Benchmarking Diffusion Annealing-Based Bayesian Inverse Problem Solvers

Area of Science:

• Computational Mathematics
• Machine Learning
• Image Processing

Background:

• Diffusion models are state-of-the-art generative models, increasingly used as priors in Bayesian inverse problems.
• Optimal integration of diffusion models with likelihood functions for posterior sampling remains an open challenge.
• Current evaluation methods struggle with rigorous uncertainty quantification due to unknown analytical priors.

Purpose of the Study:

• To introduce benchmark problems with analytically known posteriors for evaluating diffusion model-based samplers.
• To propose a general framework, Bayesian Inverse Problem Solvers through Diffusion Annealing (BIPSDA), for diffusion model-based posterior sampling.
• To enable principled assessment of uncertainty quantification in diffusion model-based Bayesian inference.

Main Methods:

• Developed three benchmark problems inspired by image inpainting, x-ray tomography, and phase retrieval.
• Introduced the Bayesian Inverse Problem Solvers through Diffusion Annealing (BIPSDA) framework, unifying existing and novel algorithms.
• Tested BIPSDA algorithms against benchmark problems to assess performance and uncertainty quantification.

Main Results:

• The benchmark problems allow for approximate ground-truth posterior sampling for performance evaluation.
• BIPSDA framework integrates various diffusion-based posterior sampling approaches.
• Evaluations provided insights into the strengths and limitations of current diffusion model-based posterior samplers.

Conclusions:

• The proposed benchmark problems offer a standardized platform for future development of diffusion model-based samplers.
• The BIPSDA framework facilitates the development and evaluation of new algorithms for Bayesian inverse problems.
• This work advances the rigorous evaluation of uncertainty quantification in generative modeling for inverse problems.