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On Wikipedia there's a statement:

When a measure such as a Euclidean distance is defined using many coordinates, there is little difference in the distances between different pairs of samples.

Is there some formal theorem that pins down what this means?

Yandle
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    The proof is outlined in the "Distance function" section of the Wikipedia article. – jbowman Jan 16 '23 at 23:45
  • @jbowman The quoted statement is from the beginning of the "Distance function" section. I have read that section but I don't see any theorem that pins down that statement (i.e. something like with high probability, the distance between all pairs of sample is less than some amount). – Yandle Jan 16 '23 at 23:57
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    The part that begins "A further development of this phenomenon...," but you're right, it's just a description of the result. The proof is in https://minds.wisconsin.edu/handle/1793/60174. – jbowman Jan 17 '23 at 01:57
  • There are lots of theorems. See, for instance, https://stats.stackexchange.com/questions/233931, https://stats.stackexchange.com/questions/451027, etc. – whuber Jan 17 '23 at 18:01

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