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

Kehan Shi, Martin Burger

TL;DR

The paper develops a hypergraph $p$-Laplacian framework for data interpolation on point clouds lacking explicit structure, introducing $oldsymbol{ ext{$\\varepsilon_n$-ball}}$ and $oldsymbol{ ext{$k_n$-NN}}$ hypergraphs and establishing variational consistency with the continuum $p$-Laplacian. It proves Γ-convergence of the discrete energies to the continuum energies under mild scaling ($\delta_n\ll\\varepsilon_n\ll1$ and $p>d$), and extends the results to the $k_n$-NN case with weight adjustments, using TL$^p$ topology and transportation maps. The numerical algorithm leverages stochastic primal-dual hybrid gradient to solve the large-scale convex, non-differentiable problem, with convergence guarantees. Empirically, hypergraph HpL regularization suppresses spiky artifacts better than graph-based regulators, improving data interpolation in 1D tests, semi-supervised learning on MNIST, and image inpainting, albeit at higher computational cost. These findings highlight the benefit of higher-order hypergraph connections for robust SSL and image restoration when explicit structure is scarce.

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.

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

TL;DR

The paper develops a hypergraph -Laplacian framework for data interpolation on point clouds lacking explicit structure, introducing \\varepsilon_n and k_n hypergraphs and establishing variational consistency with the continuum -Laplacian. It proves Γ-convergence of the discrete energies to the continuum energies under mild scaling ( and ), and extends the results to the -NN case with weight adjustments, using TL topology and transportation maps. The numerical algorithm leverages stochastic primal-dual hybrid gradient to solve the large-scale convex, non-differentiable problem, with convergence guarantees. Empirically, hypergraph HpL regularization suppresses spiky artifacts better than graph-based regulators, improving data interpolation in 1D tests, semi-supervised learning on MNIST, and image inpainting, albeit at higher computational cost. These findings highlight the benefit of higher-order hypergraph connections for robust SSL and image restoration when explicit structure is scarce.

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 -ball hypergraph and the -nearest neighbor hypergraph on a point cloud and study the -Laplacian regularization on the hypergraphs. We prove the variational consistency between the hypergraph -Laplacian regularization and the continuum -Laplacian regularization in a semisupervised setting when the number of points 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 and . 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 -Laplacian regularization outperforms the graph -Laplacian regularization in preventing the development of spikes at the labeled points.
Paper Structure (16 sections, 17 theorems, 161 equations, 6 figures, 2 tables, 4 algorithms)

This paper contains 16 sections, 17 theorems, 161 equations, 6 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related works on the graph $p$-Laplacian regularization
  3. Preliminaries
  4. Settings
  5. Hypergraphs
  6. Mathematical tools
  7. Continuum Limit of the $p$-Laplacian Regularization on the $\varepsilon_n$-ball hypergraph
  8. Nonlocal to local convergence
  9. Discrete to continuum convergence
  10. Continuum Limit of the $p$-Laplacian Regularization on the $k_n$-NN hypergraph
  11. The numerical algorithm
  12. Experimental results
  13. Experiments in 1D
  14. Semi-supervised learning
  15. Image inpainting
  16. ...and 1 more sections

Key Result

Proposition 2.2

Let $X$ be a metric space, $F_n, F: X\rightarrow [0,\infty]$ be functionals, and $F_n\stackrel{\Gamma}{\longrightarrow }F\not\equiv\infty$ as $n\rightarrow\infty$. If there exists a precompact sequence $\{x_n\}_{n\in\mathbb{N}}$ such that then and any cluster point of $\{x_n\}_{n\in\mathbb{N}}$ is a minimizer of $F$.

Figures (6)

  • Figure 1: Results of GpL and HpL with $p=2$ and different $\varepsilon_n$. First row: GpL; second row: HpL.
  • Figure 2: Results of GpL and HpL with $p=2$ and different $k_n$. First row: GpL; second row: HpL.
  • Figure 3: Results of HpL with different $p$. First row: $\varepsilon_n$-ball hypergraph with $\varepsilon_n=0.048$; second row: $k_n$-NN hypergraph with $k_n=72$.
  • Figure 4: Some images from the MNIST dataset.
  • Figure 5: Test images with size $256\times256$ for image inpainting.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 24 more