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N-dimensional medians, and a hybrid between median and average (cool-rr.com)
22 points by cool-RR on Mar 10, 2010 | hide | past | favorite | 7 comments

This seems like such a good generalization that I almost can't believe it hasn't been discovered before. (Also nice to see an article where the author wants the material to be understandable instead of trying to impress us with unnecessary complexity)

It was indeed discovered before. My goal was, exactly as you said it, to make the material easily understandable.

I think this could also be used to calculate the mode?

lim n->0 argmin(x,sum(abs(x-s)^n))

Very cool. That i variable becomes how much you want outliers to matter. i could go higher than 2 and you'd really get skewed numbers.

A relevant toy I threw together once to demonstrate finding a median/mean/centroid by minimizing a cost defined by "springs": http://www.cs.toronto.edu/~murray/04medoid/

So these are the means in the L^r norms, with r in the reals rather than the integers?

That's what I got out of it too (well, substituting "natural centrality measures" for "means"), except it bothers me that for n dimensions, it looks like he uses L₂ for the vector norm.

I'd much prefer argmin 𝐱( sum 𝐬∈Sⁿ sum j≤n |𝐬ⱼ-𝐱ⱼ|^i )

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