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A popular distance measure, cosine similarity/distance, is not a proper metric because it fails to satisfy one of the conditions (the triangle inequality). However, there is no disadvantage whatsoever in using it and it is used heavily in many applications.

Is the categorization of a distance measure as a metric useful at all?

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  • $\begingroup$ Hi and welcome to crossvalidated. Your question seems a bit unclear, I can't understand what you are asking and what is your point of discussion. Could you explain it better? Thanks $\endgroup$ – GGA Feb 8 '17 at 10:42
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    $\begingroup$ If you ask why we usually use distance functions that are metrics, the answer is simple: a metric on a space induces many topological properties which are useful and well studied. $\endgroup$ – Łukasz Grad Feb 8 '17 at 10:56
  • $\begingroup$ Welcome! I've tried to clarify your q. a little. Perhaps it might also help to mention the kind of applications you have in mind. $\endgroup$ – Scortchi Feb 8 '17 at 11:36
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    $\begingroup$ Note that if "cosine distance" or "correlation distance" is defined as $d^2=1-s$, without ignoring the sign of cosine or correlation $s$, then $d$ is euclidean distance, therefore metric. $\endgroup$ – ttnphns Feb 8 '17 at 11:42
  • $\begingroup$ Lukasz Grad, thanks for the comments. So if I understood it correctly there is no direct downside of using non-metric measure. $\endgroup$ – mitbal Feb 10 '17 at 10:55

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