# Learning From Data – A Short Course: Exercise 8.1

Assume contains two data points and . Show that:

(a) No hyperplane can tolerate noise radius greater than .

Assume such hyperplane exists and we call it .

Let is the line connecting two points and , as two points stay on different sides seperated by the hyperplane , it’s always true that crosses at some point and we call that point and:

It is also true that:

So:

Fact and contradicts each other, so the statement follows.

(b) There is a hyperplane that tolerates a noise radius

Let is the line connecting two points and , and , the hyperplane that satisfies the condition is orthogonal to and contains .