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How to prove that the cosine distance ($1-\text{cosine similarity}$) does not satisfy the four properties (Non-Negativity, Coincidence Axiom, Symmetry, Triangle Inequality) of a metric over Euclidean space?
If some properties are satisfied then how can I show it is satisfied?
We have $$D(x,y) = 1- S(x,y)$$ where $$S_c(x,y) = \frac{x \cdot y}{||x|| \, ||y||}$$ First of all, $S(\cdot, \cdot)$ is not defined when either of its inputs is $0$, so $D$ cannot be a metric on $\mathbb{R}^n$. A better question to ask is whether or not $D$ is a metric on $\mathbb{R}^n \backslash \{0\}$. The answer is still no: $$D((1,1), (2,2)) = 1 - \frac{(1,1)\cdot(2,2)}{||(1,1)|| \, ||(2,2)||}$$ $$=1- \frac{4}{\sqrt{2} \cdot 2\sqrt{2}} = 0$$ actually, the coincidence axiom is violated for any pair of nonequal vectors which are positive scalar multiples of each other. Why? Think of the definition of cosine and angles between vectors in $\mathbb{R}^n$!
