In space $ \mathcal{X} = \mathbb{R^2} $, to get all possible quadratic curves in $ \mathcal{X} $, we need feature transform $\mathbf{z} = \Phi_2(\mathbf{x})$, where $\mathbf{x} \in \mathbb{R^2}$, and $\mathbf{z}\in \mathcal{Z}$.
This transform,
$ \Phi_2(\mathbf{x}) = (1, x_1, x_2, x_1^2, x_1x_2, x_2^2) $
gives us the flexibility to represent any quadratic curve in $\mathcal{X}$ by a hyperplane in $\mathcal{Z}$. (The subscript $2$ of $\Phi$ is for polynomials of degree $2$.)
$\Phi_Q$ is called Qth order polynomial transform.
Question:
Consider the Qth order polynomial transform, $\Phi_Q$, for $ \mathcal{X} = \mathbb{R^d} $. What is the dimensionality $\tilde{d}$ of the feature space $\mathcal{Z}$ ?
I have seen this exercise in a Machine Learning related textbook which I am self studying. I am not sure which field of math it relates to.
Anyways, can anyone provide me a generic formula and also its derivation, that is valid for any $d$ and $Q$.
For $d=2$, the dimensionality, $\tilde{d}$, is given as $Q(Q+3)/2$