标题： 向量值高斯过程及其在一类度量图上的核函数 显示英文标题

标题： Vector-Valued Gaussian Processes and their Kernels on a Class of Metric Graphs

摘要： 尽管在线性网络和图上随机过程的重要性日益增加，但关于度量图上的多变量（向量值）高斯随机场的现有文献仍然很少。 本文挑战了与构建适当的矩阵值核结构相关的几个方面。 我们首先考虑可以与标量或矩阵值函数组合以实现与向量值高斯场相关有效核的矩阵值度量。 然后，我们提供了某些类矩阵值函数与单变量电阻度量组合的条件，并确保半正定性。 随后，特别关注欧几里得树，由于缺乏与依赖$\ell_1$度量的多变量核相关的文献，因此需要大量的努力。 因此，我们提供了关于依赖$\ell_1$度量的某些类矩阵值半正定函数的基础性贡献。 这一事实随后用于表征具有有限叶数的欧几里得树上的核。 其中，我们提供了某些具有紧支撑的矩阵值协方差函数类。 显示英文摘要

摘要： Despite the increasing importance of stochastic processes on linear networks and graphs, current literature on multivariate (vector-valued) Gaussian random fields on metric graphs is elusive. This paper challenges several aspects related to the construction of proper matrix-valued kernels structures. We start by considering matrix-valued metrics that can be composed with scalar- or matrix-valued functions to implement valid kernels associated with vector-valued Gaussian fields. We then provide conditions for certain classes of matrix-valued functions to be composed with the univariate resistance metric and ensure positive semidefiniteness. Special attention is then devoted to Euclidean trees, where a substantial effort is required given the absence of literature related to multivariate kernels depending on the $\ell_1$ metric. Hence, we provide a foundational contribution to certain classes of matrix-valued positive semidefinite functions depending on the $\ell_1$ metric. This fact is then used to characterise kernels on Euclidean trees with a finite number of leaves. Amongst those, we provide classes of matrix-valued covariance functions that are compactly supported.