概率测度空间上的逆问题

定义前向问题为 $\rho_y = G_\#\rho_x$，其中概率分布 $\rho_x$ 通过前向算子 $G$ 映射到另一个分布 $\rho_y$。本文我们研究对应的逆问题：给定 $\rho_y$，如何求 $\rho_x$？ Depending on whether $G$ is overdetermined or underdetermined, the solution can have drastically different behavior. In the overdetermined case, we formulate a variational problem $\min_{\rho_x} D( G_\#\rho_x, \rho_y)$, and find that different choices of the metric $D$ significantly affect the quality of the reconstruction. When $D$ is set to be the Wasserstein distance, the reconstruction is the marginal distribution, while setting $D$ to be a $\phi$-divergence reconstructs the conditional distribution. In the underdetermined case, we formulate the constrained optimization $\min_{\{ G_\#\rho_x=\rho_y\}} E[\rho_x]$. The choice of $E$ also significantly impacts the construction: setting $E$ to be the entropy gives us the piecewise constant reconstruction, while setting $E$ to be the second moment, we recover the classical least-norm solution. We also examine the formulation with regularization: $\min_{\rho_x} D( G_\#\rho_x, \rho_y) + \alpha \mathsf R[\rho_x]$, and find that the entropy-entropy pair leads to a regularized solution that is defined in a piecewise manner, whereas the $W_2$-$W_2$ pair leads to a least-norm solution where $W_2$ is the 2-Wasserstein metric.