Minkowski distance

The Minkowski distance or Minkowski metric is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance. It is named after the Polish mathematician Hermann Minkowski.

Definition

The Minkowski distance of order ${\displaystyle p}$ (where ${\displaystyle p}$ is an integer) between two points

$${\displaystyle X=(x_{1},x_{2},\ldots ,x_{n}){\text{ and }}Y=(y_{1},y_{2},\ldots ,y_{n})\in \mathbb {R} ^{n}}$$

$${\displaystyle D\left(X,Y\right)={\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}.}$$

For ${\displaystyle p\geq 1,}$ the Minkowski distance is a metric as a result of the Minkowski inequality. When ${\displaystyle p<1,}$ the distance between ${\displaystyle (0,0)}$ and ${\displaystyle (1,1)}$ is ${\displaystyle 2^{1/p}>2,}$ but the point ${\displaystyle (0,1)}$ is at a distance ${\displaystyle 1}$ from both of these points. Since this violates the triangle inequality, for ${\displaystyle p<1}$ it is not a metric. However, a metric can be obtained for these values by simply removing the exponent of ${\displaystyle 1/p.}$ The resulting metric is also an F-norm.

Minkowski distance is typically used with ${\displaystyle p}$ being 1 or 2, which correspond to the Manhattan distance and the Euclidean distance, respectively. In the limiting case of ${\displaystyle p}$ reaching infinity, we obtain the Chebyshev distance:

$${\displaystyle \lim _{p\to \infty }{{\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}}=\max _{i=1}^{n}|x_{i}-y_{i}|.}$$

Similarly, for ${\displaystyle p}$ reaching negative infinity, we have:

$${\displaystyle \lim _{p\to -\infty }{{\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}}=\min _{i=1}^{n}|x_{i}-y_{i}|.}$$

The Minkowski distance can also be viewed as a multiple of the power mean of the component-wise differences between ${\displaystyle P}$ and ${\displaystyle Q.}$

The following figure shows unit circles (the level set of the distance function where all points are at the unit distance from the center) with various values of ${\displaystyle p}$:

Applications

The Minkowski metric is very useful in the field of machine learning and AI. Many popular machine learning algorithms use specific distance metrics such as the aforementioned to compare the similarity of two data points. Depending on the nature of the data being analyzed, various metrics can be used. The Minkowski metric is most useful for numerical datasets where you want to determine the similarity of size between multiple datapoint vectors.

See also

• Generalized mean – N-th root of the arithmetic mean of the given numbers raised to the power n
• ${\displaystyle L^{p}}$ space – Function spaces generalizing finite-dimensional p norm spaces
• Norm (mathematics) – Length in a vector space
• ${\displaystyle p}$-norm – Function spaces generalizing finite-dimensional p norm spacesPages displaying short descriptions of redirect targets

External links

• Unit Balls for Different p-Norms in 2D and 3D at wolfram.com
• Unit-Norm Vectors under Different p-Norms at wolfram.com
• Simple IEEE 754 implementation in C++
• NPM JavaScript Package/Module

• v
• t
• e

Lp spaces

Basic concepts

• Banach & Hilbert spaces
• L^p spaces
• Measure
  • Lebesgue
• Measure space
• Measurable space/function
• Minkowski distance
• Sequence spaces

L¹ spaces

• Integrable function
• Lebesgue integration
• Taxicab geometry

L² spaces

• Bessel's
• Cauchy–Schwarz
• Euclidean distance
• Hilbert space
• Parseval's identity
• Polarization identity
• Pythagorean theorem
• Square-integrable function

${\displaystyle L^{\infty }}$ spaces

• Bounded function
• Chebyshev distance
• Infimum and supremum
  • Essential
• Uniform norm

Maps

• Almost everywhere
• Convergence almost everywhere
• Convergence in measure
• Function space
• Integral transform
• Locally integrable function
• Measurable function
• Symmetric decreasing rearrangement

Inequalities

• Babenko–Beckner
• Chebyshev's
• Clarkson's
• Hanner's
• Hausdorff–Young
• Hölder's
• Markov's
• Minkowski
• Young's convolution

Results

• Marcinkiewicz interpolation theorem
• Plancherel theorem
• Riemann–Lebesgue
• Riesz–Fischer theorem
• Riesz–Thorin theorem

For Lebesgue measure	Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality

Applications & related

• Bochner space
• Fourier analysis
• Lorentz space
• Probability theory
• Quasinorm
• Real analysis
• Sobolev space
• *-algebra
  • C*-algebra
  • Von Neumann