Can Diffusion Models Provide Rigorous Uncertainty Quantification for Bayesian Inverse Problems?

• URL: http://arxiv.org/abs/2503.03007v1
• Date: Tue, 04 Mar 2025 21:07:15 GMT
• Title: Can Diffusion Models Provide Rigorous Uncertainty Quantification for Bayesian Inverse Problems?
• Authors: Evan Scope Crafts, Umberto Villa,
• Abstract summary: In this work, we introduce a new framework, for diffusion model based posterior sampling, called BIPSDA.<n>The framework unifies several recently proposed diffusion model based posterior sampling algorithms and contains novel algorithms that can be realized through flexible combinations of design choices.<n>The results demonstrate that BIPSDA algorithms can provide strong performance on the image inpainting and x-ray tomography based problems.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In recent years, the ascendance of diffusion modeling as a state-of-the-art generative modeling approach has spurred significant interest in their use as priors in Bayesian inverse problems. However, it is unclear how to optimally integrate a diffusion model trained on the prior distribution with a given likelihood function to obtain posterior samples. While algorithms that have been developed for this purpose can produce high-quality, diverse point estimates of the unknown parameters of interest, they are often tested on problems where the prior distribution is analytically unknown, making it difficult to assess their performance in providing rigorous uncertainty quantification. In this work, we introduce a new framework, Bayesian Inverse Problem Solvers through Diffusion Annealing (BIPSDA), for diffusion model based posterior sampling. The framework unifies several recently proposed diffusion model based posterior sampling algorithms and contains novel algorithms that can be realized through flexible combinations of design choices. Algorithms within our framework were tested on model problems with a Gaussian mixture prior and likelihood functions inspired by problems in image inpainting, x-ray tomography, and phase retrieval. In this setting, approximate ground-truth posterior samples can be obtained, enabling principled evaluation of the performance of the algorithms. The results demonstrate that BIPSDA algorithms can provide strong performance on the image inpainting and x-ray tomography based problems, while the challenging phase retrieval problem, which is difficult to sample from even when the posterior density is known, remains outside the reach of the diffusion model based samplers.

Related papers

• VIPaint: Image Inpainting with Pre-Trained Diffusion Models via Variational Inference [5.852077003870417]
  We show that our VIPaint method significantly outperforms previous approaches in both the plausibility and diversity of imputations. We show that our VIPaint method significantly outperforms previous approaches in both the plausibility and diversity of imputations.
  arXiv  Detail & Related papers  (2024-11-28T05:35:36Z)
• Solving Video Inverse Problems Using Image Diffusion Models [58.464465016269614]
  We introduce an innovative video inverse solver that leverages only image diffusion models. Our method treats the time dimension of a video as the batch dimension image diffusion models. We also introduce a batch-consistent sampling strategy that encourages consistency across batches.
  arXiv  Detail & Related papers  (2024-09-04T09:48:27Z)
• Amortizing intractable inference in diffusion models for vision, language, and control [89.65631572949702]
  This paper studies amortized sampling of the posterior over data, $mathbfxsim prm post(mathbfx)propto p(mathbfx)r(mathbfx)$, in a model that consists of a diffusion generative model prior $p(mathbfx)$ and a black-box constraint or function $r(mathbfx)$.<n>We prove the correctness of a data-free learning objective, relative trajectory balance, for training a diffusion model that samples from
  arXiv  Detail & Related papers  (2024-05-31T16:18:46Z)
• Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction [31.503662384666274]
  In science and engineering, the goal is to infer an unknown image from a small number of measurements collected from a known forward model describing certain imaging modality. Motivated Score-based diffusion models, due to its empirical success, have emerged as an impressive candidate of an exemplary prior in image reconstruction.
  arXiv  Detail & Related papers  (2024-03-25T15:58:26Z)
• Fast Diffusion EM: a diffusion model for blind inverse problems with application to deconvolution [0.0]
  Current methods assume the degradation to be known and provide impressive results in terms of restoration and diversity. In this work, we leverage the efficiency of those models to jointly estimate the restored image and unknown parameters of the kernel model. Our method alternates between approximating the expected log-likelihood of the problem using samples drawn from a diffusion model and a step to estimate unknown model parameters.
  arXiv  Detail & Related papers  (2023-09-01T06:47:13Z)
• Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Models [98.95988351420334]
  We present the first framework to solve linear inverse problems leveraging pre-trained latent diffusion models. We theoretically analyze our algorithm showing provable sample recovery in a linear model setting.
  arXiv  Detail & Related papers  (2023-07-02T17:21:30Z)
• A Variational Perspective on Solving Inverse Problems with Diffusion Models [101.831766524264]
  Inverse tasks can be formulated as inferring a posterior distribution over data. This is however challenging in diffusion models since the nonlinear and iterative nature of the diffusion process renders the posterior intractable. We propose a variational approach that by design seeks to approximate the true posterior distribution.
  arXiv  Detail & Related papers  (2023-05-07T23:00:47Z)
• Reflected Diffusion Models [93.26107023470979]
  We present Reflected Diffusion Models, which reverse a reflected differential equation evolving on the support of the data. Our approach learns the score function through a generalized score matching loss and extends key components of standard diffusion models.
  arXiv  Detail & Related papers  (2023-04-10T17:54:38Z)
• DDFM: Denoising Diffusion Model for Multi-Modality Image Fusion [144.9653045465908]
  We propose a novel fusion algorithm based on the denoising diffusion probabilistic model (DDPM) Our approach yields promising fusion results in infrared-visible image fusion and medical image fusion.
  arXiv  Detail & Related papers  (2023-03-13T04:06:42Z)
• A deep-learning based Bayesian approach to seismic imaging and uncertainty quantification [0.4588028371034407]
  Uncertainty is essential when dealing with ill-conditioned inverse problems. It is often not possible to formulate a prior distribution that precisely encodes our prior knowledge about the unknown. We propose to use the functional form of a randomly convolutional neural network as an implicit structured prior.
  arXiv  Detail & Related papers  (2020-01-13T23:46:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.