function [grad] = gradwfixed(Sigma,indsup,Alpsup,C,Xapp,yapp,Sigmaold,pow);%GRADWFIXED Computes the gradient of an upper bound on SVM loss wrt SIGMA^POW  %  GRAD = GRADWFIXED(SIGMA,INDSUP,ALPSUP,C,XAPP,YAPP,SIGMAOLD,POW) %  is the gradient of the upper bound on the SVM loss obtained when the %  weight parameters are considered to be unaffected by SIGMA. %  %  SIGMA is the current SIGMA value%  INDSUP is the (nsup,1) index of current support vectors %  ALPSUP is the (nsup,1) vector of non-zero Lagrange multipliers%  C is the error penalty hyper-aparameter%  XAPP,YAPP are the learning examples%  SIGMAOLD is the SIGMA value corresponding to the current weights%  27/01/03 Y. Grandvalet% initializationnsup  = length(indsup);[n,d] = size(Xapp);% I) compute slack variables, old Sigma on support, new sigma on data pointsindpos = find(yapp== 1);indneg = find(yapp==-1);npos = length(indpos) ;nneg = length(indneg) ;nmin = min(npos,nneg);% I.1)  distancesXsupS0  = Xapp(indsup,:).*repmat(Sigmaold,nsup,1);XappS  = Xapp.*repmat(Sigma,n,1);Dist   = XsupS0*XappS';Dist   = Dist - repmat(0.5*sum(XsupS0.^2,2),1,n) - repmat(0.5*sum(XappS.^2,2)',nsup,1) ; % - 1/2 ||Sigma . xi - Sigmaold . xj||^2Dist   = exp(Dist) ;% I.2) slacksxipos = -sort(-(1 - Alpsup'*Dist(:,indpos))); xineg = -sort(-(1 + Alpsup'*Dist(:,indneg))); xi = sum([xineg(1:nmin) ; xipos(1:nmin)],1);% II) end: gradientind = find(xi>=0) ;grad = zeros(1,d);if ~isempty(ind);   for k=1:d;      Distk   = repmat(XappS(ind,k)',nsup,1) - repmat(XsupS0(:,k),1,length(ind)) ;      grad(k) =  Alpsup' * (Distk.*Dist(:,ind)) * (Xapp(ind,k).*yapp(ind)) ;   end;   ind = find(Sigma~=0);   grad(ind) = C*grad(ind).*(1/pow*abs(real(Sigma(ind).^(1-pow))));end;