The exclusive$-$OR is the simplest problem that cannot be solved using a linear discriminant operating directly on the features. The points $k=1,3$ at $x=(1,1{)}^t$ and $(-1,-1{)}^t$ are in category ${}_1($red in the figure$),$ while $k=2,4$ at $x=(1,-1{)}^t$ and $(-1,1{)}^t$ are in ${}_2($black in the figure$).$ Following the approach of Support Vector Machines, we preprocess the features to map them to a higher dimension space where they can be linearly separated. While many $-$functions could be used, here we use the simplest expansion up to second order: $1,\sqrt[\phantom{2}]{2}{x}_1,\sqrt[\phantom{2}]{2}{x}_2,\sqrt[\phantom{2}]{2}{x}_1{x}_2,x_1^2$ and $x_2^2,$ where the $\sqrt[\phantom{2}]{2}$ is convenient for normalization.

Using the support vector machine formulation, show that the optimum discriminant is

$g(x)=g(x_1,x_2)=x_1**x_2$