Minkowski Distance

machine-learning statistics distance-metric

Definition

Minkowski Distance

The Minkowski distance is a generalised metric in Euclidean space that parameterises a family of distances. For two vectors $\mathbf{x},\mathbf{v}\in {\mathbb{R}}^m$ and a parameter $p\ge 1$, the Minkowski distance $d(\mathbf{x},\mathbf{v})$ is defined as the $L_p$ norm of their difference:

$$d(\mathbf{x},\mathbf{v})=\Vert \mathbf{x}-\mathbf{v}{\Vert }_p={\left(\sum _{i=1}^m|x_i-v_i{|}^p\right)}^{1/p}$$

Special Cases

Manhattan Distance ($p=1$): Corresponds to the $L_1$ norm, measuring distance along grid axes.

Euclidean Distance ($p=2$): Corresponds to the $L_2$ norm, measuring the direct straight-line distance.

Chebyshev Distance ($p\to \infty$): Defined by the maximum coordinate difference: $d_{\infty }(\mathbf{x},\mathbf{v})={\max }_i|x_i-v_i|$.

Properties

The Minkowski distance satisfies the triangle inequality (Minkowski inequality) for all $p\ge 1$. For values $p<1$, the triangle inequality is violated, and the function does not constitute a formal metric.