Minkowski Distances - Yousef's Notes

Given two points $x = (x_1, x_2, \ldots, x_n)$ and $y = (y_1, y_2, \ldots, y_n)$ in $\mathbb{R}^n$, the Minkowski distance of order $p$ is defined as: $$ d_p(x, y) = \left( \sum_{i=1}^{n} |x_i - y_i|^p \right)^{\frac{1}{p}} $$

#Special Cases

Euclidean distance ($p = 2$): $d_2(x, y) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}$
Manhattan distance ($p = 1$): $d_1(x, y) = \sum_{i=1}^{n} |x_i - y_i|$
Chebyshev distance ($p = \infty$): $d_\infty(x, y) = \max_{i=1}^{n} |x_i - y_i|$

#Properties

The Minkowski distance is a metric, meaning it satisfies the properties of non-negativity, symmetry, and triangle inequality.
The Minkowski distance is sensitive to the choice of $p$, with smaller values of $p$ giving more weight to smaller differences and larger values of $p$ giving more weight to larger differences.

#Applications

Minkowski distances are used in various fields, including:

Data analysis and machine learning
Image and signal processing
Clustering and classification algorithms
Optimization problems