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：我们提出了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）。这包括许多在图节点上定义的最流行的核函数示例。我们的算法相对于节点数量具有次二次时间复杂度，克服了精确图核评估众所周知的立方级计算瓶颈。该算法还可以轻松跨机器分布式执行，从而支持在更大规模的网络上进行学习。算法的核心是一个调制函数，它根据随机游走的长度对其贡献进行加权或降权。我们证明，通过使用神经网络对其进行参数化，可以获得能提供更高质量核估计或实现高效、可扩展核学习的u-GRFs。我们提供了稳健的理论分析，并通过实验验证了我们的发现，包括固定图核的逐点估计、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。