https://piazza.com/class_profile/get_resource/ln18bjs43q41tr/ln18n411vo070m
Motivation: Basis Expansion
$$ \text{boundary}\\ x_1 -x_2^2 -5 = 0\\
w * \phi(x) + b = 0 \\
w = [1, 0, 0, -1, 0] \\ b = -5 $$
$$ \text{Quadratic in x = Linear in } \phi(x) $$
Suppose
$$ x = (x_1, x_2, x_3) \\ $$
$$ \phi(x) = (x_1,x_2,x_3, \\ x_1^2, x_2^2, x_3^2,\\ x_1x_2, x_1x_3, x_2x_3) $$
Then our enhanced function is 9-d
Suppose
$$ x = (x_1 ...x_d) \\
\phi(x) = (x_1,...,x_d, \\ x_1^2,...,x_d^2,\\ x_1x_2,..., x_{d-1}x_d) $$
Then any quadratic function is just a linear function in the expanded basis