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

Kehan Shi

TL;DR

The paper analyzes the continuum limit of hypergraph $p$-Laplacian equations on point clouds in a semi-supervised learning setting. It establishes that the discrete hypergraph operator $L^{H_n}_{p,\varepsilon_n}$ is consistent with a nonlocal $p$-Laplacian and, under suitable scaling $\varepsilon_n$ and density assumptions, the solutions converge uniformly (almost surely) to the viscosity solution of the weighted $p$-Laplacian $\Delta^\rho_p u=0$ in $\Omega\setminus\mathcal{O}$ with Dirichlet data on $\mathcal{O}$ and Neumann data on $\partial\Omega$. The key contributions include a pointwise consistency analysis connecting the discrete operator to a continuum $L_p$ in the interior and near the boundary, a Hölder regularity estimate for discrete solutions, and a full discrete-to-continuum convergence result in the viscosity framework, justifying the hypergraph discretization as a faithful representation of the continuum problem on point clouds. These results provide theoretical grounding for hypergraph-based discretizations of the $p$-Laplacian in semi-supervised learning tasks on high-dimensional data geometries.

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 $Ω\subset\mathbb{R}^d$, 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.

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

TL;DR

The paper analyzes the continuum limit of hypergraph -Laplacian equations on point clouds in a semi-supervised learning setting. It establishes that the discrete hypergraph operator is consistent with a nonlocal -Laplacian and, under suitable scaling and density assumptions, the solutions converge uniformly (almost surely) to the viscosity solution of the weighted -Laplacian in with Dirichlet data on and Neumann data on . The key contributions include a pointwise consistency analysis connecting the discrete operator to a continuum in the interior and near the boundary, a Hölder regularity estimate for discrete solutions, and a full discrete-to-continuum convergence result in the viscosity framework, justifying the hypergraph discretization as a faithful representation of the continuum problem on point clouds. These results provide theoretical grounding for hypergraph-based discretizations of the -Laplacian in semi-supervised learning tasks on high-dimensional data geometries.

Abstract

This paper studies a class of -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 , 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 in the viscosity solution framework, that the continuum limit is a weighted -Laplacian equation subject to mixed Dirichlet and Neumann boundary conditions. The result provides a new discretization of the -Laplacian on point clouds.
Paper Structure (6 sections, 12 theorems, 109 equations, 1 figure)

This paper contains 6 sections, 12 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Assumptions and the main result
  3. Consistency for the hypergraph $p$-Laplacian operator
  4. Discrete to continuum convergence
  5. Hölder regularity of the hypergraph $p$-Laplacian equation
  6. Proof of Theorem \ref{['th:convergence']}

Key Result

Theorem 1

Let $p\geq 2$, $n>N>0$ be fixed, and $\varepsilon_n$ be large such that the hypergraph $H_n$ is connected. Then equation eq:2.1 admits a unique solution $u_n\in\mathbb{R}^n$.

Figures (1)

  • Figure 1: Interpolation results of the graph and hypergraph $p$-Laplacians for a one-dimensional signal with $n=1280$ points (drawn uniformly from the interval $[0,1]$) and $N=6$ labels (denoted by red circles). For each point, we select its nearest 72 points to construct edges or a hyperedge. The weights for the graph and the hypergraph are set to be 1.

Theorems & Definitions (21)

  • Theorem 1
  • Proposition 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 11 more