Using the kernel trick introduced in L$5$ to extend the regu$-$

larized linear regression model $($L$3)$ to solve nonlinear regression problems. Derive a

closed$-$form solution $($i$.$e$.,$ to derive a kernelized version of the closed$-$form solution on

Given image$).$Closed$-$form Solution $($cont$.)$

$({\displaystyle \underset{i=1}{\overset{N}{}}}(x_i{x}_i^T))w-{\displaystyle \underset{i=1}{\overset{N}{}}}{y}_i{x}_i+w=0$

$x{x}^{T}w-xy+Iw=0$

$(x{x}^T+I)w-xy=0$

Always invertible $as$

long $asis$ positive

$(x{x}^T+I)w=$xylongrightarroww$=(x{x}^T+I{)}^{-1}xy$