%SVO Support Vector Optimizer%%   [V,J] = SVO(K,NLAB,C)%% INPUT%   K     Similarity matrix%   NLAB  Label list consisting of -1/+1%   C     Scalar for weighting the errors (optional; default: 10)%% OUTPUT%   V     Vector of weights for the support vectors%   J     Index vector pointing to the support vectors%% DESCRIPTION% A low level routine that optimizes the set of support vectors for a 2-class% classification problem based on the similarity matrix K computed from the% training set. SVO is called directly from SVC. The labels NLAB should indicate % the two classes by +1 and -1. Optimization is done by a quadratic programming. % If available, the QLD function is used, otherwise an appropriate Matlab routine.%% SEE ALSO% SVC% Revisions:% DR1, 07-05-2003%      Sign error in calculation of offset% Copyright: D.M.J. Tax, D. de Ridder, R.P.W. Duin, duin@ph.tn.tudelft.nl% Faculty of Applied Sciences, Delft University of Technology% P.O. Box 5046, 2600 GA Delft, The Netherlands% $Id: svo.m,v 1.16 2005/02/01 16:07:34 duin Exp $function [v,J] = svo(K,y,C)	prtrace(mfilename);	if (nargin < 3)		prwarning(3,'Third parameter not specified, assuming 10.');    C = 10;	end	vmin = 1e-9;		% Accuracy to determine when an object becomes the support object.	% Set up the variables for the optimization.	n  = size(K,1);	D  = (y*y').*K;	f  = -ones(1,n);	A  = y';	b  = 0;	lb = zeros(n,1);	ub = repmat(C,n,1);	p  = rand(n,1);	% Make the kernel matrix K positive definite.	i = -30;	while (pd_check (D + (10.0^i) * eye(n)) == 0)		i = i + 1;	end	if (i > -30),		prwarning(2,'K is not positive definite. The diagonal is regularized by 10.0^(%d)*I',i);	end	i = i+2;	D = D + (10.0^(i)) * eye(n);	% Minimization procedure initialization:	% 'qp' minimizes:   0.5 x' K x + f' x	% subject to:       Ax <= b	%	if (exist('qld') == 3)		v = qld (D, f, -A, b, lb, ub, p, 1);	elseif (exist('quadprog') == 2)		prwarning(1,'QLD not found, the Matlab routine QUADPROG is used instead.')		opt = optimset; opt.LargeScale='off'; opt.Display='off';		v = quadprog(D, f, [], [], A, b, lb, ub,[],opt);	else 		prwarning(1,'QLD not found, the Matlab routine QP is used instead.')		verbos = 0;		negdef = 0;		normalize = 1;		v = qp(D, f, A, b, lb, ub, p, 1, verbos, negdef, normalize);	end		% Find all the support vectors.	J = find(v > vmin);	% First find the SV on the boundary	I = find((v > vmin) & (v < C-vmin));		if (isempty(v) | isempty(I) )		prwarning(1,'Quadratic Optimization failed. Pseudo-Fisher is computed instead.');		v = pinv([K ones(n,1)])*y;		J = [1:n]';		return;	end	v = y.*v;	% and now, find the output for these SVs:	%DR1 out = sum(repmat(v(J)',length(I),1).*K(I,J),2)-y(I);	out = y(I)-sum(repmat(v(J)',length(I),1).*K(I,J),2);	v = [v(J); mean(out)];return;