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.

翻译：预训练的无条件扩散模型结合后采样或最大后验估计技术，无需任务特定训练或微调即可解决任意反问题。然而，现有的后采样与最大后验估计方法通常依赖建模近似且计算成本较高。本研究提出一种新的最大后验估计策略，利用预训练无条件扩散模型解决反问题。具体而言，我们引入变分寻模损失，并证明其在每个反向扩散步骤中的最小化能引导生成样本趋近最大后验估计解（实践中对应分布众数）。该损失源于最小化扩散后验$p(\mathbf{x}_0|\mathbf{x}_t)$与测量后验$p(\mathbf{x}_0|\mathbf{y})$间KL散度的新视角，其中$\mathbf{y}$表示观测数据。值得注意的是，对于线性反问题，VML可通过解析推导获得且无需任何建模近似。基于进一步的理论分析，我们提出VML-MAP算法——一种通过VML最小化解决反问题的实证有效方法，并在多个数据集的多样化图像复原任务中，通过大量实验验证了其在性能与计算时间方面的优越性。