Can polynomial regression be used to handle data with more than one dimension?

Question

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Here, $x$ no longer represents the entries of a vector $x \in \mathbb{R}^n$.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.

Answer 1

Here is an example 3D surface quadratic polynomial and data.

Answer 2

I will quote wikipedia because it is really well said.

you can find this article here : https://en.wikipedia.org/wiki/Polynomial_regression

You will see here how you obtain it : http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html

and here on a more mathematic point of view: http://home.iitk.ac.in/~shalab/regression/Chapter12-Regression-PolynomialRegression.pdf