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.

The Minkowski distance of order

is an integer) between two points

is defined as:

the Minkowski distance is a metric as a result of the Minkowski inequality.

the distance between

but the point

is at a distance

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

The resulting metric is also an F-norm.

Minkowski distance is typically used with

being 1 or 2, which correspond to the Manhattan distance and the Euclidean distance, respectively.

[2] In the limiting case of

reaching infinity, we obtain the Chebyshev distance:

lim

max

Similarly, for

reaching negative infinity, we have:

lim

min

The Minkowski distance can also be viewed as a multiple of the power mean of the component-wise differences between

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

: 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.