Good similarity measure based on euclidean distance?

Question

I am calculating the Euclidean distances of a few pairs feature vectors $p, q$. Usually, I get distances in the range $[0, 2]$ however I've had some occasions (outliers) when the distance was more than $2$. I want to find a similarity metric between that depends on the distance between the two vectors and differentiates the distances considerably ranging in this range. For instance, something like

$$\frac{1}{1+dist(p, q)}$$

or

$$\frac{1}{e^{dist(p,q)}}$$

Although it doesn't give that much 'spread' I would hope for. How can I generate a function that gives the most spread for $[0,2]$ but at the same time maintains that for any distance, it is between $0$ and $1$?

Edit:

An example of such similarity would be, for a given small $\epsilon >0$, the similarity score is $\in [\epsilon, 1]$ when $dist(p,q)\in [0,2]$ and $ \in [0, \epsilon]$ when $dist(p, q)\in [2, +\infty]$.

Answer 1

You can use the Gaussian function: $e^{-||p - q||^2 / 2\sigma^2}$. Whenever the distance (in the exponent) is 0, the result is 1. Larger distances result in smaller results, and the decreasing speed is controlled by the variance $\sigma^2$. Just set it according to your problem to get a reasonable range of values and achieve your desired spread.