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Feature Maps
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Polynomial Fit Models
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Polynomial Regressors


A simple example of an embedding.1


Fix $d \in \N $. A polynomial feature map of degree $d$ is a function $\phi : \R \to \R ^d$ with

\[ \phi (x) = \pmat{1 & x^2 & \cdots & x^d}^\top . \]

For $x \in \R $, we call $\phi (x)$ the polynomial embedding of $x$.

A polynomial regressor is a least squares linear predictor using a polynomial feature embedding (of any degree, but to be precise one must specify the degree). The task of consructing a linear predictor is often referred to as polynomial regression.

Given a dataset of paired records $(x^1, y^1), \dots , (x^n, y^n) \in \R ^2$, one can construct a predictor $g: \R \to \R $ for $y$ by embedding the dataset $(\phi (x^1), \dots , \phi (x^n))$ and finding the least squares linear regressor $f: \R ^d \to \R $ for $y$. One defines the predictor $g: \R \to \R $ by $g(\phi (x))$.

  1. Future editions will expand, or perhaps collapse this sheet. ↩︎
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