On Feb 11, 9:34 am, Andrey Savov <andrey.sa...@gmail.com> wrote: > On Monday, February 11, 2013 9:17:56 AM UTC-8, Andrey Savov wrote: >> Was wondering if you guys can point me in the right direction. >> >> Are there any known/studied methods to calculate a centroid >> (geometric center) of finite set of points in n-dimensional >> real Euclidean space by only knowing a distance metric f(x,y): >> R^n x R^n -> R ? > > Should have been a bit more clear. I am looking to find the > point in R^ that minimizes the square of the given metric between > itself and the given points. For Euclidean distance metric that > point would be sum(x_i)/k.
No, that's for *squared* Euclidean distance. In one dimension, minimizing the sum of the Euclidean distances gets you the median. However, in more than one dimension, there are also other definitions of the median. Google multidimensional median and multivariate median. In addition to what you find there, see
H.Oja (1983), Descriptive Statistics for Multivariate Distributions, Statistics and Probability Letters, 1, 327-332.
H.Oja & J.Niinimaa (1985), Asymptotic Properties of the Generalized Median in the Case of Multivariate Normality, Journal of the Royal Statistical Society, Series B, 47, 372-377.