On the Posterior Distribution in Denoising: Application to Uncertainty Quantification

TL;DR

This paper establishes a fundamental relation between the moments of the posterior distribution in Gaussian denoising and the derivatives of the posterior mean, enabling efficient uncertainty quantification without retraining denoisers.

Contribution

It introduces a novel theoretical relation linking posterior moments to mean derivatives and applies it to fast, memory-efficient uncertainty quantification in denoising tasks.

Findings

Efficient computation of principal components of the posterior distribution.

Approximation of the full marginal distribution along specified directions.

Method does not require retraining or fine-tuning of denoisers.

Abstract

Denoisers play a central role in many applications, from noise suppression in low-grade 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 (\ie the minimum MSE denoiser) with the score of the data distribution. Here, we derive a fundamental relation between the higher-order central moments of the posterior distribution, and the higher-order derivatives of the posterior mean. We harness this result for uncertainty quantification of pre-trained denoisers. 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 webpage in https://hilamanor.github.io/GaussianDenoisingPosterior/ .