Cosine Similarity

• Lp Regularization l2norm aka p = 2
• $S_c(A,B):=cos(\theta )=\frac{A\cdot B}{||A||||B||}=\frac{{\Sigma }_{i=1}^n{A}_i{B}_i}{\sqrt{{\Sigma }_{i=1}^n{A}_i^2}\sqrt{{\Sigma }_{i=1}^n{B}_i^2}}$
• ranges from -1 : exactly opposite, 1 : exactly same, 0: orthogonal/not correlated, intermediate
• [Cosine Distance](Cosine Distance.md)
• Cosine similarity is $-\mathrm{sum}\left(l2norm\left(y\right)\cdot l2norm\left(\hat{y}\right)\right)$
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• magnitude of vectors is not taken into account, merely their direction
• In practice, this means that the differences in values are not fully taken into account
• If you take a Recommender System, for example, then the cosine similarity does not take into account the difference in rating scale between different users
• high-dimensional data and when the magnitude of the vectors is not of importance