# Energy-based models for inverse imaging problems

arXiv: https://arxiv.org/abs/2507.12432 · HTML: https://mlsanity.com/paper/2507.12432/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jul 2025.

## Built on by (most cited first, 4 of 4)

- [Annealed Langevin Posterior Sampling (ALPS): A Rapid Algorithm for Image Restoration with Multiscale Energy Models](https://mlsanity.com/paper/2601.02594/) (2026)
- [Time-Inhomogeneous Preconditioned Langevin Dynamics](https://mlsanity.com/paper/2605.06091/) (2026)
- [Product-of-Gaussian-Mixture Diffusion Models for Joint Nonlinear MRI Reconstruction](https://mlsanity.com/paper/2605.10629/) (2026)
- [Composing Flow-Matching Energies with Known Physics: Generation, OOD Detection, and Inversion on PDE Fields](https://mlsanity.com/paper/2608.18004/) (2026)

## Abstract (arXiv)

In this chapter we provide a thorough overview of the use of energy-based models (EBMs) in the context of inverse imaging problems. EBMs are probability distributions modeled via Gibbs densities p(x) propto exp-E(x) with an appropriate energy functional E. Within this chapter we present a rigorous theoretical introduction to Bayesian inverse problems that includes results on well-posedness and stability in the finite-dimensional and infinite-dimensional setting. Afterwards we discuss the use of EBMs for Bayesian inverse problems and explain the most relevant techniques for learning EBMs from data. As a crucial part of Bayesian inverse problems, we cover several popular algorithms for sampling from EBMs, namely the Metropolis-Hastings algorithm, Gibbs sampling, Langevin Monte Carlo, and Hamiltonian Monte Carlo. Moreover, we present numerical results for the resolution of several inverse imaging problems obtained by leveraging an EBM that allows for the explicit verification of those properties that are needed for valid energy-based modeling.