Suppose I have drawn a set, $G$, of 3-dimensional points (x, y, z) from a multivariate normal gaussian.

Is there a way that I can determine a set of sampling probabilities for the points in $G$ such that if I now drew samples from $G$ with these probabilities I would obtain a uniform distribution across each of the 3 dimensions?

I have attempted a number of efforts at casting this as an optimization problem with the per-sample probabilities as parameters of the optimization. I have failed at most efforts, and only succeeded when I computed a parameter for each joint probability. Unfortunately, that scales exponentially, thus suitable only for toy problems.

In cases that I succeeded at defining an appropriate loss function with weights per sample I've ended up with high dimensional non-convex problems that don't optimize well.

Is there hope of an optimal solution?


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