Suppose that $Y$ follows multivariate normal $\mathbb{N}(\mu, \Sigma)$.

And we know that $f(Y)$ follows $\mathbb{U}(0,1)$, where $f: \mathbb{R}^n \to \mathbb{R}$

Can we find exact form of $f$ given this information? I think that we can't, but I can sample pairs $(Y, f(Y))$ and use that to approximate $f$

Given some basis functions $\phi_1, \phi_2, ... , \phi_k$ I could approximate $f$ by using least squares:

$$\min\sum (f(Y_i) - a_1\phi_1(Y_i) - a_2\phi_2(Y_i)-... - a_k\phi_k(Y_i))^2 $$

My question is: Are there any smart ways to choose basis $\Phi$, given that I know distribution of $Y$ and $f(Y)$


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