Basics for the positive support vectors in the support vector machine

Question

Can anyone explain in detail why $wx^++b=1$ for positive support vectors in SVM and $wx^-+b=-1$ for negative support vectors in SVM?

Answer 1

The decision boundary of SVM is $wx+b=0$. When you shift this line to positive and negative sides by some amount, you'll get parallel lines of the form: $wx+b=\pm d$. $wx+b=d$ will hit positive SVs and $wx+b=-d$ will hit negative SVs. Instead of using a third variable ($d$) here, we could scale $w$ and $b$ such that our line does not change. For example, $2x+3y=0$ and $4x+6y=0$ are still the same. Similarly, one could search for $w$ and $b$ such that $wx+b=\pm1$ at SVs without loss of generality.