0
$\begingroup$

I am a beginner at SVM and I am currently reading Introduction to Statistical Learning.

Our aim is to maximize the margin of the of hyperplane boundary subject to certain conditions. My question is, why is one of the restrictions $\sum \beta_i^2 = 1$? Or did I misunderstand the statement?

$\endgroup$
1
$\begingroup$

In short - you have misunderstood something. There is no such condition in SVM formulation, nor is there any $\beta$ parameter.

Original (linear) formulation goes like

$$ \min_{w,b} \frac{1}{2}\|w\|^2 + C\sum_{i=1}^N\xi_i $$ subject to $$ y_i(\langle w, x_i\rangle -b) \geq 1- \xi_i, i=1, \dots, N $$

And in kernel formulation

$$ \max_{\alpha} \sum_{i=1}^N \alpha_i - \frac{1}{2} \sum_{i,j=1}^N \alpha_i \alpha_j y_i y_j K(x_i, x_j) $$

subject to

$$ 0 \leq \alpha_i \leq C $$ $$ \sum_{i=1}^N \alpha_iy_i = 0 $$

and these conditions come directly from KKT conditions for solutions of dual problem in convex optimization with linear constraints.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.