How to convert Euclidean distance to range 0 and 1 like Cosine Similarity?

Question

Want to map Euclidean distance to the range [0, 1], somewhat like the cosine similarity of vectors.

For instance

input  output
  0      1.0
  1      0.9  approximate
  2      0.8 to 0.9 somewhere
 inf     0.0

I tried the formula 1/(1+d), but that falls away from 1.0 too quickly.

Answer 1

It seems that you want the fraction's denominator to grow more slowly (the denominator is the bottom part, which you have as (d+1) so far). There are various ways to handle this. For instance, try a lower power for d, such as

1 / (1 + d**(0.25))

... or an exponential decay in the denominator, such as

1 / (1.1 ** d)

... or using a trig function to temper your mapping, such as

1 - tanh(d)

Would something in one of these families work for you?