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:我们建议一种原则性的方法来定义高斯进程在各种未加权图集中的前期:定向或未定向,有或无循环。我们建议一种有效的计算技术,用于评估这些前期内核的表面棘手问题,使这些高斯进程在通常的工具箱和下游应用中能够使用。我们进一步考虑非加权图的等值分类,并界定其前期的适当版本。我们证明了一个困难的结果,表明在此情况下,精确的内核计算无法有效进行。然而,我们提出一个简单的蒙特卡洛近似法,用于处理中度大小的案件。我们根据化学应用,我们演示了在小型数据系统中进行实际分子属性预测的拟议技术。</s>