svm optimization problem

Question

Suppose we have the dataset:

{(3,1),(3-1),(6,1),(6,-1)} {(1,0),(0,1),(0,-1),(-1,0)} the first set represent the positive label, and de second the negative.

I want manually find the support vectors, solve the equations and determine the coefficients W and bias w0 of the separating hyperplane.

I know that the SVs are {s1=(1,0),s2=(3,1),s3=(3-1)}

after i wrote a system of equations:

α1k(s1,s1) + α2k(s2,s1) + α3(s3,s1) = -1

α1k(s1,s2) + α2k(s2,s2) + α3(s3,s2) = -1

α1k(s1,s3) + α2k(s2,s3) + α3(s3,s3) = -1

Where k(x,y) is the dot product xy

the resolution of this system is impossible, can someone tell me where I'm wrong?  

Answer 1

I think that this system of equation is incorrect. If you know that (3, -1), (3, 1) and (1, 0) are support vectors then you need to solve the next system:

3*w1 - 1*w2 - w0 = ±1
3*w1 + 1*w2 - w0 = ±1
1*w1 + 0*w2 - w0 = ±1

Support vectors on the same side of the separation line have the same sign on the right side of the equation (same sign here: ±1). In this example, (1, 0) locates on one side, while (3, -1) and (3, 1) are on the other side. Thus, the system will be the following:

3*w1 - 1*w2 - w0 = 1
3*w1 + 1*w2 - w0 = 1
1*w1 + 0*w2 - w0 = -1

The solution is: w0 = 2, w1 = 1, w2 = 0. You can plot all the points and check that this solution is correct.