Problem $4-$ Nonlinear SVM $[10$ pts bonus$]$

Before starting, note that sections $4,5$ are above the expected level in this course.

We now introduce the normalized linear kernel:

$K(x,y)=\frac{x^{t}y}{||x||*||y||}$

Find the feature vector mapping, i$.$e$.,$ the $$ that maps each sample to its feature space.

Given the following data, map the points to their new feature representations using the figure as the feature space. Draw the $6$new points $($with colors$)$ on the figure.

Draw the resulting margin decision boundary in the feature space. Also draw the separating line. You can use the following link.

For this section, work only with the new feature space.

Given that the separating hyperplane is defined by $l$:$-x-y+1\mathrm{.}378=0,$ so $w=(-1,-1),b=1\mathrm{.}378,$ find the alphas for each sample.

Guide: start from the rule that $w={\displaystyle \underset{i=1}{\overset{3}{}}}{y}_i{}_i{x}_i$ to get $2$ equations. Obtain the third equation by the rule ${\displaystyle \underset{i=1}{\overset{3}{}}}{y}_i{}_i=0$ and solve system of $3$ equations $($no need to show, but write the final alphas$).$

Note: most of the time, it is not that simple, but since we have only $3$ points in $2d,$ it works.

Draw the decision boundary in the original input space, resulting from the kernel. Recall that the nonlinear hyperplane given by:

${\displaystyle \underset{i=1}{\overset{3}{}}}{y}_i{}_{i}K(x,x_i)+b=0$

Where $b$ is given to you, you found the alphas earlier.

Use the following diagram:

Tip: try to write to yourselves how this equation looks like before going to desmos.