We propose a novel random walk-based algorithm for unbiased estimation of arbitrary functions of a weighted adjacency matrix, coined universal graph random features (u-GRFs). This includes many of the most popular examples of kernels defined on the nodes of a graph. Our algorithm enjoys subquadratic time complexity with respect to the number of nodes, overcoming the notoriously prohibitive cubic scaling of exact graph kernel evaluation. It can also be trivially distributed across machines, permitting learning on much larger networks. At the heart of the algorithm is a modulation function which upweights or downweights the contribution from different random walks depending on their lengths. We show that by parameterising it with a neural network we can obtain u-GRFs that give higher-quality kernel estimates or perform efficient, scalable kernel learning. We provide robust theoretical analysis and support our findings with experiments including pointwise estimation of fixed graph kernels, solving non-homogeneous graph ordinary differential equations, node clustering and kernel regression on triangular meshes.

翻译：我们提出一种新颖的基于随机游走的算法，用于无偏估计加权邻接矩阵的任意函数，称为通用图随机特征（u-GRFs）。这包括定义在图节点上的许多最流行的核函数实例。我们的算法在节点数量上具有次二次时间复杂度，克服了精确图核计算中众所周知的难以承受的三次复杂度。该算法还可以轻松分布式部署在多台机器上，支持在更大规模网络上进行学习。算法的核心是一个调制函数，它根据随机游走长度不同而增强或减弱其贡献。我们证明，通过使用神经网络对其进行参数化，可以获得u-GRFs，从而得到更高质量的核估计，或执行高效、可扩展的核学习。我们提供了稳健的理论分析，并通过实验支持我们的发现，实验包括固定图核的点态估计、求解非齐次图常微分方程、节点聚类以及三角网格上的核回归。