I have a multi-output black box function $f: x \rightarrow y$, where $x \in R^{M}$ and $y \in R^N$. Both $M$ and $N$ are greater than 1. For example, $M=4$, and $N=3$. My goal is to sample a set $\mathcal{X}$, so that the projected set $\mathcal{Y}$ is nearly uniform in the space $R^N$. The problem is that the mapping between $X$ and $Y$ is not isomorphism, so that sampling $X$ uniformly won't give me a uniform distribution in $Y$. Note that, here I'm trying to achieve uniform sampling for the black-box function in the $N$-dim target space rather than doing multi-objective optimization. I'd appreciate it a lot if someone could suggest methods for doing this. Thanks!
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