MLG 032 Cartesian Similarity Metrics

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Show notes at ocdevel.com/mlg/32.

L1/L2 norm, Manhattan, Euclidean, cosine distances, dot product

Normed distances link

• A norm is a function that assigns a strictly positive length to each vector in a vector space. link
• Minkowski is generalized. p_root(sum(xi-yi)^p). "p" = ? (1, 2, ..) for below.
• L1: Manhattan/city-block/taxicab. abs(x2-x1)+abs(y2-y1). Grid-like distance (triangle legs). Preferred for high-dim space.
• L2: Euclidean. sqrt((x2-x1)^2+(y2-y1)^2. sqrt(dot-product). Straight-line distance; min distance (Pythagorean triangle edge)
• Others: Mahalanobis, Chebyshev (p=inf), etc

Dot product

• A type of inner product. Outer-product: lies outside the involved planes. Inner-product: dot product lies inside the planes/axes involved link. Dot product: inner product on a finite dimensional Euclidean space link

Cosine (normalized dot)