Assume we have a real random variable $Y\in \mathbb R^p$, such that $Y= g(X)$ where $X \in\mathbb R^n $, $n< p$, is some random variable with continuous distribution $P$ and $g:\mathbb R^n \rightarrow\mathbb R^p$ is some smooth function (with continuous derivatives and second derivatives).
Obviously, the representation of $Y$ is not unique, we can find $\tilde g$ and $\tilde X$ satisfying $Y =\tilde g(\tilde X)$ just by defining an invertible map $H$ with $\tilde g = g\circ H$ and $\tilde X = H^{-1}(X)$.
I would like to know if there are other smooth $\tilde g$ and $\tilde X$ satisfying $Y =\tilde g(\tilde X)$ ?