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Hypergraph $p$-Laplacian equations for data interpolation and semi-supervised learning

Kehan Shi, Martin Burger

TL;DR

The paper tackles the difficulty of hypergraph learning with $p$-Laplacian regularization by addressing non-differentiability and minimizer non-uniqueness. It derives a hypergraph $p$-Laplacian equation from the subdifferential of $F_H^{con}$ and then proposes a simplified, well-posed surrogate that uses a uniform diffusion coefficient on the full vertex set, enabling a stable fixed-point solver. The key contributions include a single-valued operator from the subdifferential, a computationally light surrogate with a proven comparison principle and unique solvability, and numerical demonstrations showing suppression of spiky interpolations and competitive or superior SSL accuracy with substantial speedups. This approach enhances the scalability and robustness of hypergraph-based interpolation and semi-supervised learning for large-scale data. The methods hold potential for broad adoption in applications requiring efficient modeling of higher-order relationships.

Abstract

Hypergraph learning with $p$-Laplacian regularization has attracted a lot of attention due to its flexibility in modeling higher-order relationships in data. This paper focuses on its fast numerical implementation, which is challenging due to the non-differentiability of the objective function and the non-uniqueness of the minimizer. We derive a hypergraph $p$-Laplacian equation from the subdifferential of the $p$-Laplacian regularization. A simplified equation that is mathematically well-posed and computationally efficient is proposed as an alternative. Numerical experiments verify that the simplified $p$-Laplacian equation suppresses spiky solutions in data interpolation and improves classification accuracy in semi-supervised learning. The remarkably low computational cost enables further applications.

Hypergraph $p$-Laplacian equations for data interpolation and semi-supervised learning

TL;DR

The paper tackles the difficulty of hypergraph learning with -Laplacian regularization by addressing non-differentiability and minimizer non-uniqueness. It derives a hypergraph -Laplacian equation from the subdifferential of and then proposes a simplified, well-posed surrogate that uses a uniform diffusion coefficient on the full vertex set, enabling a stable fixed-point solver. The key contributions include a single-valued operator from the subdifferential, a computationally light surrogate with a proven comparison principle and unique solvability, and numerical demonstrations showing suppression of spiky interpolations and competitive or superior SSL accuracy with substantial speedups. This approach enhances the scalability and robustness of hypergraph-based interpolation and semi-supervised learning for large-scale data. The methods hold potential for broad adoption in applications requiring efficient modeling of higher-order relationships.

Abstract

Hypergraph learning with -Laplacian regularization has attracted a lot of attention due to its flexibility in modeling higher-order relationships in data. This paper focuses on its fast numerical implementation, which is challenging due to the non-differentiability of the objective function and the non-uniqueness of the minimizer. We derive a hypergraph -Laplacian equation from the subdifferential of the -Laplacian regularization. A simplified equation that is mathematically well-posed and computationally efficient is proposed as an alternative. Numerical experiments verify that the simplified -Laplacian equation suppresses spiky solutions in data interpolation and improves classification accuracy in semi-supervised learning. The remarkably low computational cost enables further applications.

Paper Structure

This paper contains 9 sections, 6 theorems, 61 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Hypergraph $p$-Laplacian equations
  3. The property for the minimizer of $F_H^{con}$
  4. The subdifferential of $F_H^{con}$ and the hypergraph $p$-Laplacian equation
  5. A simplified hypergraph $p$-Laplacian equation
  6. Numerical experiments
  7. Data interpolation in 1D
  8. Semi-supervised learning
  9. Conclusion

Key Result

Lemma 2.2

Let $u$ be a minimizer of $F_H^{con}$ and $x_i\in D(u)\backslash L$. There exist two hyperedges $e_k, e_l\in E$ such that

Figures (6)

  • Figure 1: A hypergraph with 6 vertices and 4 hyperedges. Let $p=2$, $w_k=1$, $k=1,\cdots,4$, and $x_1, x_3, x_5, x_6\in L$ be labeled vertices. Then $u=(4,\frac{5}{2},0,\frac{5}{2},3,3)^T$ is a minimizer of $F_H^{con}$ and $\beta_2(x_2)=\frac{3}{5}$, $\beta_2(x_4)=\frac{2}{5}$.
  • Figure 2: A hypergraph with 7 vertices and 3 hyperedges. Let $p=2$, $w_k=1$, $k=1,2,3$, and $x_1, x_7\in L$ be labeled vertices. Then $u=(0,1,\frac{3}{2},2,2,\frac{5}{2},3)^T$ is both a solution of equation \ref{['AE']} and a minimizer of $F_H^{con}$.
  • Figure 3: Results of graph Laplacian and hypergraph Laplacian for different ${k_n}$. From top to bottom: $F_{G}^{con}$, $F_{CE}^{con}$, $F_{H}^{con}$, equation \ref{['AE']}.
  • Figure 4: The running time of different methods with respect to the relative $\ell_2$ error.
  • Figure 5: Solutions of different algorithms for the dataset Mushroom with 1% labeling rate. The first row and second row are the solution $u_1$ and $u_2$ in Algorithm \ref{['alg1']} respectively. Vertices with values $\pm 1$ are the labeled points.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6: Comparison principle
  • proof
  • ...and 2 more