$3.$ Consider the ridge regression problem:

$\backslash$beta $$

ridge $=$ argmin

$\backslash$beta

$($

X

N

i$=1$

$$

yi $\backslash$beta $0$

X

p

j$=1$

xi j$\backslash$beta j

$!2$

$+\backslash$lambda

X

p

j$=1$

$\backslash$beta

$2$

j

$)$

Show that this problem is equivalent to the problem

$\backslash$beta $$

c $=$ argmin

$\backslash$beta $\backslash$alpha

$($

X

N

i$=1$

$"$

yi $\backslash$beta

c

$0$

X

p

j$=1$

$$

xi j $$ x$$ j

$$

$\backslash$beta

c

j

#$2$

$+\backslash$lambda

X

p

j$=1$

$\backslash$beta

c$2$

j

$)$

$.$

Winter $2024$ p$\mathrm{.}1/3$

STAT $541$ Assignment $1$ Due Date: Jan $26,2024$ by $18$:$00$

Give the correspondence between $\backslash$beta

c and the original $\backslash$beta in ridge regression. Characterize

the solution to this modified criterion. Show that a similar result holds for the lasso. $(15$

marks$)$