Minkowski distance

Not to be confused with the pseudo-Euclidean metric of theMinkowski space.

TheMinkowski distanceorMinkowski metricis ametricin anormed vector spacewhich can be considered as a generalization of both theEuclidean distanceand theManhattan distance.It is named after the Polish mathematicianHermann Minkowski.

Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard

Definition

The Minkowski distance of order$p$(where$p$is an integer) between two points $$X=(x_{1},x_{2},\ldots,x_{n}){\text{ and }}Y=(y_{1},y_{2},\ldots,y_{n})\in \mathbb {R} ^{n}$$ is defined as: $$D\left(X,Y\right)={\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}.$$

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

Minkowski distance is typically used with$p$being 1 or 2, which correspond to theManhattan distanceand theEuclidean distance,respectively.In the limiting case of$p$reaching infinity, we obtain theChebyshev distance: $$\lim _{p\to \infty }{{\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}}=\max _{i=1}^{n}|x_{i}-y_{i}|.$$

Similarly, for$p$reaching negative infinity, we have: $$\lim _{p\to -\infty }{{\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}}=\min _{i=1}^{n}|x_{i}-y_{i}|.$$

The Minkowski distance can also be viewed as a multiple of thepower meanof the component-wise differences between$P$and$Q.$

The following figure shows unit circles (thelevel setof the distance function where all points are at the unit distance from the center) with various values of$p$:

Applications

The Minkowski metric is very useful in the field ofmachine learningandAI.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
• $L^{p}$space– Function spaces generalizing finite-dimensional p norm spaces
• Norm (mathematics)– Length in a vector space
• $p$-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 3Dat wolfram.com
• Unit-Norm Vectors under Different p-Normsat wolfram.com
• Simple IEEE 754 implementation in C++
• NPM JavaScript Package/Module