How do I convert between a measure of similarity and a measure of difference (distance)?

Question

Is there a general way to convert between a measure of similarity and a measure of distance?

Consider a similarity measure like the number of 2-grams that two strings ha

Answer 1

Yes, there is a most general way to change between similarity and distance: a strictly monotone decreasing function f(x).

That is, with f(x) you can make similarity = f(distance) or distance = f(similarity). It works in both directions. Such function works, because the relation between similarity and distance is that one decreases when the other increases.

Examples:

These are some well-known strictly monotone decreasing candidates that work for non-negative similarities or distances:

• f(x) = 1 / (a + x)
• f(x) = exp(- x^a)
• f(x) = arccot(ax)

You can choose parameter a>0 (e.g., a=1)