kernelsk.m

this a SVM toolbox,it is very useful for someone who just learn SVM.In order to be undestood easily,

function [Alpha,bias,sol,t,flps,margin,up,lo,trn_err]=...
    kernelsk(X,I,epsilon,ker,arg,tmax,C,up,lo)
% KERNELSK kernel Schlesinger-Kozinec's algorithm, SVM (L2). 
% [Alpha,bias]=kernelsk(X,I,epsilon,ker,arg,tmax,C)
%
% [Alpha,bias,sol,t,flps,margin,up,lo,trm_err]=...
%    kernelsk(X,I,epsilon,ker,arg,tmax,C)
%
% This algorithm solves the SVM problem with the quadratic 
% penalizition of classification violations using the 
% kernel Schlesinger-Kozinec's algorithm.
% 
% Inputs:
%   X [NxL] training patterns, N is dimension and L number of patterns.
%   I [1xL] labels, 1 for 1st class and 2 for 2nd class.
%   epsilon [1x1] precision of found solution. The margin of found 
%       hyperplane is less than the optimal margin at most by epsilon. 
%   ker [string] kernel, see 'help kernel'.
%   arg [...] argument of given kernel, see 'help kernel'.
%   tmax [int] maximal number of iterations.
%   C [real] trade-off between margin and training error.
%  
% Outputs:
%   Alpha [1xL] Weights (Lagrangians) of patterns.
%   bias [real] bias (threshold) of found decision rule.
%   sol [int] 1 solution is found
%             0 algorithm stoped (t == tmax) before converged.
%            -1 hyperplane with margin greater then epsilon 
%               does not exist.
%   t [int] number of iterations.
%   margin [real] margin between classes.
%   flps [int] number of used floating point operations.
%   up [1,t] evolution of the upper bound on the optimal margin.
%   lo [1,t] evolution of the lower bound on the optimal margin.
%   trn_err [real] training error (empirical risk).
%
% See also KERNELSKF, SVM.
%

% Statistical Pattern Recognition Toolbox, Vojtech Franc, Vaclav Hlavac
% (c) Czech Technical University Prague, http://cmp.felk.cvut.cz
% Written Vojtech Franc (diploma thesis) 02.11.1999, 13.4.2000
% Modifications
% 23-October-2001, V.Franc
% 19-Septemberr-2001, V.Franc, comments changed.
% 18-August-2001, V.Franc, m^* - m <= epsilon  instead of  <= epsilon/2
%  added uppper and lower bound on 
% 17-August-2001, V.Franc, renamed to KERNELSK.
% 13-July-2001, V.Franc, comments
% 12-July-2001, V.Franc, C, bias and normal vect. normalized.
% 11-July-2001, V.Franc, Rosta Horcik proved that the computation 
%                        of threshold is OK.
% 10-July-2001, V.Franc, derived from kekozinec2

flops(0);

% set default values of the input argiments
if nargin < 7,
  C = inf;
end

% maximal number of iteraions
if nargin < 6,
  tmax = inf;
end

% indexes of pattens in the 1st and 2nd class
xinx1 = find(I == 1);
xinx2 = find(I == 2);

X1=X(:,xinx1);      % patters from 1st class
X2=X(:,xinx2);      % patters from 1st class

l1 = size(X1,2);    % number os patterns
l2 = size(X2,2);

% compute kernel matrices
K1 = kernel( X1, X1, ker, arg );   % [l1 x l1]
K2 = kernel( X2, X2, ker, arg );   % [l2 x l2]
K12 = kernel( X1, X2, ker, arg );  % [l1 x l2]

% make problem lin-separable in high dimensional space

if C ~= 0,
  CD1=eye(l1,l1)/(2*C);    % additional diagonal matrix 
  CD2=eye(l2,l2)/(2*C);
  K1=K1+CD1;
  K2=K2+CD2;
end

