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 (i.e., 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's webpage in
this https URL

通过计算后验分布的高阶中心矩与后验均值的高阶导数之间的基本关系，提出了一种用于预训练的降噪器的不确定性估计方法，可有效计算图像区域的主成分以及在任意一维方向上的近似全边际分布。该方法快速、内存效率高，并且无需训练或微调降噪器。

关于去噪的后验分布：应用于不确定性量化

On the Posterior Distribution in Denoising: Application to Uncertainty  Quantification

Spectral methods have been the mainstay in several domains such as machine
learning and scientific computing. They involve finding a certain kind of
spectral decomposition to obtain basis functions that can capture important
structures for the problem at hand. The most common spectral method is the
principal component analysis (PCA). It utilizes the top eigenvectors of the
data covariance matrix, e.g. to carry out dimensionality reduction. This data
pre-processing step is often effective in separating signal from noise. PCA and
other spectral techniques applied to matrices have several limitations. By
limiting to only pairwise moments, they are effectively making a Gaussian
approximation on the underlying data and fail on data with hidden variables
which lead to non-Gaussianity. However, in most data sets, there are latent
effects that cannot be directly observed, e.g., topics in a document corpus, or
underlying causes of a disease. By extending the spectral decomposition methods
to higher order moments, we demonstrate the ability to learn a wide range of
latent variable models efficiently. Higher-order moments can be represented by
tensors, and intuitively, they can encode more information than just pairwise
moment matrices. More crucially, tensor decomposition can pick up latent
effects that are missed by matrix methods, e.g. uniquely identify
non-orthogonal components. Exploiting these aspects turns out to be fruitful
for provable unsupervised learning of a wide range of latent variable models.
We also outline the computational techniques to design efficient tensor
decomposition methods. We introduce Tensorly, which has a simple python
interface for expressing tensor operations. It has a flexible back-end system
supporting NumPy, PyTorch, TensorFlow and MXNet amongst others, allowing
multi-GPU and CPU operations and seamless integration with deep-learning
functionalities.

该研究论文介绍了基于张量分解方法的高阶时刻谱方法及其计算技术，也提出了一个名为 Tensorly 的 Python 数组操作工具， 具有良好的灵活性和兼容性，能够与多种深度学习框架结合使用来学习广泛的潜在变量模型。

矩阵和张量的谱学习

Spectral Learning on Matrices and Tensors

Moment closure methods appear in myriad scientific disciplines in the
modelling of complex systems. The goal is to achieve a closed form of a large,
usually even infinite, set of coupled differential (or difference) equations.
Each equation describes the evolution of one "moment", a suitable
coarse-grained quantity computable from the full state space. If the system is
too large for analytical and/or numerical methods, then one aims to reduce it
by finding a moment closure relation expressing "higher-order moments" in terms
of "lower-order moments". In this brief review, we focus on highlighting how
moment closure methods occur in different contexts. We also conjecture via a
geometric explanation why it has been difficult to rigorously justify many
moment closure approximations although they work very well in practice.

本文简要回顾了瞬时闭合方法在不同上下文中所发挥的作用，同时通过几何解释揭示了为何很难严格证明许多瞬时封闭方法的近似，尽管它们在实践中非常有效。