We propose a principled way to define Gaussian process priors on various sets of unweighted graphs: directed or undirected, with or without loops. We endow each of these sets with a geometric structure, inducing the notions of closeness and symmetries, by turning them into a vertex set of an appropriate metagraph. Building on this, we describe the class of priors that respect this structure and are analogous to the Euclidean isotropic processes, like squared exponential or Mat\'ern. We propose an efficient computational technique for the ostensibly intractable problem of evaluating these priors' kernels, making such Gaussian processes usable within the usual toolboxes and downstream applications. We go further to consider sets of equivalence classes of unweighted graphs and define the appropriate versions of priors thereon. We prove a hardness result, showing that in this case, exact kernel computation cannot be performed efficiently. However, we propose a simple Monte Carlo approximation for handling moderately sized cases. Inspired by applications in chemistry, we illustrate the proposed techniques on a real molecular property prediction task in the small data regime.

翻译：我们提出了一种原则性方法，用于在多种无权图集合（有向/无向、带环/不带环）上定义高斯过程先验。通过将这些集合转化为适当度量图的顶点集，我们赋予它们几何结构，从而引入邻近性和对称性概念。在此基础上，我们描述了尊重这种结构且类似于欧几里得各向同性过程（如平方指数过程或马特恩过程）的先验类。针对评估这些先验核函数这一看似棘手的问题，我们提出了一种高效计算技术，使得此类高斯过程可在标准工具包和下游应用中使用。我们进一步考虑了无权图等价类集合，并在其上定义了相应的先验版本。我们证明了硬度结果，表明在此情况下精确核函数计算无法高效完成。然而，我们提出了一种简单的蒙特卡洛近似方法以处理中等规模情形。受化学应用启发，我们在小数据场景下的真实分子性质预测任务中展示了所提出的技术。