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Topic: Question: Centroid given a distance metric
Replies: 14   Last Post: Feb 12, 2013 1:04 PM

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quasi

Posts: 10,396
Registered: 7/15/05
Re: Question: Centroid given a distance metric
Posted: Feb 12, 2013 3:49 AM
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Andrey Savov wrote
>
>In other words, all I can do is, given 2 points, tell the
>distance between them. I'm looking for known algorithms to
>locate a point that minimizes all distances between itself
>and the original points.


Minimizes all distances? That makes no sense.

What you previously said and what I think you meant to say
above is that you want to find a point which minimizes the
sum of the squares of the distances from the point to the
points of the given set.

For a general distance function, the sum of squares of the
distances may have no geometric significance whatsoever, so
calling a point which minimizes that sum "a centroid" is, in
my opinion, a poor choice of terminology.

>Pretty sure I can prove existence and uniqueness of these
>assuming only metric definition:
>

<http://en.wikipedia.org/wiki/Metric_(mathematics)#Definition>

Can you prove that a metric, regarded as a function from
R^n x R^n -> R^n, is continuous? If not, existence of the point
you seek may not be guaranteed.

And even if you assume the metric is continuous, uniqueness
may fail. For example, with the discrete metric where
d(p,q) = 1 for all distinct points p,q, any point of the
original set qualifies as a point which minimizes the sum of
the squares of the distances to the set.

quasi



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