On the Posterior Distribution in Denoising: Application to Uncertainty Quantification

• URL: http://arxiv.org/abs/2309.13598v2
• Date: Mon, 19 Feb 2024 10:48:06 GMT
• Title: On the Posterior Distribution in Denoising: Application to Uncertainty Quantification
• Authors: Hila Manor and Tomer Michaeli
• Abstract summary: Tweedie's formula links the posterior mean in Gaussian denoising with the score of the data distribution. We show how to efficiently compute the principal components of the posterior distribution for any desired region of an image. Our method is fast and memory-efficient, as it does not explicitly compute or store the high-order moment tensors.
• Score: 28.233696029453775
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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/ .

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