Inverse problems with diffusion models: MAP estimation via mode-seeking loss

A pre-trained unconditional diffusion model, combined with posterior sampling or maximum a posteriori (MAP) estimation techniques, can solve arbitrary inverse problems without task-specific training or fine-tuning. However, existing posterior sampling and MAP estimation methods often rely on modeling approximations and can also be computationally demanding. In this work, we propose a new MAP estimation strategy for solving inverse problems with a pre-trained unconditional diffusion model. Specifically, we introduce the variational mode-seeking loss (VML) and show that its minimization at each reverse diffusion step guides the generated sample towards the MAP estimate (modes in practice). VML arises from a novel perspective of minimizing the Kullback-Leibler (KL) divergence between the diffusion posterior $p(\mathbf{x}_0|\mathbf{x}_t)$ and the measurement posterior $p(\mathbf{x}_0|\mathbf{y})$, where $\mathbf{y}$ denotes the measurement. Importantly, for linear inverse problems, VML can be analytically derived without any modeling approximations. Based on further theoretical insights, we propose VML-MAP, an empirically effective algorithm for solving inverse problems via VML minimization, and validate its efficacy in both performance and computational time through extensive experiments on diverse image-restoration tasks across multiple datasets.

💡 Research Summary

This paper introduces a novel MAP‑estimation framework for solving inverse problems using a pre‑trained unconditional diffusion model, eliminating the need for task‑specific training or fine‑tuning. Existing approaches fall into two categories: posterior‑sampling methods, which aim to draw samples proportional to the posterior p(x₀|y) but require approximations of the conditional score ∇ₓₜ log p(xₜ|y) (Gaussian approximations, Monte‑Carlo, or special designs like DAPS), and MAP‑estimation methods, which directly seek the most probable solution but rely on heuristic variable‑splitting (HQS/ADMM) or consistency models that demand additional gradient steps and often a separate consistency network. Both families are computationally heavy and involve modeling approximations.

The authors propose the Variational Mode‑Seeking Loss (VML), defined at each diffusion time step t as the reverse KL divergence D_KL(p(x₀|xₜ)‖p(x₀|y)). Here p(x₀|xₜ) is the distribution induced by the reverse diffusion process; as t→0 its variance σₜ² shrinks, making p(x₀|xₜ) concentrate around the current noisy sample xₜ. Minimizing VML therefore forces the mode of p(x₀|xₜ) to align with the mode of the posterior, i.e., the MAP estimate. Crucially, for linear inverse problems y = Hx₀ + η (η∼N(0,σ_y²I)), the KL can be expanded analytically without any approximations, yielding a closed‑form expression:

VML(xₜ,t) = −log p(xₜ) −‖D(xₜ,t)−xₜ‖²/(2σₜ²) − ½σₜ⁻¹ Tr