Continuum limit of hypergraph $p$-Laplacian equations on point clouds

TL;DR

This paper analyzes the asymptotic behavior of hypergraph $p$-Laplacian equations on point clouds, showing they converge to a weighted $p$-Laplacian PDE as data points grow, bridging discrete learning models and continuous PDEs.

Contribution

It establishes the continuum limit of hypergraph $p$-Laplacian equations on point clouds, providing a new PDE-based discretization method for semi-supervised learning.

Findings

Convergence of hypergraph $p$-Laplacian to a weighted $p$-Laplacian PDE

Validity of the continuum limit for $p > d$

Characterization of boundary conditions in the limit

Abstract

This paper studies a class of $p$-Laplacian equations on point clouds that arise from hypergraph learning in a semi-supervised setting. Under the assumption that the point clouds consist of independent random samples drawn from a bounded domain $\Omega\subset\mathbb{R}^d$, we investigate the asymptotic behavior of the solutions as the number of data points tends to infinity, with the number of labeled points remains fixed. We show, for any $p>d$ in the viscosity solution framework, that the continuum limit is a weighted $p$-Laplacian equation subject to mixed Dirichlet and Neumann boundary conditions. The result provides a new discretization of the $p$-Laplacian on point clouds.