Similarity Notes

1 Similarity Metrics Many recommendation algorithms employ some form of similarity metric in the generation of ratings predictions. Similarity metrics are often associated with some form of distance measure. Definition 1.0.1. Let δ be a function δ : R × R → R. Let x,y, z ∈ R. Then δ is a distance measure if it satisfies the following four properties. (d1) δ(x,y) ≥ 0 (no negative distances). (d2) δ(x,y) = 0 if and only if x = y (different vectors can’t be in the same position). (d3) δ(x,y) = δ(x,y) (distance is symmetric). (d4) δ(x,y) ≤ δ(z,x) + δ(z,y) (triangle inequality). 1.1 Manhattan Distance The Manhattan Distance, δM , between two vectors x,y ∈ R is the sum of the magnitudes of the differences in each dimension, i.e.,