# Inverse Problems Over Probability Measure Space

arXiv: https://arxiv.org/abs/2504.18999 · HTML: https://mlsanity.com/paper/2504.18999/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Apr 2025.

## Built on by (most cited first, 6 of 6)

- [Generative Modeling from Black-box Corruptions via Self-Consistent Stochastic Interpolants](https://mlsanity.com/paper/2512.10857/) (2025)
- [Efficient Deconvolution in Populational Inverse Problems](https://mlsanity.com/paper/2505.19841/) (2025)
- [Learning interacting particle systems from unlabeled data](https://mlsanity.com/paper/2604.02581/) (2026)
- [Cryo-EM as a Stochastic Inverse Problem](https://mlsanity.com/paper/2509.05541/) (2025)
- [Nesterov acceleration in optimizing over probability measures](https://mlsanity.com/paper/2607.23008/) (2026)
- [Copula Transformations for Data-Consistent Inversion](https://mlsanity.com/paper/2609.02832/) (2026)

## Abstract (arXiv)

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.