Annealed Langevin Posterior Sampling (ALPS): A Rapid Algorithm for Image Restoration with Multiscale Energy Models

Abstract: Solving inverse problems in imaging requires models that support efficient inference, uncertainty quantification, and principled probabilistic reasoning. Energy-Based Models (EBMs), with their interpretable energy landscapes and compositional structure, are well-suited for this task but have historically suffered from high computational costs and training instability. To overcome the historical shortcomings of EBMs, we introduce a fast distillation strategy to transfer the strengths of pre-trained diffusion models into multi-scale EBMs. These distilled EBMs enable efficient sampling and preserve the interpretability and compositionality inherent to potential-based frameworks. Leveraging EBM compositionality, we propose Annealed Langevin Posterior Sampling (ALPS) algorithm for Maximum-A-Posteriori (MAP), Minimum Mean Square Error (MMSE), and uncertainty estimates for inverse problems in imaging. Unlike diffusion models that use complex guidance strategies for latent variables, we perform annealing on static posterior distributions that are well-defined and composable. Experiments on image inpainting and MRI reconstruction demonstrate that our method matches or surpasses diffusion-based baselines in both accuracy and efficiency, while also supporting MAP recovery. Overall, our framework offers a scalable and principled solution for inverse problems in imaging, with potential for practical deployment in scientific and clinical settings. ALPS code is available at the GitHub repository \href{https://github.com/JyoChand/ALPS}{ALPS}.