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

2025/12/11 by Sai Bharath Chandra Gutha, Ricardo Vinuesa, Gutha, Sai Bharath Chandra +3
Engineering · Computer Science · Mathematics · #Sparse and Compressive Sensing Techniques #Medical Image Segmentation Techniques #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2512.10524

Abstract

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(x₀|x_t) and the measurement posterior p(x₀|y), where 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.

Citations

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Related

• Inverse Problems with Diffusion Models: A MAP Estimation Perspective
• Efficient Diffusion Posterior Sampling for Noisy Inverse Problems
• A Variational Perspective on Solving Inverse Problems with Diffusion Models
• MAP-based Problem-Agnostic diffusion model for Inverse Problems
• Diffusion models for inverse problems