$1.$ Support Vector Machines: $(4$ marks$)$ In the derivation for the Support Vector Machine,

we assumed that the margin boundaries are given by w$.$x$+$b $=+1$ and w$.$x$+$b $=1.$ Show

that, if the $+1$ and $-1$ on the right$-$hand side were replaced by some arbitrary constants $+\backslash$gamma

and $\backslash$gamma where $\backslash$gamma $>0,$ the solution for the maximum margin hyperplane is unchanged. $($You

can show this for the hard$-$margin SVM without any slack variables.$)$

$2.$ Support Vector Machines: $(4$ marks$)$ Consider the half$-$margin of maximum$-$margin

SVM defined by $\backslash$rho $,$ i$.$e$.\backslash$rho $=$

$1$

$||$w$||.$ Show that $\backslash$rho is given by:

$1$

$\backslash$rho

$2$

$=$

X

N

i$=1$

$\backslash$alpha i

$1$

where $\backslash$alpha i are the Lagrange multipliers given by the SVM dual $($as on Slide $30$ of the SVM

lecture uploaded on Piazza$).($Hint: The answer involves just $3-4$ steps, if you are thinking

of something longer, re$-$think!$)$

$3.$ Kernels: $(5$ marks$)$ Let k$1$ and k$2$ be valid kernel functions. Comment about the validity

of the following kernel functions, and justify your answer with proof or counter$-$examples as

required:

$($a$)$ k$($x$,$ z$)=$ k$1($x$,$ z$)+$ k$2($x$,$ z$)$

$($b$)$ k$($x$,$ z$)=$ k$1($x$,$ z$)$k$2($x$,$ z$)$

$($c$)$ k$($x$,$ z$)=$ h$($k$1($x$,$ z$))$ where h is a polynomial function with positive co$-$efficients

$($d$)$ k$($x$,$ z$)=$ exp$($k$1($x$,$ z$))$

$($e$)$ k$($x$,$ z$)=$ exp

$$kx$$zk

$2$

$2$

$\backslash$sigma $2$