Hypergraph $p$-Laplacian regularization on point clouds for data interpolation
<|reference_start|>Hypergraph $p$-Laplacian regularization on point clouds for data interpolation: 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.<|reference_end|>
@article{shi2024hypergraph,
    title={Hypergraph $p$-Laplacian regularization on point clouds for data
  interpolation},
    author={Kehan Shi, Martin Burger},
    journal={arXiv preprint arXiv:2405.01109},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.01109},
    primaryClass={math.NA cs.LG cs.NA math.AP}
}