Date: Feb 13, 2013 7:25 AM
Author: quasi
Subject: Re: Question: Centroid given a distance metric

Andrey Savov wrote:
>Ray Vickson wrote:
>>I would ask: why do you want to minimize the sum of squares?
>>For Euclidean distance, that F(x) has some physical and
>>statistical meaning, and furthermore leads to a simple
>>solution. However, for other norms such as d(x,y) = |x|+|y|
>>or d(x,y) = max(|x|,|y|), or for a p-norm with 1 < p < 2,
>>what significance can one attach to the sum of squares?
>>Certainly it makes _some_ problems much harder instead of
>>easier (for example, when d(x,y) = |x| + |y|).

>The norms I had in mind were actually much nicer than the
>ones you mention.

Actually, _you_ were the one who mentioned the taxicab norm
when I asked for a concrete example of a norm other than the
standard Euclidean one to be used as a test case to discuss
the questions you raised.

>They were continuous and even convex functions on a subset
>of R^n, so for them that point has meaning similar to the
>Euclidean norm.

Let's see an example.

>I over-generalized when I posted the question.


So why not try to fix it?

Clarify your assumptions and state what you think is true
based on those assumptions.