If the design matrix is not the unit matrix for the generalized Lasso, then for

x$^(+)$:$=($x$^($T$)$x$)^(-1)$x$^($T$),$tilde$($y$)$:$=$xx$^(+)$y$,$tilde$($D$)$:$=$Dx$^(+),$ the problem reduces to mini$-$

mizing

$(1)/(2)||$tilde$($y$)-$tilde$($D$)\backslash$alpha $||\_(2)^(2)$

Extend the functions fused$\_($d$)$ual and fused$\_($p$)$rime to fused$\_($d$)$ual$\_()$

general and fused$\_($p$)$rime$\_($g$)$eneral$,$ and execute the following procedure.$()/()$

n $=20$

p $=10$

beta $=$ randn$($p$+1)$

X $=$ randn$($n$*$p$).$reshape$(($n$,$ p$))$

x$\_(0)=$ np$\mathrm{.}1$nsert $($x $,0,$ np$.$ones$($n$),$ ax $1$s$=1)$

y $=$ x$\_(0)$ beta $+$ randn$($n$)$

m $-$ p $-1$

D $=($np$.$eye$($p$)+$ np$.$d$1$ag$([-1]*($p$-1),$k$-1))[$:$-1]$

alpha, lambda$\_($s$)$eq $=$ rused$\_($d$)$ual$\_($g$)$eneral$($x$,$ y$,$ D$)$

lambda$\_($m$)$ax $=$ np$.$max$(1$ambda$\_($s$)$eq$)$

alpha$\_($n$)1$n $=$ np$.$min$($alpha$)$

alpha$\_($n$)$ax $=$ np$.$max$($alpha$)$

plt$.$xl$1$n$(0,$ lanbda$\_($n$)$ax$)$

plt$.$yl$1$m$($alpha$\_($m$)1$n$,$ alpha$\_($m$)$ax$)$

plt$.$xlabel$($r$"")$

plt$.$ylabel$()$

for $1$ in range$(0,$ n$)$:

plt$.$plot$($lambda$\_($s$)$eq$,$ alpha$[$:$,1])$

beta, lambda$\_($s$)$eq $=$ rused$\_($p$)$rine$\_($g$)$eneral $($x$,$ y$,$ D$)$

beta$\_($n$)1$n $=$ np$.$min$($beta$)$

beta$\_($n$)$ax $=$ np$.$max$($beta$)$

plt$.$xl$1$n$(0,1$anbda$\_($n$)$ax$)$

plt$.$yl$1$m$($beta$\_($n$)1$n$.$ beta$\_($m$)$ax$)$

plt$.$xlabel$($r$"")$

plt$.$ylabel$()$

for $1$ in range$(0,$ p$)$:

plt$.$plot$($lambda$\_($s$)$eq$,$ beta$[$:$,1])$