Metric

math analysis geometry

Definition

Metric

A metric (or distance function) on a set $X$ is a function $d:X\times X\to \mathbb{R}$ that satisfies the following axioms for all $x,y,z\in X$:

1. Non-negativity: $d(x,y)\ge 0$
2. Identity of Indiscernibles: $d(x,y)=0⟺x=y$
3. Symmetry: $d(x,y)=d(y,x)$
4. Triangle Inequality: $d(x,z)\le d(x,y)+d(y,z)$

A set $X$ equipped with a metric $d$ is called a metric space.

Applications in Machine Learning

The choice of metric determines the “closeness” of instances in the instance space. It is the fundamental component of distance-based learning algorithms (e.g., kNN, k-Means), where predictions are derived from the geometric proximity of data points.