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.

翻译：在求解贝叶斯反问题时，设定先验分布是至关重要的一步。先验体现了对解的性质的信念，从而对问题进行正则化。本文完整刻画了一种高斯先验，该先验编码了"解是结构化张量"这一信念。我们首先定义了(A,b)-约束张量的概念，并证明其能描述多种不同结构，如Hankel矩阵、循环矩阵、三角矩阵、对称矩阵等。随后，我们通过指定其均值向量与协方差矩阵，完整刻画了此类张量的高斯概率分布。此外，本文证明了其条目在置换下保持不变的张量的协方差矩阵的显式表达式。这些成果为贝叶斯反问题开启了一类全新的先验。我们展示了如何设计并高效计算新的核函数，并将研究成果应用于两个具体的贝叶斯反问题：从少量噪声观测中补全Hankel矩阵，以及手写数字图像分类器的学习。针对这两个问题均验证了所提出先验的有效性。所有应用均已通过Julia语言中的响应式Pluto笔记本实现。