Cosine Similarity

Definition

Cosine Similarity

Let $x,y\in {\mathbb{R}}^d$ be non-zero vectors. Their cosine similarity is the inner product normalised by the product of their norms:

$${\mathrm{sim}}_{\cos }(x,y)=\frac{⟨x,y⟩}{\Vert x\Vert \Vert y\Vert }$$

If $\theta$ is the angle between $x$ and $y$, then

$${\mathrm{sim}}_{\cos }(x,y)=\cos \theta \in [-1,1].$$

Thus $1$ means the vectors point in the same direction, $0$ means they are orthogonal, and $-1$ means they point in opposite directions. Cosine similarity compares direction while ignoring scale.