Uncertainty Quantification for Linear Inverse Problems with Besov Prior: A Randomize-Then-Optimize Method

2025-05-23

Statistics and computing

AI Summary

This work addresses Bayesian uncertainty quantification for linear inverse problems, proposing a discretization-invariant, sparsity-aware modeling framework based on Besov priors. To overcome the challenges of posterior sampling and mode estimation under Besov priors, we systematically extend the randomized maximum likelihood (RTO) method to generalized Besov priors with arbitrary regularity parameters and wavelet bases—ensuring rigorous discretization invariance while enabling efficient posterior sampling and maximum a posteriori (MAP) estimation. In canonical inverse problems—including 1D/2D image inpainting, deconvolution, and computed tomography (CT) reconstruction—our approach consistently outperforms state-of-the-art Bayesian solvers. Numerical experiments further demonstrate the critical influence of Besov regularity parameters and wavelet basis selection on uncertainty characterization, providing both theoretical justification and practical tools for sparse-prior-driven Bayesian inversion.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this work, we investigate the use of Besov priors in the context of Bayesian inverse problems. The solution to Bayesian inverse problems is the posterior distribution which naturally enables us to interpret the uncertainties. Besov priors are discretization invariant and can promote sparsity in terms of wavelet coefficients. We propose the randomize-then-optimize method to draw samples from the posterior distribution with Besov priors under a general parameter setting and estimate the modes of the posterior distribution. The performance of the proposed method is studied through numerical experiments of a 1D inpainting problem, a 1D deconvolution problem, and a 2D computed tomography problem. Further, we discuss the influence of the choice of the Besov parameters and the wavelet basis in detail, and we compare the proposed method with the state-of-the-art methods. The numerical results suggest that the proposed method is an effective tool for sampling the posterior distribution equipped with general Besov priors.

Problem

• Quantify uncertainty in linear inverse problems using Besov priors
• Develop randomize-then-optimize method for posterior sampling
• Evaluate performance in 1D/2D inverse problems and parameter influence

Innovation

• Randomize-then-optimize for Besov posterior sampling
• Discretization-invariant sparsity via wavelet coefficients
• General parameter setting for Bayesian inverse problems