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

TL;DR

This paper introduces a simplified hypergraph p-Laplacian equation for data interpolation and semi-supervised learning, improving computational efficiency and accuracy by suppressing spiky solutions.

Contribution

It derives a well-posed, computationally efficient hypergraph p-Laplacian equation and demonstrates its effectiveness in data interpolation and semi-supervised learning.

Findings

Suppresses spiky solutions in data interpolation

Improves classification accuracy in semi-supervised learning

Has remarkably low computational cost

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.