I am sure this is trivial, but I am looking for a transformation that nonuniformly discretizes all values of a range into several bins. The bins should be variant and I'd like them to be smaller around the middle of my distribution and grows larger around both ends. Say, instead of one fixed \delta as the step size for a uniform binning as below:
I'd like to have the following binning:
Where around the middle of the distribution, I have $\delta_{min}$ and they grow gradually (linearly) to $\delta_{max}$ on both ends.
Do we have a transformation/regularization which does the following without the need to know the value of each element inside the bin and can generate all variable deltas at runtime? For instance, for the uniform binning I can write $d=\frac{maxValue- minValue}{\#ofBins}$ and I can update the values of all members of each bin (W) to the middle of each bin.