I don't know if this is a well-studied problem (I hope it is!):
Suppose I have some points {$X_i\}$ that are generated from a probability distribution on $\mathbb{R}^2$, for example. We do not know this probability distribution's analytical form. It could be uniform over a convex set, or just a $2$-D Gaussian distribution.
Is there a method that maps these points to some points $\{X^\prime_i\}$ within the $[0, 1] \times [0, 1]$ $2$-D cube, so that it looks like these transformed points are generated from a uniform distribution on $[0, 1] \times [0, 1]$.
I know the very last part of my question is not quite rigorous (what do I mean by close to a uniform distribution?), but does anyone know which field I should look into to get any hints on this type of question? Thanks.
Answering my own question in case anyone is interested. The above question is well-studied in the field of optimal transport. Suppose my ultimate goal is transforming an unknown distribution (but I can sample from) to a, say, uniform distribution. Then all I need to do is sample a bunch of samples from the two distributions and find the optimal transport between them.
For example, in this paper the authors discuss some methods of estimating such a mapping using wavelet and Gaussian kernel.