Inverse Problems Over Probability Measure Space

Define a forward problem as ρy = G_#ρx, where the probability distribution ρx is mapped to another distribution ρy using the forward operator G. In this work, we investigate the corresponding inverse problem: Given ρy, how to find ρx? Depending on whether G is overdetermined or underdetermined, the solution can have drastically different behavior. In the overdetermined case, we formulate a variational problem minρx D( G_#ρx, ρy), 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 φ-divergence reconstructs the conditional distribution. In the underdetermined case, we formulate the constrained optimization min_ G_#ρx=ρy E[ρx]. 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: minρx D( G_#ρx, ρy) + α mathsf R[ρx], and find that the entropy-entropy pair leads to a regularized solution that is defined in a piecewise manner, whereas the W2-W2 pair leads to a least-norm solution where W2 is the 2-Wasserstein metric.