On the Posterior Distribution in Denoising: Application to Uncertainty Quantification

Denoisers play a central role in many applications, from noise suppression in lowgrade imaging sensors, to empowering score-based generative models. The latter category of methods makes use of Tweedie's formula, which links the posterior mean in Gaussian denoising (i.e., the minimum MSE denoiser) with the score of the data distribution. Here, we derive a fundamental relation between the higherorder central moments of the posterior distribution, and the higher-order derivatives of the posterior mean. Particularly, we show how to efficiently compute the principal components of the posterior distribution for any desired region of an image, as well as to approximate the full marginal distribution along those (or any other) one-dimensional directions. Our method is fast and memory efficient, as it does not explicitly compute or store the high-order moment tensors and it requires no training or fine tuning of the denoiser. Code and examples are available on the project's webpage. Denoisers serve as key ingredients in solving a wide range of tasks. Indeed, along with their traditional use for noise suppression (Krull et al., 2019; Liang et al., 2021; Zhang et al., 2017a; 2021), the last decade has seen a steady increase in their use for solving other tasks. For example, the plug-and-play method (Venkatakrishnan et al., 2013) demonstrated how a denoiser can be used in an iterative manner to solve arbitrary inverse problems (e.g., deblurring, inapinting). This approach was adopted and extended by many, and has led to state-of-the-art results on various restoration tasks (Brifman et al., 2016; Romano et al., 2017; Tirer & Giryes, 2018; Zhang et al., 2017b).