Determining SVMs by hand

Disclaimer

This is not in the study design, however, our teacher has done this in a previous SAC.

Find the equation of an actual line that is a decent hyperplane: Here, a good hyperplane would be a linear line in the middle of the points (2,3) and (4,5). Note that the gradient of the line would be $- 1$ (perpendicular to the line that joins the two support vectors). Hence, note that our line would be: $x_{2} - 3 = - (x_{1} - 2)$ . Rearranging, we get $x_{1} + x_{2} - 7 = 0$ .
Note down the proportion of the weights and find a relationship between the weights: The weights $(w_{1}, w_{2})$ must be proportional to the coefficients of the line. Hence, $(w_{1}, w_{2}) = k (1, 1), k \in R$ . Therefore, $w_{1} = w_{2}$ .
Create new equations that satisfy the class +1 and -1 equations: Recall that support vectors must satisfy the margin equations:
- $w_{1} x_{1} + w_{2} x_{2} + b = + 1$ - positive class vectors
- $w_{1} x_{1} + w_{2} x_{2} + b = - 1$ - negative class vectors Substituting points, this yields:
- $4 w_{1} + 5 w_{2} + b = - 1$
- $2 w_{1} + 3 w_{2} + b = 1$
Solve the equations for the weights and biases, by using the relation from step 2: Solving yields $w_{1} = w_{2} = \frac{1}{2}, b = - \frac{7}{2}$ .