% convex coeficients defining normal of the decision hyperplane
% (they correspond to the Lagrangian multiplyers).
s1 = zeros(l1, 1);
s2 = zeros(l2, 1);

% initial solution
s1(1)=1;       % take the 1st pattern from the 1st class
s2(1)=1;       % take the 2nd pattern from the 2st class

sol=0;
t = 0;

up=[];
lo=[];

% main cycle
while sol == 0 & tmax > t,

   t = t + 1;
   sol = 1;
   
   % -- compute auxciliary variables --
   a = s1'*K1*s1;
   b = s2'*K2*s2;
   c = s1'*K12*s2;
   
   f = K2*s2;
   e = K1*s1;
   
   d1 = e - K12*s2;
   d2 = f - (s1'*K12)';   
   
   [d1min,inx1] = min(d1);
   [d2min,inx2] = min(d2);
      
   % projection x \in X_1 on (w_1 - w_2)
   proj1 = (d1min + b -c )/sqrt(a-2*c+b);
   
   % projection x \in X_2 on (w_2 - w_1)
   proj2 = (d2min + a - c)/sqrt(a-2*c+b);
      
   if sqrt( a -2*c +b) <= 0,    
     % algorithm has converged to the zero vector --> classes overlap
     sol = -1;
     break;
   end

   % --- compute stop condition for the alpha1 (1st class) -----
   
   % (proj1 < proj2) ~ the worst point will be used for update 
   if (proj1 < proj2) & (proj1 <= (sqrt(a-2*c+b) - epsilon)),
    
     % -- Adaptation phase of vector alpha1 ----------------------------
     
     k = (a - d1min - c)/(a + K1(inx1,inx1) - 2*e(inx1) );
  
     k = min( 1, k );
    
     s1 = s1 * (1-k);
     s1(inx1) = s1(inx1) + k;
    
     sol = 0;
          
   else        
      % --- compute stop condition for the alpha2 (2st class) ------   
      
      if proj2 <= (sqrt(a-2*c+b) - epsilon ),

         % -- Adaptation phase ----------------------------------

         k = (b - d2min -c)/(b + K2(inx2,inx2) - 2*f(inx2) );

         k = min( 1, k );
    
         s2 = s2 * (1-k);
         s2(inx2) = s2(inx2) + k;

         sol = 0;
         
      end  
      
   end   
   
   % store up=||w||/2 and current margin m(w1-w2,theta) = min( m1, m2) 
   m = min([proj1,proj2]) - 0.5*sqrt(a-2*c+b);
   up = [up,sqrt(a-2*c+b)/2];       
   lo = [lo,m ];
         
end

% does the found hyperplane separate the classes or not ?
if sol == 1 & ((d2min + a - c) < 0 | (d1min + b -c ) < 0),
  sol = -2;
end

% --- determine threshold --------

%theta = 0.5*( s1'*K1*s1 - s2'*K2*s2 );

% sqared margin in transfromed space
margin2 = s1'*K1*s1 - 2*s1'*K12*s2 + s2'*K2*s2;

% threshold after normalization
theta = ( s1'*K1*s1 - s2'*K2*s2 )/margin2;

% solution (normal vect. in the transformed space) after normalization
s1=2*s1/margin2;
s2=2*s2/margin2;

% --- compute margin and classify training patterns
if C~=0,
  K1=K1-CD1;
  K2=K2-CD2;
end

margin = 1/sqrt( s1'*K1*s1 - 2*s1'*K12*s2 + s2'*K2*s2 );

dpred1 = (K1*s1 - K12*s2 )' - theta; 
dpred2 = (K12'*s1 -K2*s2 )' - theta; 

% classification error on the traning set
trn_err = length( find([dpred1,-dpred2] < 0) )/(l1+l2);

%----------------------
% make SVM-like output

Alpha=zeros(1,l1+l2);
Alpha(xinx1)=s1;
Alpha(xinx2)=s2;

bias = -theta;

% overall number of used float point operations
flps=flops;

return;