Distant Functions

A distant function works on two elements (i.e. vectors, sets…).

A given distance function $d(s,t)$ needs to fulfill properties:

1. $d(s,t) \geq 0$ (non-negative)
2. $d(s,s) = 0$ (distance to itself is zero)
3. $d(s,t) = d(t,s)$ (symmetric)
4. $d(s,t) + d(t,r) \geq d(s,r)$ (triangle inequality)

Metric

A metric requires an additional property in addition to the four properties of a distant function:

• $d(x, y) = 0$ if and only if $x = y$ (coincidence axiom)

Two given elements with no distance to each other must be the same element

Jaccard Similarity

$Sim(A,B) = \frac{|A \cap B|}{|A \cup B|}$

Jaccard Distance

$d(A,B) = 1 - \frac{|A \cap B|}{|A \cup B|}$