Minkowski distance

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

$$X\; =\; (x\_1,x\_2,\backslash ldots,x\_n)\; \backslash text\{\; and\; \}\; Y\; =\; (y\_1,y\_2,\backslash ldots,y\_n)\; \backslash in\; \backslash R^n$$

is defined as:

$$D\backslash left(X,Y\backslash right)\; =\; \backslash biggl(\backslash sum\_\{i=1\}^n\; |x\_i-y\_i|^p\backslash biggr)^\{\backslash frac\{1\}\{p\}\}.$$

For $p\; \backslash geq\; 1,$ the Minkowski distance is a metric as a result of the Minkowski inequality.{{citation

| last = Şuhubi | first = Erdoğan S.

| contribution = Chapter V: Metric Spaces

| doi = 10.1007/978-94-017-0141-9_5

| isbn = 9789401701419

| pages = 261–356

| publisher = Springer Netherlands

| title = Functional Analysis

| year = 2003}} When 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 the triangle inequality, for 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 an F-norm.

Minkowski distance is typically used with $p$ being 1 or 2, which correspond to the Manhattan distance and the Euclidean distance, respectively.{{citation

| last1 = Zezula | first1 = Pavel

| last2 = Amato | first2 = Giuseppe

| last3 = Dohnal | first3 = Vlastislav

| last4 = Batko | first4 = Michal

| contribution = Chapter 1, Foundations of Metric Space Searching, Section 3.1, Minkowski Distances

| doi = 10.1007/0-387-29151-2

| isbn = 9780387291512

| page = 10

| publisher = Springer

| series = Advances in Database Systems

| title = Similarity Search: The Metric Space Approach

| year = 2006}} In the limiting case of $p$ reaching infinity, we obtain the Chebyshev distance:

$$\backslash lim\_\{p\; \backslash to\; \backslash infty\}\{\backslash biggl(\backslash sum\_\{i=1\}^n\; |x\_i-y\_i|^p\backslash biggr)^\backslash frac\{1\}\{p\}\}\; =\; \backslash max\_\{i=1\}^n\; |x\_i-y\_i|.$$

Similarly, for $p$ reaching negative infinity, we have:

$$\backslash lim\_\{p\; \backslash to\; -\backslash infty\}\{\backslash biggl(\backslash sum\_\{i=1\}^n\; |x\_i-y\_i|^p\backslash biggr)^\backslash frac\{1\}\{p\}\}\; =\; \backslash min\_\{i=1\}^n\; |x\_i-y\_i|.$$

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

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 $p$:

{{panorama|image=File:2D unit balls.svg|height=200|alt=Unit circles using different Minkowski distance metrics.}}

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 one wants to determine the similarity of size between multiple datapoint vectors.

• {{annotated link|Generalized mean}}
• {{annotated link|Lp space|$L^p$ space}}
• {{annotated link|Norm (mathematics)}}
• {{annotated link|p-norm|$p$-norm}}

{{reflist}}

• [https://demonstrations.wolfram.com/UnitBallsForDifferentPNormsIn2DAnd3D/ Unit Balls for Different p-Norms in 2D and 3D] at wolfram.com
• [https://demonstrations.wolfram.com/UnitNormVectorsUnderDifferentPNorms/ Unit-Norm Vectors under Different p-Norms] at wolfram.com
• [https://gist.github.com/pallas/5565528 Simple IEEE 754 implementation in C++]
• [https://www.npmjs.com/package/compute-minkowski-distance NPM JavaScript Package/Module]

{{Lp spaces}}

Category:Distance

Category:Hermann Minkowski

Category:Metric geometry

Category:Normed spaces