Exercise $17\mathrm{.}1[$Fitting an SVM classifier by hand $*]$

$($Source: Jaakkola.$)$ Consider a dataset with $2$ points in $1$d: x$1=0$ with label y$1=1$ and x$2=2$ with

label y$2=1.$ Consider mapping each point to $3$d using the feature vector $\backslash$phi $($x$)=[1,2$x$,$ x$2]$T $.($This is Author: Kevin P$.$ Murphy. $($C$)$ MIT Press. CC$-$BY$-$NC$-$ND license

$600$ Chapter $17.$ Kernel Methods $*$ equivalent to using a second order polynomial kernel.$)$ The max margin classifier has the form

min $||$w$||2$ s$.$t$.$ y$1($wT $\backslash$phi $($x$1)+$ w$0)>=1$ y$2($wT $\backslash$phi $($x$2)+$ w$0)>=1$

$(17\mathrm{.}119)(17\mathrm{.}120)(17\mathrm{.}121)$

a$.$ Write down a vector that is parallel to the optimal vector w$.$ Hint: recall from Figure $17\mathrm{.}12($a$)$ that w is perpendicular to the decision boundary between the two points in the $3$d feature space.

b$.$ What is the value of the margin that is achieved by this w$?$ Hint: recall that the margin is the distance from each support vector to the decision boundary. Hint $2$: think about the geometry of $2$ points in space, with a line separating one from the other.

c$.$ Solve for w$,$ using the fact that the margin is equal to $1/||$w$||.$

d$.$ Solve for w$0$ using your value for w and Equations $17\mathrm{.}119$ to $17\mathrm{.}121.$ Hint: the points will be on the

decision boundary, so the inequalities will be tight.

e$.$ Write down the form of the discriminant function f $($x$)=$ w$0+$ wT $\backslash$phi $($x$)$ as an explicit function of x$.$