Editing Triangle inequality (section)

==Triangle inequality for cosine similarity==
{{further|Cosine similarity}}
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that<ref>{{cite conference |last=Schubert |first=Erich |title=A Triangle Inequality for Cosine Similarity |conference=International Conference on Similarity Search and Applications |date=2021 |location=Dortmund |doi=10.1007/978-3-030-89657-7_3 |publisher=Springer |arxiv=2107.04071 }}</ref>

<math display="block">\operatorname{sim}(u,w) \geq \operatorname{sim}(u,v) \cdot \operatorname{sim}(v,w) - \sqrt{\left(1-\operatorname{sim}(u,v)^2\right)\cdot\left(1-\operatorname{sim}(v,w)^2\right)}</math>

and

<math display="block">\operatorname{sim}(u,w) \leq \operatorname{sim}(u,v) \cdot \operatorname{sim}(v,w) + \sqrt{\left(1-\operatorname{sim}(u,v)^2\right)\cdot\left(1-\operatorname{sim}(v,w)^2\right)}\,.</math>

With these formulas, one needs to compute a [[square root]] for each triple of vectors {{math|{''u'', ''v'', ''w''}<nowiki/>}} that is examined rather than {{math|arccos(sim(''u'',''v''))}} for each pair of vectors {{math|{''u'', ''v''}<nowiki/>}} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.