rand('state',0);
N = 100;
M = 120;
X = 2 * rand(2,N) - 1;
X = X * diag(0.9*rand(1,N)./sqrt(sum(X.^2)));
Y = 2 * rand(2,M) - 1;
Y = Y * diag((1.1+rand(1,M))./sqrt(sum(Y.^2)));
d = sqrt(sum((X-[1.1;0]*ones(1,N)).^2));
Y = [ Y, X(:,d<0.9) ];
X = X(:,d>1);
N = size(X,2);
M = size(Y,2);
p1 = [0,0,1,0,1,2,0,1,2,3]';
p2 = [0,1,1,2,2,2,3,3,3,3]'-p1;
np = length(p1);
op = ones(np,1);
monX = X(op,:) .^ p1(:,ones(1,N)) .* X(2*op,:) .^ p2(:,ones(1,N));
monY = Y(op,:) .^ p1(:,ones(1,M)) .* Y(2*op,:) .^ p2(:,ones(1,M));
fprintf(1,'Finding the optimal polynomial of order 4 that separates the 2 classes...');
cvx_begin
variables a(np) t(1)
minimize ( t )
a'*monX <= t;
a'*monY >= -t;
cvx_end
fprintf(1,'Done! \n');
nopts = 2000;
angles = linspace(0,2*pi,nopts);
cont = zeros(2,nopts);
for i=1:nopts
v = [cos(angles(i)); sin(angles(i))];
l = 0; u = 1;
while ( u - l > 1e-3 )
s = (u+l)/2;
x = s * v;
if a' * ( x(op,:) .^ p1 .* x(2*op) .^ p2 ) > 0,
u = s;
else
l = s;
end
end;
s = (u+l)/2;
cont(:,i) = s*v;
end;
graph=plot(X(1,:),X(2,:),'o', Y(1,:), Y(2,:),'o', cont(1,:), cont(2,:), '-');
set(graph(2),'MarkerFaceColor',[0 0.5 0]);
title('No cubic polynomial can separate the 2 classes')
disp('-----------------------------------------------------------------');
disp('As seen on the figure, the 2 sets of points are not separated. ');
disp('There exists no cubic polynomial that can separate these 2 sets.');
Finding the optimal polynomial of order 4 that separates the 2 classes...
Calling sedumi: 211 variables, 11 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 11, order n = 212, dim = 212, blocks = 1
nnz(A) = 2321 + 0, nnz(ADA) = 121, nnz(L) = 66
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 4.26E+02 0.000
1 : -3.39E-01 1.69E+02 0.000 0.3957 0.9000 0.9000 1.99 1 1 9.7E+01
2 : -5.79E-02 1.23E+02 0.000 0.7309 0.9000 0.9000 9.56 1 1 1.3E+01
3 : -3.90E-03 5.32E+01 0.000 0.4318 0.9000 0.9000 9.12 1 1 1.6E+00
4 : -2.19E-03 2.29E+01 0.000 0.4309 0.9000 0.9000 0.93 1 1 7.3E-01
5 : -1.10E-03 1.10E+01 0.000 0.4807 0.9000 0.9000 1.56 1 1 2.7E-01
6 : -3.97E-04 5.29E+00 0.000 0.4804 0.9000 0.9000 1.77 1 1 9.5E-02
7 : -1.54E-04 2.84E+00 0.000 0.5359 0.9000 0.9000 1.99 1 1 3.6E-02
8 : -3.21E-05 8.44E-01 0.000 0.2975 0.9000 0.9000 1.89 1 1 7.1E-03
9 : -6.12E-07 1.86E-02 0.000 0.0221 0.9900 0.9900 1.29 1 1 1.4E-04
10 : -3.35E-12 1.03E-07 0.088 0.0000 1.0000 1.0000 1.01 1 1
iter seconds digits c*x b*y
10 0.2 17.6 0.0000000000e+00 -3.3652985170e-26
|Ax-b| = 5.2e-16, [Ay-c]_+ = 1.7E-22, |x|= 4.8e-01, |y|= 2.8e-23
Detailed timing (sec)
Pre IPM Post
3.000E-02 1.700E-01 1.000E-02
Max-norms: ||b||=1, ||c|| = 0,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 13.8426.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +3.3653e-26
Done!
-----------------------------------------------------------------
As seen on the figure, the 2 sets of points are not separated.
There exists no cubic polynomial that can separate these 2 sets.