Specifying a prior distribution is an essential part of solving Bayesian
inverse problems. The prior encodes a belief on the nature of the solution and
this regularizes the problem. In this article we completely characterize a
Gaussian prior that encodes the belief that the solution is a structured
tensor. We first define the notion of (A,b)-constrained tensors and show that
they describe a large variety of different structures such as Hankel,
circulant, triangular, symmetric, and so on. Then we completely characterize
the Gaussian probability distribution of such tensors by specifying its mean
vector and covariance matrix. Furthermore, explicit expressions are proved for
the covariance matrix of tensors whose entries are invariant under a
permutation. These results unlock a whole new class of priors for Bayesian
inverse problems. We illustrate how new kernel functions can be designed and
efficiently computed and apply our results on two particular Bayesian inverse
problems: completing a Hankel matrix from a few noisy measurements and learning
an image classifier of handwritten digits. The effectiveness of the proposed
priors is demonstrated for both problems. All applications have been
implemented as reactive Pluto notebooks in Julia.

通过特定的高斯先验，完全描述了结构化张量解的概率分布，包括均值向量和协方差矩阵，为贝叶斯反问题提供了一种新的先验方法。通过设计新的核函数并有效计算，实现了对汉克尔矩阵完成和手写数字图像分类器学习的两个具体问题的解决方案的可行性。

构建贝叶斯逆问题的结构张量先验

Constructing structured tensor priors for Bayesian inverse problems

Modern Generative Adversarial Networks are capable of creating artificial,
photorealistic images from latent vectors living in a low-dimensional learned
latent space. It has been shown that a wide range of images can be projected
into this space, including images outside of the domain that the generator was
trained on. However, while in this case the generator reproduces the pixels and
textures of the images, the reconstructed latent vectors are unstable and small
perturbations result in significant image distortions. In this work, we propose
to explicitly model the data distribution in latent space. We show that, under
a simple nonlinear operation, the data distribution can be modeled as Gaussian
and therefore expressed using sufficient statistics. This yields a simple
Gaussian prior, which we use to regularize the projection of images into the
latent space. The resulting projections lie in smoother and better behaved
regions of the latent space, as shown using interpolation performance for both
real and generated images. Furthermore, the Gaussian model of the distribution
in latent space allows us to investigate the origins of artifacts in the
generator output, and provides a method for reducing these artifacts while
maintaining diversity of the generated images.

本文研究了生成式对抗网络中的潜在向量空间中数据分布模型，提出了一个简单的高斯先验模型，用于规范图像向潜在空间的映射，这样生成的图像更加平滑和稳定，同时维持多样性和减少生成图像中的伪像。