# Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow

arXiv: https://arxiv.org/abs/2410.00229 · HTML: https://mlsanity.com/paper/2410.00229/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 34 references that are in the map.
- First posted on arXiv 30 Sep 2024.

## Builds on (most cited first, 7 of 7)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [On Linear Optimization over Wasserstein Balls](https://mlsanity.com/paper/2004.07162/) (2020)
- [Differential-Equation Constrained Optimization With Stochasticity](https://mlsanity.com/paper/2305.04024/) (2023)
- [Adjoint DSMC for nonlinear Boltzmann equation constrained optimization](https://mlsanity.com/paper/2009.01363/) (2020)
- [Wasserstein sensitivity of Risk and Uncertainty Propagation](https://mlsanity.com/paper/2003.03129/) (2020)
- [Building Population-Informed Priors for Bayesian Inference Using Data-Consistent Stochastic Inversion](https://mlsanity.com/paper/2407.13814/) (2024)
- ["Stochastic Inverse Problems" and Changes-of-Variables](https://mlsanity.com/paper/2211.15730/) (2022)

## Built on by (most cited first, 8 of 8)

- [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)
- [Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime](https://mlsanity.com/paper/2606.08654/) (2026)
- [Cryo-EM as a Stochastic Inverse Problem](https://mlsanity.com/paper/2509.05541/) (2025)
- [Unfolding with a Wasserstein Loss](https://mlsanity.com/paper/2603.20903/) (2026)
- [On the Unique Recovery of Transport Maps and Vector Fields from Finite Measure-Valued Data](https://mlsanity.com/paper/2604.07671/) (2026)
- [Copula Transformations for Data-Consistent Inversion](https://mlsanity.com/paper/2609.02832/) (2026)

## Abstract (arXiv)

Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with measurements. Consequently, these problems are naturally framed as stochastic inverse problems. In this paper, we explore three aspects of this problem: direct inversion, variational formulation with regularization, and optimization via gradient flows, drawing parallels with deterministic inverse problems. A key difference from the deterministic case is the space in which we operate. Here, we work within probability space rather than Euclidean or Sobolev spaces, making tools from measure transport theory necessary for the study. Our findings reveal that the choice of metric -- both in the design of the loss function and in the optimization process -- significantly impacts the stability and properties of the optimizer.