As a generalization of graphs, hypergraphs are widely used to model higher-order relations in data. This paper explores the benefit of the hypergraph structure for the interpolation of point cloud data that contain no explicit structural information. We define the $\varepsilon_n$-ball hypergraph and the $k_n$-nearest neighbor hypergraph on a point cloud and study the $p$-Laplacian regularization on the hypergraphs. We prove the variational consistency between the hypergraph $p$-Laplacian regularization and the continuum $p$-Laplacian regularization in a semisupervised setting when the number of points $n$ goes to infinity while the number of labeled points remains fixed. A key improvement compared to the graph case is that the results rely on weaker assumptions on the upper bound of $\varepsilon_n$ and $k_n$. To solve the convex but non-differentiable large-scale optimization problem, we utilize the stochastic primal-dual hybrid gradient algorithm. Numerical experiments on data interpolation verify that the hypergraph $p$-Laplacian regularization outperforms the graph $p$-Laplacian regularization in preventing the development of spikes at the labeled points.

翻译：作为图的推广，超图被广泛用于建模数据中的高阶关系。本文探讨了超图结构在无显式结构信息的点云数据插值中的优势。我们在点云上定义了$\varepsilon_n$-球超图和$k_n$-最近邻超图，并研究了超图上的$p$-拉普拉斯正则化。在半监督场景下，当点数$n$趋向无穷大而标记点数量保持不变时，我们证明了超图$p$-拉普拉斯正则化与连续$p$-拉普拉斯正则化之间的变分一致性。与图情形相比，一个关键改进在于结果依赖于对$\varepsilon_n$和$k_n$上界更弱的假设。为解决凸但不可微的大规模优化问题，我们采用了随机原始-对偶混合梯度算法。数据插值数值实验验证了超图$p$-拉普拉斯正则化在抑制标记点处尖峰发展方面优于图$p$-拉普拉斯正则化。