Question

upport Vector Machines 1. Show mathematically that weight vector is orthogonal to the decision boundary. 2. Show that the distance from a point vector x on to a decision boundary line y(x)- +bishere | lwl| is the Euclidean norm. y(xq) llwll

Answer 1

Let the decision boundary be defined as $w^Tx+b=0$. Consider the points $x_a$ and , which lie on the decision boundary. This gives us two equations:

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$w^Tx_a+b=0$

$w^Tx_b+b=0$

Subtracting these two equations gives us $w^T\cdot(x_a - x_b)=0$ .

Note that the vector lies on the decision boundary, and it is directed from
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to $x_a$. Since the dot product $w^T\cdot(x_a - x_b)$ is zero, $w^T$ must be orthogonal to , and in turn, to the decision boundary.