function [model,Z]=greedykpca(X,options)
% GREEDYKPCA Greedy Kernel Principal Component Analysis.
%
% Synopsis:
% model = greedykpca(X)
% model = greedykpca(X,options)
%
% Description:
% This function implements a greedy kernel PCA algorithm.
% The input data X are first approximated by GREEDYKPCA in the
% feature space and second the ordinary PCA is applyed on the
% approximated data. This algorithm has the same objective function
% as the ordinary Kernel PCA but, in addition, the number of data in
% the resulting kernel expansion is limited.
%
% For more info refer to V.Franc: Optimization Algorithms for Kernel
% Methods. Research report. CTU-CMP-2005-22. CTU FEL Prague. 2005.
% ftp://cmp.felk.cvut.cz/pub/cmp/articles/franc/Franc-PhD.pdf .
%
% Input:
% X [dim x num_data] Input column vectors.
%
% options [struct] Control parameters:
% .ker [string] Kernel identifier. See 'help kernel' for more info.
% .arg [1 x narg] Kernel argument.
% .m [1x1] Maximal number of base vectors (Default m=0.25*num_data).
% .p [1x1] Depth of search for the best basis vector (p=m).
% .mserr [1x1] Desired mean squared reconstruction errors of approximation.
% .maxerr [1x1] Desired maximal reconstruction error of approximation.
% See 'help greedyappx' for more info about the stopping conditions.
% .verb [1x1] If 1 then some info is displayed (default 0).
%
% Output:
% model [struct] Kernel projection:
% .Alpha [nsv x new_dim] Multipliers defining kernel projection.
% .b [new_dim x 1] Bias the kernel projection.
% .sv.X [dim x num_data] Seleted subset of the training vectors..
% .nsv [1x1] Number of basis vectors.
% .kercnt [1x1] Number of kernel evaluations.
% .MaxErr [1 x nsv] Maximal reconstruction error for corresponding
% number of base vectors.
% .MsErr [1 x nsv] Mean square reconstruction error for corresponding
% number of base vectors.
%
% Example:
% X = gencircledata([1;1],5,250,1);
% model = greedykpca(X,struct('ker','rbf','arg',4,'new_dim',2));
% X_rec = kpcarec(X,model);
% figure;
% ppatterns(X); ppatterns(X_rec,'+r');
% ppatterns(model.sv.X,'ob',12);
%
% See also
% KERNELPROJ, KPCA, GREEDYAPPX.
%
% About: Statistical Pattern Recognition Toolbox
% (C) 1999-2003, Written by Vojtech Franc and Vaclav Hlavac
% <a href="http://www.cvut.cz">Czech Technical University Prague</a>
% <a href="http://www.feld.cvut.cz">Faculty of Electrical Engineering</a>
% <a href="http://cmp.felk.cvut.cz">Center for Machine Perception</a>
% Modifications:
% 09-sep-2005, VF
% 19-feb-2005, VF
% 10-jun-2004, VF
% 05-may-2004, VF
% 14-mar-2004, VF
start_time = cputime;
[dim,num_data]=size(X);
if nargin < 2, options = []; else options=c2s(options); end
if ~isfield(options,'ker'), options.ker = 'linear'; end
if ~isfield(options,'arg'), options.arg = 1; end
if ~isfield(options,'m'), options.m = fix(0.25*num_data); end
if ~isfield(options,'p'), options.p = options.m; end
if ~isfield(options,'maxerr'), options.maxerr = 1e-6; end
if ~isfield(options,'mserr'), options.mserr = 1e-6; end
if ~isfield(options,'verb'), options.verb = 0; end
[inx,Alpha,Z,kercnt,MsErr,MaxErr] = ...
greedyappx(X,options.ker,options.arg,...
options.m,options.p,options.mserr,options.maxerr,options.verb);
mu = sum(Z,2)/num_data;
Z=Z-mu*ones(1,num_data);
S = Z*Z';
[U,D,V]=svd(S);
model.eigval=diag(D);
sum_eig = triu(ones(size(Z,1),size(Z,1)),1)*model.eigval;
model.MsErr = MsErr(end)+sum_eig/num_data;
options.new_dim = min([options.new_dim,size(Z,1)]);
V = V(:,1:options.new_dim);
model.Alpha = Alpha'*V;
model.nsv = length(inx);
model.b = -V'*mu;
model.sv.X= X(:,inx);
model.sv.inx = inx;
model.kercnt = kercnt;
model.GreedyMaxErr = MaxErr;
model.GreedyMsErr = MsErr;
model.options = options;
model.cputime = cputime - start_time;
model.fun = 'kernelproj';
return;