In "Efficient Estimation of Word Representations in Vector Space" Mikolov et.al argue that any mapping of words into vectors should satisfy approximate constraints, such as

$vector(''Paris'') - vector(''France'') + vector(''Poland'') \approx vector(''Warsaw'')$

Then, on page 5 they choose the Cosine distance to measure the vector proximity. Unfortunately, cosine distance is not a metric. It hints however that perhaps the problem setup has to be transferred into Euclidean Projective space. Unfortunately, there is no identity element in projective space. Why the authors didn't use a conventional norm, such as $l^2$?



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