It is very common tu use softmax function for converting an array of values in an array of probabilities. In general, the function amplifies the probability of the greater values of the array.
However, this function is not scale invariant. Let us consider an example:
If we take an input of [1, 2, 3, 4, 1, 2, 3], the softmax of that is [0.024, 0.064, 0.175, 0.475, 0.024, 0.064, 0.175]. The output has most of its weight where the '4' was in the original input. That is, softmax highlights the largest values and suppress values which are significantly below the maximum value. However, if the input were [0.1, 0.2, 0.3, 0.4, 0.1, 0.2, 0.3] (which sums to 1.6) the softmax would be [0.125, 0.138, 0.153, 0.169, 0.125, 0.138, 0.153]. This shows that for values between 0 and 1 softmax, in fact, de-emphasizes the maximum value (note that 0.169 is not only less than 0.475, it is also less than the initial proportion of 0.4/1.6=0.25).
I would need a function that amplifies differences between values in an array, emphasizing the greatest values and that is not so affected by the scale of the numbers in the array.
Can you suggest some function with these properties?