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Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems

Lorenzo Baldassari, Ali Siahkoohi, Josselin Garnier, Knut Solna, Maarten V. de Hoop

TL;DR

This paper tackles Bayesian nonlinear inverse problems in infinite-dimensional spaces by introducing ∞-PMC-RED, an approach that uses infinite-dimensional score-based diffusion priors within a Langevin-type MCMC on a Hilbert space $H$. The method bypasses the need for log-concavity and provides a dimension-free convergence bound that explicitly depends on the score-matching accuracy through the parameter $ au$ and the score approximation error. Theoretical contributions include extending finite-dimensional PMC-RED analysis to function spaces and establishing a non-asymptotic, measure-theoretic KL/Fisher-information framework that yields a bound with error terms $A_1 au^2$ and $A_2 obreak\epsilon_ au^2$. Empirically, the authors validate the framework via stylized 2D Rosenbrock sampling and a nonlinear PDE-based seismic imaging problem, demonstrating discretization-invariant posterior sampling and meaningful uncertainty quantification. The work advances discretization-invariant Bayesian inversion by combining diffusion priors with function-space MCMC and highlights practical challenges in score learning and computational cost, outlining directions for future work on scalability and broader inverse problems.

Abstract

This work introduces a sampling method capable of solving Bayesian inverse problems in function space. It does not assume the log-concavity of the likelihood, meaning that it is compatible with nonlinear inverse problems. The method leverages the recently defined infinite-dimensional score-based diffusion models as a learning-based prior, while enabling provable posterior sampling through a Langevin-type MCMC algorithm defined on function spaces. A novel convergence analysis is conducted, inspired by the fixed-point methods established for traditional regularization-by-denoising algorithms and compatible with weighted annealing. The obtained convergence bound explicitly depends on the approximation error of the score; a well-approximated score is essential to obtain a well-approximated posterior. Stylized and PDE-based examples are provided, demonstrating the validity of our convergence analysis. We conclude by presenting a discussion of the method's challenges related to learning the score and computational complexity.

Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems

TL;DR

This paper tackles Bayesian nonlinear inverse problems in infinite-dimensional spaces by introducing ∞-PMC-RED, an approach that uses infinite-dimensional score-based diffusion priors within a Langevin-type MCMC on a Hilbert space . The method bypasses the need for log-concavity and provides a dimension-free convergence bound that explicitly depends on the score-matching accuracy through the parameter and the score approximation error. Theoretical contributions include extending finite-dimensional PMC-RED analysis to function spaces and establishing a non-asymptotic, measure-theoretic KL/Fisher-information framework that yields a bound with error terms and . Empirically, the authors validate the framework via stylized 2D Rosenbrock sampling and a nonlinear PDE-based seismic imaging problem, demonstrating discretization-invariant posterior sampling and meaningful uncertainty quantification. The work advances discretization-invariant Bayesian inversion by combining diffusion priors with function-space MCMC and highlights practical challenges in score learning and computational cost, outlining directions for future work on scalability and broader inverse problems.

Abstract

This work introduces a sampling method capable of solving Bayesian inverse problems in function space. It does not assume the log-concavity of the likelihood, meaning that it is compatible with nonlinear inverse problems. The method leverages the recently defined infinite-dimensional score-based diffusion models as a learning-based prior, while enabling provable posterior sampling through a Langevin-type MCMC algorithm defined on function spaces. A novel convergence analysis is conducted, inspired by the fixed-point methods established for traditional regularization-by-denoising algorithms and compatible with weighted annealing. The obtained convergence bound explicitly depends on the approximation error of the score; a well-approximated score is essential to obtain a well-approximated posterior. Stylized and PDE-based examples are provided, demonstrating the validity of our convergence analysis. We conclude by presenting a discussion of the method's challenges related to learning the score and computational complexity.
Paper Structure (25 sections, 8 theorems, 109 equations, 2 figures)

This paper contains 25 sections, 8 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contribution
  4. Background
  5. The Bayesian Approach to Inverse Problems
  6. Formulation of PMC-RED
  7. Score-Based Diffusion Priors in Infinite Dimensions
  8. Our Method: $\infty$-PMC-RED
  9. Measure-Theoretic Definitions of the KL Divergence and the Fisher Information
  10. Theoretical Convergence Analysis
  11. Numerical Experiments
  12. Stylized example
  13. Nonlinear wave-equation based imaging example
  14. Discussion and Conclusion
  15. Proof of Proposition \ref{['prop:score-gaussian']}
  16. ...and 10 more sections

Key Result

Proposition 1

Let Assumption assumption-C-Cmu hold. Then where $x^{(j)}:=\langle x,e_j\rangle$, $p_0^{(j)}:=\frac{\lambda_j}{\mu_{0j}}$, and $C_\tau := e^{-\tau}C_{\mu_0} + (1-e^{-\tau})C$.

Figures (2)

  • Figure 1: Comparison of samples from (a) the true 2D Rosenbrock distribution and (b) samples obtained using $\infty$-PMC-RED.
  • Figure 2: Nonlinear wave-equation-based imaging and uncertainty quantification. (a) Ground-truth squared-slowness model. (b) Conditional (posterior) mean. (c) Absolute error between Figures \ref{['fig:true_model']} and \ref{['fig:conditional_mean']}. (d) Pointwise standard deviation. (e) Vertical profiles of the ground-truth squared-slowness mode, conditional mean estimate, and the $99\%$ confidence interval at two lateral positions.

Theorems & Definitions (25)

  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • Corollary 1
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 15 more