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Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow

Qin Li, Maria Oprea, Li Wang, Yunan Yang

TL;DR

This work extends deterministic inverse problems into the realm of probability measures, formulating stochastic inverse problems where the unknown is a distribution $\rho_u$ and the data distribution is $\rho_y$, related by the forward push-forward $\mathcal{G}\#\rho_u$. It analyzes three core problems—direct inversion, variational formulations with regularization, and gradient-flow dynamics in probability space—revealing that the choice of metric (Wasserstein vs $f$-divergence) critically shapes stability, convergence, and equilibrium behavior. The paper provides rigorous results: stability bounds for invertible and under-determined maps; existence of minimizers for variational problems and regularization-by-design leading to Tikhonov-like behavior; and exponential convergence of KL-based Wasserstein gradient flows under log-concavity assumptions, with clear distinctions between conditional and marginal recovers depending on the discrepancy used. Collectively, these findings offer a principled framework for solving stochastic inverse problems with measure-valued parameters, and highlight practical implications for stability and reconstruction in applications where the parameter is inherently random.

Abstract

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.

Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow

TL;DR

This work extends deterministic inverse problems into the realm of probability measures, formulating stochastic inverse problems where the unknown is a distribution and the data distribution is , related by the forward push-forward . It analyzes three core problems—direct inversion, variational formulations with regularization, and gradient-flow dynamics in probability space—revealing that the choice of metric (Wasserstein vs -divergence) critically shapes stability, convergence, and equilibrium behavior. The paper provides rigorous results: stability bounds for invertible and under-determined maps; existence of minimizers for variational problems and regularization-by-design leading to Tikhonov-like behavior; and exponential convergence of KL-based Wasserstein gradient flows under log-concavity assumptions, with clear distinctions between conditional and marginal recovers depending on the discrepancy used. Collectively, these findings offer a principled framework for solving stochastic inverse problems with measure-valued parameters, and highlight practical implications for stability and reconstruction in applications where the parameter is inherently random.

Abstract

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.
Paper Structure (12 sections, 12 theorems, 91 equations, 3 figures)

This paper contains 12 sections, 12 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem I: direct inversion, wellposedness and stability
  3. Direct inversion in the deterministic setting
  4. Direct inversion in the stochastic setting
  5. When $\mathcal{G}$ is invertible
  6. Under-determined case
  7. Problem II: variational formulation
  8. Existence of the solution to the variational framework
  9. Variational formulation with regularization
  10. Problem III: gradient flow
  11. Characterizations of the equilibrium
  12. Exponential convergence

Key Result

Theorem 1

\newlabelthm:inv_stability0 Consider the push-forward of a map $\mathcal{G}:\mathcal{D}\longrightarrow\mathcal{R}$eqn:G_def and assume $\mathcal{G}$ is invertible, with its inverse $\mathcal{G}^{-1}$ being $\beta$-continuous for a constant $C_{\mathcal{G}^{-1}}$; see eqn:holder_cont. Then given tw

Figures (3)

  • Figure 1: A diagram showing the relations between deterministic inverse problem \ref{['eqn:inversion_det']} and the stochastic inverse problem \ref{['eqn:inversion_sto']} formulated based on the push-forward map.
  • Figure 1: Orthogonal decomposition of the domain of $\mathsf{A}$.
  • Figure 1: In the over-determined case, the $\mathcal{W}_2$ gradient flow of $\text{KL}$ divergence and the squared $\mathcal{W}_2$ metric between $\mathsf{A} \# \rho_u$ and $\rho_y^\delta$ have two different steady states $\rho_y^{\infty}$. The $\text{KL}$ divergence recovers the conditional distribution of $\rho_y^\delta$ on $\text{Col}(\mathsf{A})$ while the squared $\mathcal{W}_2$ metric recovers the marginal distribution of $\rho_y^\delta$ on $\text{Col}(\mathsf{A})$. \newlabelfig:compare0

Theorems & Definitions (26)

  • Remark 1
  • Theorem 1
  • Proof 1
  • Theorem 2: Measure disintegration disintegration
  • Theorem 3
  • Proof 2: Proof of \ref{['eq:Wasserstein_stability_inf']}
  • Proof 3: Proof of \ref{['eqn:f_divergence_stability']}
  • Theorem 1
  • Theorem 2
  • Proof 4
  • ...and 16 more