$(40$ points$)$ Fitting an SVM by hand.

For this problem you will solve an SVM without the help of a computer, relying instead on

principled rules and properties of these classifiers. Consider a dataset with the following $7$ data

points $(x_i,y_i)$ :

$\{(-3,+1),(-2,+1),(-1,-1),(0,-1),(+1,-1),(+2,+1),(+3,+1)\}$

Here $y_i=-1$ or $+1$ indicates the data point i belongs to either class.

Consider mapping these points to $2$ dimensions using the feature vector $(x)=(x,x^2).$ The

hard margin classifier training problem is:

min$||{|}_2^2$

such that $y_i({}^T(x_i)+{}_0)1,$AAiin$\{1,$dots,$n\}$

This question has been broken down into a series of questions, each providing a part of the

solution. Make sure to follow the logical structure of the exercise when composing your answer

and to justify each step.

a$.$ Plot the training data and draw the decision boundary of the max margin classifier.

b$.$ What is the value of the margin achieved by the optimal decision boundary?

c$.$ What is a vector that is orthogonal to the decision boundary?

d$.$ Considering discriminant ${}^T(x_i)+{}_0,$ what value of $$ and ${}_0$ will achieve optimal

hyperplane?

e$.$ What are the support vectors of the classifier? Confirm that the support vectors are indeed

solutions to the constraint $y_i({}^T(x_i)+{}_0)1$ using estimated $$ and ${}_0.$