Point-hyperplane distance
May 29, 2023
Here’s a super simple proof I did that I thought was quite neat during the course of CS 289A: Introduction to Machine Learning at Berkeley!
Theorem. The distance between a point $x \in \mathbb{R}^n$ and a hyperplane $H = \{v \in \mathbb{R}^n \mid w \cdot v + \alpha = 0 \}$ for $w \in \mathbb{R}^n$ and $\alpha \in \mathbb{R}$ is $\frac{|w\cdot x + \alpha|}{||w||}$.
Proof. Suppose $\lambda\in \mathbb{R}$ such that $x - \lambda w \in H$. Then, as $w$ is normal to $H$, the distance between $x$ and $H$ is $||\lambda w||$. Moreover, as $x - \lambda w \in H$, we have
$$ 0 = w\cdot (x - \lambda w )+ \alpha = w \cdot x - \lambda ||w||^2 + \alpha, $$
so $\lambda = \frac{w\cdot x + \alpha}{||w||^2}$. □