Let $x$ denote an $nd$ matrix where rows are training points, $y$ denotes an $n1$ vector of corresponding output value, $w$ denotes a $d1$ parameter vector and $w^{***}$ denotes the optimal parameter vector. To make the analysis easier we will consider the special case where the training data is whitened $($i$.$e$.,x^{TT}x=$I $).$ For lasso regression, the optimal parameter vector is given by

$w^{***}=argmi{n}_w\frac{1}{2}||y-xw|{|}^2+||w|{|}_1$

where $>0.$ a$.$ Assume that wi$*>0,$ what is the value of wi$*$ in this case?b$.$ Assume that wi what is the value of wi$*$ in this case $?$ c$.$ From a and b what is the condition for wi$*$ to be $0?$ How can you interpret that condition?d$.$ Now consider ridge regression where the regularization term is replaced by $(1/2)\backslash$lambda $||$w$||2.$ What is the condition for wi$*=0?$ How does it differ from the condition you obtained in c$?$