Please do not use chargpt$/$AI$.$ Thanks.

Consider a binary classification problem with feature space $x.$ Let $$ be a feature mapping of $x$ into some Hilbert feature space $V,$ and $K$:$xxR$ be a kernel function that implements inner products in the feature space $V$ so that $K(x,x^{\text{'}})=($:$(x),(x^{\text{'}})$:$)$ for all $x,x^{\text{'}}$inx.

A classification algorithm $h$ that predicts the label of an unseen instance according to the class with the closest average. Formally, given $n$ training data $D=\{(x_1,y_1),$dots,$(x_n,y_n)\}$ with

$y_{i}in\{+-1\}.$ For each class yin$\{+-1\},$ we define the number of points as $m_y=|\{i$:$y_i=y\}|$ and the average feature as

$c_y=\frac{1}{m_y}{\displaystyle \underset{i\text{:}{y}_i=y}{\overset{m}{}}}(x_i)$

$($We assume that $m_{+}$and $m_{-}$are nonzero.$)$ Then, the algorithm outputs the following decision rule:

$($a$)$ Let $w=c_{+}-c_{-}$and let $b=\frac{1}{2}(||c_{-}|{|}^2-||c_{+}|{|}^2).$ Show that $h(x)=$sign$(($:$w,(x)$:$)+b).$

$($b$)$ However, if we cannot access $($and thus $c_y,w$ and $b),$ show how to express $h(x)$ in terms of the kernel function $K.$