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Question $1.$ Kernel Power Consider the following $2-$dimensional data$-$set, where

y

denotes the class of each point. Throughout this question, you may use any desired packages to answer the questions. $($a$)$ Use the transformation

x$=($x

$1$

$$

$,$x

$2$

$$

$)($

$1$

$$

$($x$),$

$2$

$$

$($x$))$

where

$$

$1$

$$

$($x$)=2$x

$2$

$2$

$$

$4$x

$1$

$$

$+1$

and

$$

$2$

$$

$($x$)=$

x

$1$

$2$

$$

$2$x

$2$

$$

$3$

$.$ What is the equation of the best separating hyper$-$plane in the new feature space? Provide a plot with the data set and hyperplane clearly shown. What to submit: a single plot, the equation of the separating hyperplane, a screen shot of your code, a copy of your code in your $.$py file for this question. $($b$)$ Fit a hard margin linear SVM to the transformed data$-$set in the previous part

$1$

$.$ What are the estimated values of

$(\backslash$alpha

$1$

$$

$,...,\backslash$alpha

$7$

$$

$)$

$.$ Based on this, which points are the support vectors? What error does your computed SVM achieve? What to submit: the indices of your identified support vectors, the train error of your SVM$,$ the computed

$\backslash$alpha

$\text{'}$s $($rounded to $3$ d$.$p$.),$ a screen shot of your code, a copy of your code in your $.$py file for this question. $($c$)$ Consider now the kernel

k$($x$,$z$)=(2+$x

$$

z$)$

$2$

$.$ Run a hard$-$margin kernel SVM on the original $($un$-$transformed$)$ data given in the table at the start of the question. What are the estimated values of

$(\backslash$alpha

$1$

$$

$,...,\backslash$alpha

$7$

$$

$)$

$.$ Based on this, which points are the support vectors? What error does your computed SVM achieve? What to submit: the indices of your identified support vectors, the train error of your SVM$,$ the computed

$\backslash$alpha

$\text{'}$s $($rounded to $3$ d$.$p$.),$ a screen shot of your code, a copy of your code in your $.$py file for this question. $($d$)$ Provide a detailed argument explaining your results in parts $($i$),($ii$)$ and $($iii$).$ Your argument should explain the similarities and differences in the answers found. In particular, is your answer in $($iii$)$ worse than in $($ii$)?$ Why? To get full marks, be as detailed as possible, and use mathematical arguments or extra plots if necessary. What to submit: some commentary and$/$or plots. If you use any code here, provide a screen shot of your code, and a copy of your code in your $.$py file for this question.

$$