Diffusion generative models unlock new possibilities for inverse problems as they allow for the incorporation of strong empirical priors in scientific inference. Recently, diffusion models are repurposed for solving inverse problems using Gaussian approximations to conditional densities of the reverse process via Tweedie's formula to parameterise the mean, complemented with various heuristics. To address various challenges arising from these approximations, we leverage higher order information using Tweedie's formula and obtain a statistically principled approximation. We further provide a theoretical guarantee specifically for posterior sampling which can lead to a better theoretical understanding of diffusion-based conditional sampling. Finally, we illustrate the empirical effectiveness of our approach for general linear inverse problems on toy synthetic examples as well as image restoration. We show that our method (i) removes any time-dependent step-size hyperparameters required by earlier methods, (ii) brings stability and better sample quality across multiple noise levels, (iii) is the only method that works in a stable way with variance exploding (VE) forward processes as opposed to earlier works.

翻译：扩散生成模型为反问题开辟了新的可能性，因为它们能够在科学推断中融入强大的经验先验。最近，扩散模型被重新用于解决反问题，其方法是通过Tweedie公式对反向过程的条件密度进行高斯近似以参数化均值，并辅以各种启发式策略。为了应对这些近似带来的各种挑战，我们利用Tweedie公式的高阶信息，获得了一种统计上更严谨的近似。我们进一步为后验采样提供了专门的理论保证，这有助于深化对基于扩散的条件采样方法的理论理解。最后，我们在玩具合成示例以及图像恢复任务中，展示了我们方法对于一般线性反问题的实证有效性。我们证明，我们的方法（i）消除了早期方法所需的时间依赖性步长超参数，（ii）在多个噪声水平下带来了更高的稳定性和更好的样本质量，（iii）是唯一能够以稳定方式与方差爆炸（VE）前向过程协同工作的方法，而早期方法则无法做到这一点。