Learning Normalized Energy Models for Linear Inverse Problems

2026-05-14

AI Summary

This work addresses the sampling bias in generative diffusion models for image inverse problems, which arises from implicit prior representation and approximate likelihood modeling. The authors propose an energy-based normalized posterior inference framework that trains a denoising network with covariance regularization, enabling—for the first time without retraining—the exact computation of normalized posterior densities for a wide range of linear inverse problems. This formulation facilitates energy-guided adaptive sampling, unbiased Metropolis–Hastings correction, and blind estimation of degradation operators. Experiments on ImageNet, CelebA, and AFHQ for inpainting and deblurring demonstrate that the proposed method achieves performance on par with or superior to current state-of-the-art approaches.

Abstract

Generative diffusion models can provide powerful prior probability models for inverse problems in imaging, but existing implementations suffer from two key limitations: $(i)$ the prior density is represented implicitly, and $(ii)$ they rely on likelihood approximations that introduce sampling biases. We address these challenges by introducing a new energy-based model trained for denoising with a covariance-based regularization term that enforces consistency across different measurement conditions. The trained model can compute normalized posterior densities for diverse linear inverse problems, without additional retraining or fine tuning. In addition to preserving the sampling capabilities of diffusion models, this enables previously unavailable capabilities: energy-guided adaptive sampling that adjusts schedules on-the-fly, unbiased Metropolis-Hastings correction steps, and blind estimation of the degradation operator via Bayes rule. We validate the method on multiple datasets (ImageNet, CelebA, AFHQ) and tasks (inpainting, deblurring), demonstrating competitive or superior performance to established baselines.

Problem

• inverse problems
• diffusion models
• energy-based models
• posterior inference
• sampling bias

Innovation

• energy-based model
• normalized posterior
• covariance regularization
• adaptive sampling
• blind deconvolution