Optimal search, finding a point where the sum of distances to other points is the smallest

Question

I have k points and I would like to find a (different) point closest to them (the sum of distances between the new point and the given points is the smallest)

On the plane for eight points

How to write a program that will get me such a point for k given points in n dimensional space (eg 16 points in a 10-dimensional space)

How to write such a solver?

However, I would not like to use ready functions, although I will accept such a solution

Answer 1

The point for which the sum of the distances to other points is minimum is called the spatial median of these points, also called the geometric median . It can be derived by the Weiszfeld algorithm , which is implemented in the R package Gmedian .

Let's try with your example:

library(Gmedian)

X <- rbind(
  c(3, 4),
  c(5, 3),
  c(1, 8),
  c(4, 6),
  c(6, 6),
  c(0, 1),
  c(4, 6),
  c(2, 1)
)

W <- Weiszfeld(X)

---

> W
$median
         [,1]     [,2]
[1,] 3.472091 4.607492

$iter
[1] 76

Here is how you can get the sum of the distances:

smedian <- c(W$median)
sum(
  apply(X, 1, function(x){
    sqrt(crossprod(x-smedian))
  })
)
# 21.95253

As you can see, the spatial median is different than the one you got ( (4,4) ), and the sum of the distances is smaller than the one you got ( 22.84819 ).