Polynomial regression is said to be a subclass of linear regression. However, it seems that only linear regression can handle data input with more than one dimension, whereas polynomial regression cannot. Is my understanding correct?
For example,
Given a set of $N$ data pairs $\{(x_k,t_k)\}_{k = 1}^N, x_k \in \mathbb{R}^n, t_k\in \mathbb{R}$
A linear regression uses the model
$y(x,t) = \sum\limits_{i = 1}^n w_i x_i$ where $w_i$ are the coefficients of my model
Then the empirical loss is written as
$E = \|Xw - t\|$^2
where $X = \begin{bmatrix} 1 & x_1^T \\ \vdots & \vdots \\ 1 & x_N^T \end{bmatrix}$, $w = \begin{bmatrix} 1 & w_1 & \ldots & w_n\end{bmatrix}^T$
In the Polynomial regression case, my model is,
$y(x,t) =\sum\limits_{i = 0}^n w_ix^i = w_0 + w_1x + w_2x^2 + \ldots w_nx^n$
Here, $x$ no longer represents the entries of a vector $x \in \mathbb{R}^n$. But just a single point $x \in \mathbb{R}$.
I think the issue here is that powers are not defined for non-1D vectors.