  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.
