FB-GDM: Fully-Bayesian Guided Diffusion Models for High-Dimensional Linear Inverse Problems via Unsupervised Variational Inference
Abstract
Diffusion models are powerful priors for linear inverse problems, but the reference guidance methods, Diffusion Posterior Sampling (DPS) and Pseudoinverse-Guided Diffusion Models (ΠGDM), rely on scalar hyperparameters tuned per task, usually against the ground truth. We introduce FB-GDM, a fully-Bayesian guided diffusion method that removes this calibration step. Starting from the Gaussian approximation of ΠGDM, we derive a closed-form conditional score that depends on two precision parameters (inverse variances), one associated with the denoising approximation and one with the observation likelihood, and treat them as latent variables inferred by variational inference at each reverse step. A separable factorization makes each update scale linearly with the number of pixels, so the inference stays tractable at full image resolution, at a cost comparable to one ΠGDM run. FB-GDM requires neither the noise level nor the ground truth: its only inputs are the observation and the forward operator. Experiments on CelebA-HQ inverse problems establish two results. (i) The precision parameters, inferred from the observation alone, allow FB-GDM to outperform ΠGDM at its nominal setting, even when the latter is given the true noise level, by up to 14 dB depending on the operator, and to match the ground-truth-calibrated ΠGDM oracle within 0.1 dB. (ii) FB-GDM is robust when the forward operator, the noise level, or the image distribution changes: it stays close to a per-problem ΠGDM oracle throughout and does not exhibit the hallucinations observed with DPS, whereas DPS substantially degrades at a fixed scale and ΠGDM stays competitive only if it is re-tuned against the ground truth for each new problem. When the prior is applied to images outside its training set, this re-balancing between data and prior keeps FB-GDM faithful where a fixed face-prior guidance can otherwise hallucinate.
Related
- A Hierarchical Bayesian Cox Model with Variational Inference for High-Dimensional Survival Analysis
- Asymptotically Valid Laplace Variational Inference for Bayesian Nonparametric Clustering of Marked Point Patterns
- Variational Bayesian inference for logistic regression models for group testing data: Comparing a new hybrid variational inference approach to traditional methods.
- Variational Bayesian Inference Based Progressive Multi-Target MIMO Sensing with Nuisance Parameters
- Rényi-Regularized Flow-Based Variational Inference for Bayesian Calibration
BibTeX
@misc{seguy2026fbgdm,
title = {{FB-GDM: Fully-Bayesian Guided Diffusion Models for High-Dimensional Linear Inverse Problems via Unsupervised Variational Inference}},
author = {Gatien Séguy and Thomas Rodet},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29216},
eprint = {2609.29216},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05758586v1},
}
RIS
TY - UNPB
TI - FB-GDM: Fully-Bayesian Guided Diffusion Models for High-Dimensional Linear Inverse Problems via Unsupervised Variational Inference
AU - Gatien Séguy
AU - Thomas Rodet
PY - 2026
DA - 2026/09/23
PB - HAL
DO - 10.48550/arxiv.2609.29216
UR - https://hal.science/hal-05758586v1
AB - Diffusion models are powerful priors for linear inverse problems, but the reference guidance methods, Diffusion Posterior Sampling (DPS) and Pseudoinverse-Guided Diffusion Models (ΠGDM), rely on scalar hyperparameters tuned per task, usually against the ground truth. We introduce FB-GDM, a fully-Bayesian guided diffusion method that removes this calibration step. Starting from the Gaussian approximation of ΠGDM, we derive a closed-form conditional score that depends on two precision parameters (inverse variances), one associated with the denoising approximation and one with the observation likelihood, and treat them as latent variables inferred by variational inference at each reverse step. A separable factorization makes each update scale linearly with the number of pixels, so the inference stays tractable at full image resolution, at a cost comparable to one ΠGDM run. FB-GDM requires neither the noise level nor the ground truth: its only inputs are the observation and the forward operator. Experiments on CelebA-HQ inverse problems establish two results. (i) The precision parameters, inferred from the observation alone, allow FB-GDM to outperform ΠGDM at its nominal setting, even when the latter is given the true noise level, by up to 14 dB depending on the operator, and to match the ground-truth-calibrated ΠGDM oracle within 0.1 dB. (ii) FB-GDM is robust when the forward operator, the noise level, or the image distribution changes: it stays close to a per-problem ΠGDM oracle throughout and does not exhibit the hallucinations observed with DPS, whereas DPS substantially degrades at a fixed scale and ΠGDM stays competitive only if it is re-tuned against the ground truth for each new problem. When the prior is applied to images outside its training set, this re-balancing between data and prior keeps FB-GDM faithful where a fixed face-prior guidance can otherwise hallucinate.
ER -