Inverse Problems Over Probability Measure Space

Define a forward problem as $ρ_y = G_\#ρ_x$, where the probability distribution $ρ_x$ is mapped to another distribution $ρ_y$ using the forward operator $G$. In this work, we investigate the corresponding inverse problem: Given $ρ_y$, how to find $ρ_x$? Depending on whether $ G$ is overdetermined or