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

TL;DR

This paper introduces hypergraph $p$-Laplacian regularization for point cloud data interpolation, demonstrating its theoretical consistency and superior performance over graph-based methods in numerical experiments.

Contribution

It defines hypergraph $p$-Laplacian regularization for point clouds and proves its variational consistency with continuum models, improving upon graph-based approaches under weaker assumptions.

Findings

Hypergraph $p$-Laplacian regularization outperforms graph $p$-Laplacian in data interpolation.

The method prevents spike development at labeled points.

Theoretical consistency is established for large point sets.

Abstract

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.