Euclidean distance between two n-dimensional vectors

Question

What's an easy way to find the Euclidean distance between two n-dimensional vectors in Julia?

Zina

Answer 1

Here is a simple way

n = 10
x = rand(n)
y = rand(n)
d = norm(x-y)  # The euclidean (L2) distance

For Manhattan/taxicab/L1 distance, use norm(x-y,1)

Timbuktu

In Julia 1.0 you have to call using LinearAlgebra first. – Octangular

Answer 2

This is easily done thanks to the lovely Distances package:

Pkg.add("Distances") #if you don't have it
using Distances
one7d = rand(7)
two7d = rand(7)
dist = euclidean(one7d,two7d)

Also if you have say 2 matrices of 9d col vectors, you can get the distances between each corresponding pair using colwise:

thousand9d1 = rand(9,1000)
thousand9d2 = rand(9,1000)
dists = colwise(Euclidean(), thousand9d1, thousand9d2)
#returns: 1000-element Array{Float64,1}

You can also compare to a single vector e.g. the origin (if you want the magnitude of each column vector)

origin9 = zeros(9)
mags = colwise(Euclidean(), thousand9ds1, origin9)
#returns: 1000-element Array{Float64,1}

Other distances are also available:

• Squared Euclidean
• Cityblock
• Chebyshev
• Minkowski
• Hamming
• Cosine
• Correlation
• Chi-square
• Kullback-Leibler divergence
• Jensen-Shannon divergence
• Mahalanobis
• Squared Mahalanobis
• Bhattacharyya
• Hellinger

More details at the package's github page here.

Zina

+1 Thanks for bringing up the Distances package. I think OP's question is sufficiently answered by norm(), but it is useful to know of the package. – Heyward

This also works for dimensions above 2 (norm only supports 1- or 2-dimensional arrays). E.g., norm(rand((4,4,4)) - rand((4,4,4))) will fail – Foregather

Answer 3

To compute the norm without any imports, simply

norm(x) = sqrt(sum(x.^2))

More generic for other L1, L2, L3, ... norms

norm(x; L=2) = sum(abs.(x).^L) ^ (1/L)

For Euclidean distance between two n-dimensional vectors just call norm(x-y).

Stillness

sqrt(sum(x.^2)) creates an unnecessary intermediate array. Better to write sqrt(sum(abs2, x)) – Floccus