# Hypergraph p-Laplacian regularization on point clouds for data interpolation

arXiv: https://arxiv.org/abs/2405.01109 · HTML: https://mlsanity.com/paper/2405.01109/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 53 references that are in the map.
- First posted on arXiv 2 May 2024.

## Builds on (most cited first, 14 of 14)

- [Poisson Learning: Graph Based Semi-Supervised Learning At Very Low Label Rates](https://mlsanity.com/paper/2006.11184/) (2020)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited](https://mlsanity.com/paper/1312.5179/) (2013)
- [Stochastic Primal-Dual Hybrid Gradient Algorithm with Arbitrary Sampling and Imaging Applications](https://mlsanity.com/paper/1706.04957/) (2017)
- [Submodular Hypergraphs: p-Laplacians, Cheeger Inequalities and Spectral Clustering](https://mlsanity.com/paper/1803.03833/) (2018)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [Properly-weighted graph Laplacian for semi-supervised learning](https://mlsanity.com/paper/1810.04351/) (2018)
- [Algorithms for Lipschitz Learning on Graphs](https://mlsanity.com/paper/1505.00290/) (2015)
- [Asymptotic behavior of ℓp-based Laplacian regularization in semi-supervised learning](https://mlsanity.com/paper/1603.00564/) (2016)
- [Consistency of Lipschitz learning with infinite unlabeled data and finite labeled data](https://mlsanity.com/paper/1710.10364/) (2017)
- [Learning Hypergraph-regularized Attribute Predictors](https://mlsanity.com/paper/1503.05782/) (2015)
- [On the convergence of stochastic primal-dual hybrid gradient](https://mlsanity.com/paper/1911.00799/) (2019)
- [Continuum Limit of Lipschitz Learning on Graphs](https://mlsanity.com/paper/2012.03772/) (2020)
- [Variational limits of k-NN graph based functionals on data clouds](https://mlsanity.com/paper/1607.00696/) (2016)

## Built on by (most cited first, 1 of 1)

- [Hypergraph p-Laplacian equations for data interpolation and semi-supervised learning](https://mlsanity.com/paper/2411.12601/) (2024)

## Abstract (arXiv)

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 varepsilonn-ball hypergraph and the kn-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 varepsilonn and kn. 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.