In this paper we give a broad overview of the intersection of partial differential equations (PDEs) and graph-based semi-supervised learning. The overview is focused on a large body of recent work on PDE continuum limits of graph-based learning, which have been used to prove well-posedness of semi-supervised learning algorithms in the large data limit. We highlight some interesting research directions revolving around consistency of graph-based semi-supervised learning, and present some new results on the consistency of $p$-Laplacian semi-supervised learning using the stochastic tug-of-war game interpretation of the $p$-Laplacian. We also present the results of some numerical experiments that illustrate our results and suggest directions for future work.

翻译：本文系统综述了偏微分方程与基于图的半监督学习之间的交叉领域。综述重点聚焦于近期关于图学习PDE连续极限的大量研究，这些研究被用于证明大数据极限下半监督学习算法的适定性。我们围绕基于图的半监督学习一致性问题指出若干值得关注的研究方向，并利用p-Laplacian算子的随机拔河博弈解释，提出了关于p-Laplacian半监督学习一致性的若干新结果。同时展示了数值实验结果，这些结果既验证了我们的理论发现，也为未来研究指明了方向。