit'}
            if (strmatch(Options{i+1},{'M'; 'corr'; 'mse'; 'mse_change'},'exact'));
                STOPCRIT    = Options{i+1};  
            else error('stopCrit must be char string [M, corr, mse, mse_change]. Exiting.'); end 
        case {'stopTol'}
            if isa(Options{i+1},'numeric') ; STOPTOL     = Options{i+1};   
            else error('stopTol must be number. Exiting.'); end
        case {'P_trans'} 
            if isa(Options{i+1},'function_handle'); Pt = Options{i+1};   
            else error('P_trans must be function _handle. Exiting.'); end
        case {'maxIter'}
            if isa(Options{i+1},'numeric'); MAXITER     = Options{i+1};             
            else error('maxIter must be a number. Exiting.'); end
        case {'verbose'}
            if isa(Options{i+1},'logical'); verbose     = Options{i+1};   
            else error('verbose must be a logical. Exiting.'); end 
        case {'start_val'}
            if isa(Options{i+1},'numeric') & length(Options{i+1}) == m ;
                s_initial     = Options{i+1};   
                initial_given=1;
            else error('start_val must be a vector of length m. Exiting.'); end
        case {'GradSteps'}
            if isa(Options{i+1},'numeric') || strcmp(Options{i+1},'auto') ;
                GradSteps     = Options{i+1};   
            else error('start_val must be a vector of length m. Exiting.'); end
        otherwise
            error('Unrecognised option. Exiting.') 
   end
end


if strcmp(STOPCRIT,'M') 
    maxM=STOPTOL;
else
    maxM=MAXITER;
end

if nargout >=2
    err_mse = zeros(maxM,1);
end
if nargout ==3
    iter_time = zeros(maxM,1);
end



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Make P and Pt functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

if          isa(A,'float')      P =@(z) A*z;  Pt =@(z) A'*z;
elseif      isobject(A)         P =@(z) A*z;  Pt =@(z) A'*z;
elseif      isa(A,'function_handle') 
    try
        if          isa(Pt,'function_handle'); P=A;
        else        error('If P is a function handle, Pt also needs to be a function handle. Exiting.'); end
    catch error('If P is a function handle, Pt needs to be specified. Exiting.'); end
else        error('P is of unsupported type. Use matrix, function_handle or object. Exiting.'); end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Do we start from zero or not?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

if initial_given ==1;
    IN          = find(s_initial);
    Residual    = x-P(s_initial);
    s           = s_initial;
    oldERR      = Residual'*Residual/n;
else
    IN          = [];
    Residual    = x;
    s           = s_initial;
    sigsize     = x'*x/n;
    oldERR      = sigsize;
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                 Random Check to see if dictionary is normalised 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

        mask=zeros(m,1);
        mask(ceil(rand*m))=1;
        nP=norm(P(mask));
        if abs(1-nP)>1e-3;
            display('Dictionary appears not to have unit norm columns.')
        end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Main algorithm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if verbose
   display('Main iterations...') 
end
tic
t=0;
p=zeros(m,1);
DR=Pt(Residual);
[v I]=max(abs(DR));
IN=[IN I];
done = 0;
iter=1;
while ~done

    
    % Select new element
    if isa(GradSteps,'char')
        if strcmp(GradSteps,'auto')
             
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Iteration to automatic selection of the number of gradient steps
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

            finished=0;    
            while ~finished
                    p(IN)=DR(IN);
                 % Step size
                     Dp=P(p);
                     a=Residual'*Dp/(Dp'*Dp);

                 % Update coefficients   
                      s=s+a*p;

                 % New Residual and inner products
                     Residual=Residual-a*Dp;
                     DR=Pt(Residual);
                 % select new element
                     [v I]=max(abs(DR));

                 if isempty(find (IN==I))
                    IN=[IN I];
                    finished=1;
                 end
            end
        else
            
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                           Is option known?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
            error('Undefined option for GradSteps, use ''auto'' or an integer.')
        end
    elseif isa(GradSteps,'numeric') 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               Iteration for fixed number of gradient steps
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    
                
        % Do GradSteps gradient steps
        count=1;
        while count<=GradSteps
                 p(IN)=DR(IN);
             % Step size
                 Dp=P(p);
                 a=Residual'*Dp/(Dp'*Dp);

             % Update coefficients   
                 s=s+a*p;

             % New Residual and inner products
                 Residual=Residual-a*Dp;
                 DR=Pt(Residual);
                 count=count+1;
        end
             % select new element
                 [v I]=max(abs(DR));
                 IN=[IN I];
                
    else
          error('Undefined option for GradSteps, use ''auto'' or an integer.')
    end


        
    ERR=Residual'*Residual/n;
     if comp_err
         err_mse(iter)=ERR;
     end
     
     if comp_time
         iter_time(iter)=toc;
     end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Are we done yet?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     
     if strcmp(STOPCRIT,'M')
         if iter >= STOPTOL
             done =1;
         elseif verbose && toc-t>10
            display(sprintf('Iteration %i. --- %i iterations to go',iter ,STOPTOL-iter)) 
            t=toc;
         end
    elseif strcmp(STOPCRIT,'mse')
         if comp_err
            if err_mse(iter)<STOPTOL;
                done = 1; 
            elseif verbose && toc-t>10
                display(sprintf('Iteration %i. --- %i mse',iter ,err_mse(iter))) 
                t=toc;
            end
         else
             if ERR<STOPTOL;
                done = 1; 
             elseif verbose && toc-t>10
                display(sprintf('Iteration %i. --- %i mse',iter ,ERR)) 
                t=toc;
             end
         end
     elseif strcmp(STOPCRIT,'mse_change') && iter >=2
         if comp_err && iter >=2
              if ((err_mse(iter-1)-err_mse(iter))/sigsize <STOPTOL);
                done = 1; 
             elseif verbose && toc-t>10
                display(sprintf('Iteration %i. --- %i mse change',iter ,(err_mse(iter-1)-err_mse(iter))/sigsize )) 
                t=toc;
             end
         else
             if ((oldERR - ERR)/sigsize < STOPTOL);
                done = 1; 
             elseif verbose && toc-t>10
                display(sprintf('Iteration %i. --- %i mse change',iter ,(oldERR - ERR)/sigsize)) 
                t=toc;
             end
         end
     elseif strcmp(STOPCRIT,'corr') 
          if max(abs(DR)) < STOPTOL;
             done = 1; 
          elseif verbose && toc-t>10
                display(sprintf('Iteration %i. --- %i corr',iter ,max(abs(DR)))) 
                t=toc;
          end
     end
     
    % Also stop if residual gets too small or maxIter reached
     if comp_err
         if err_mse(iter)<1e-16
             display('Stopping. Exact signal representation found!')
             done=1;
         end
     else


         if iter>1
             if ERR<1e-16
                 display('Stopping. Exact signal representation found!')
                 done=1;
             end
         end
     end

     if iter >= MAXITER
         display('Stopping. Maximum number of iterations reached!')
         done = 1; 
     end
     
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                    If not done, take another round
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
     if ~done
        iter=iter+1;
        oldERR=ERR;
     end
end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                  Only return as many elements as iterations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

if nargout >=2
    err_mse = err_mse(1:iter);
end
if nargout ==3
    iter_time = iter_time(1:iter);
end

if verbose
   display('Done') 
end

% Change history
%
% 8 of Februray: Algo does no longer stop if dictionary is not normaliesd.