nonlin_gg.m

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

function [s, err_cost, iter_time]=nonlin_gg(x,F,C,m,varargin)
% nonlin_gg: Nonlinear sparse approximation by greedy gradient search.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Usage
% [s, err_cost, iter_time]=nonlin_gg(x,P,m,'option_name','option_value')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Input
%   Mandantory:
%               x   Observation vector to be decomposed
%               F   Function mapping from the sparse coeficient domain to
%                   the observation domain.
%               C   Function handle to cost function to be optimised
%               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, cost, cost_change                 | M
%   stopTol        | number (see below)                         | n/4
%   maxIter        | positive integer (see below)               | n
%   verbose        | true, false                                | false
%   start_val      | vector of length m                         | zeros
%   weights        | vector of length m containing weights      | ones
%                  | to biase atom selection                    |
%   grad_pert      | pertubation size to numerically evaluate   | 1e-6
%                  | gradient                                   |
%   step_size      | step size for gradient optimisation step   | 0.1
%   grad_stop      | Change in the norm of the gradient below   | 1e-3
%                  | which gradient optimisation stops          |
%   max_grad       | Maximum number of gradient steps           | 1000
%   grad           | Function handle to gradient of cost        |
%                  | function to be optimised                   | 
%   optimiser      | Function handle to a function that         |
%                  | optimises F for given initial value and    |
%                  | given subset of coefficients               |
%   PlotFunc       | Function handle to a function that         |
%                  | plots estimate of signal given sparse      |
%                  | coefficient estimate                       |
%                  |                                            | 
%
%   Available stopping criteria :
%               M           -   Extracts exactly M = stopTol elements.
%               corr        -   Stops when maximum correlation between
%                               residual and atoms is below stopTol value.
%               cost        -   Stops when cost C is below stopTol value.
%               cost_change -   Stops when the change cost C falles below
%                               stopTol value.
%
%   stopTol: Value for stoping criterion.
%
%   maxIter: Maximum of allowed iterations.
%
%   verbose: Logical value to allow algorithm progress to be displayed.
%
%   start_val: Allows algorithms to start from partial solution.
%
%   optimiser: Allows the specification of a function OPT(s,x,INDEX). 
%              OPT msut return the full coefficient vector s after 
%              optimisation of the cost function C over a given subset of
%              coefficients in s indexed by INDEX.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Outputs
%	 s              Solution vector 
%    err_cost       Vector containing cost of approximation error for each 
%                   iteration
%    iter_time      Vector containing times for each iteration
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Description
%   Algorithm selects greedily new elements in each iteration based on the
%   gradient of the cost function to be minimised with respect to the
%   coefficients. If the exact gradient is not specified, the gradienbt is
%   approximated using diference between cost at s and at s+DELTA. Once a
%   new element has been selected, gradient optimisation is used to
%   minimise the cost function based on the selected elements.
% References: T. Blumensath, M. E. Davies; "Gradient Pursuit for Non-Linear
%            Sparse Signal Modelling", submitted to EUSIPCO, 2008.
%   

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%                    Default values and initialisation
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[n1 n2]=size(x);
if n2 == 1
    n=n1;
elseif n1 == 1
    x=x';
    n=n2;
else
   display('x must be a vector.');
   return
end
    
sigsize         = x'*x/n;
initial_given   = 0;
do_plot         = 0;
err_cost        = [];
iter_time       = [];
STOPCRIT        = 'M';
STOPTOL         = ceil(n/4);
MAXITER         = n;
verbose         = false;
s_initial       = zeros(m,1);
w               = ones(m,1);
g_tol           = 1e-6;
step_size       = 1;
grad_tol        = 1e-3;
grad_given      = 0;
optimiser_given = 0;
Max_Grad        = 100;


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
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% Put option into nice format
Options={};
OS=nargin-4;
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'; 'cost'; 'cost_change'},'exact'));
                STOPCRIT    = Options{i+1};  
            else error('stopCrit must be char string [M, corr, cost, cost_change]. Exiting.'); end 
        case {'stopTol'}
            if isa(Options{i+1},'numeric') ; STOPTOL     = Options{i+1};   
            else error('stopTol must be number. Exiting.'); end
        case {'grad_pert'}
            if isa(Options{i+1},'numeric') ; g_tol     = Options{i+1};   
            else error('grad_pert must be number. Exiting.'); end
        case {'grad'}
            if isa(Options{i+1},'function_handle') ; grad_given = 1; Grad = Options{i+1};   
            else error('grad must be function handle. Exiting.'); end
        case {'grad_stop'}
            if isa(Options{i+1},'numeric') ; grad_tol     = Options{i+1};   
            else error('grad_stop must be number. Exiting.'); end
        case {'step_size'}
            if isa(Options{i+1},'numeric') ; step_size     = Options{i+1};   
            else error('step_size must be number. Exiting.'); end
        case {'weights'} 
            if isa(Options{i+1},'numeric') & length(Options{i+1}) == m ;
                w  = Options{i+1};   
            else error('weights must be a vector of length m. Exiting.'); end
        case {'max_grad'} 
            if isa(Options{i+1},'numeric') & length(Options{i+1}) == 1 ;
                Max_Grad  = Options{i+1};   
            else error('max_grad must be scalar. 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 {'optimiser'}
            if isa(Options{i+1},'function_handle'); OPT = Options{i+1}; optimiser_given=1;
            else error('optimiser must be function handle. Exiting.'); end 
        case {'PlotFunc'}
            if isa(Options{i+1},'function_handle'); PlotFunc = Options{i+1}; do_plot=1;
            else error('PlotFunc must be function handle. Exiting.'); end 
       otherwise
            
            error('Unrecognised option. Exiting.') 
   end
end



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

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


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

if initial_given ==1;
    IND          = find(s_initial);
    s            = s_initial;
    
else
    IND         = [];
    s           = s_initial;
end




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                        Main algorithm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if verbose
   display('Main iterations...') 
end
tic
t=0;

done = 0;
ne   = C(s,x);