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

2024-11-19

arXiv.org

AI Summary

This work addresses the numerical challenges in hypergraph $p$-Laplacian regularization for data interpolation and semi-supervised learning—namely, non-smoothness of the objective functional and non-uniqueness of minimizers. We first rigorously derive a variational equation based on the subdifferential of the $p$-Laplacian energy and construct a mathematically well-posed, computationally efficient simplified model. This model circumvents both non-smoothness and solution non-uniqueness, effectively suppressing spurious spikes and oscillations in interpolation while improving classification accuracy in semi-supervised tasks. It exhibits low time complexity and strong scalability. Our core contribution is the systematic integration of subdifferential analysis into hypergraph signal processing, yielding the first numerically implementable $p$-Laplacian framework for large-scale hypergraph learning that simultaneously ensures theoretical rigor and practical efficiency.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Machine learning and data science for the Web

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.

Problem

• Develops fast numerical methods for hypergraph p-Laplacian learning
• Addresses non-differentiability and non-uniqueness in optimization
• Proposes simplified equation for efficient data interpolation and classification

Innovation

• Hypergraph p-Laplacian for higher-order data relationships
• Simplified well-posed p-Laplacian equation for efficiency
• Low-cost solution enhancing interpolation and classification