Manhattan Distance

geometry

Definition

Manhattan Distance

The Manhattan distance (also known as city block distance or $L_1$ distance) is a metric defined as the sum of the absolute differences of the coordinates of two points. Formally, for two vectors $\mathbf{x},\mathbf{v}\in {\mathbb{R}}^m$, the Manhattan distance $d(\mathbf{x},\mathbf{v})$ is:

$$d(\mathbf{x},\mathbf{v})=\Vert \mathbf{x}-\mathbf{v}{\Vert }_1=\sum _{i=1}^m|x_i-v_i|$$

Properties

Metric Axioms: The Manhattan distance satisfies all formal requirements of a metric, including the triangle inequality.

Taxicab Geometry: The distance corresponds to the length of a path between two points in a grid-based system where movement is restricted to axes-parallel steps.

Regularisation: In machine learning, the $L_1$ norm is frequently utilised in regularisation (e.g., Lasso) to promote sparsity in model parameters.