Hypergraph $p$-Laplacian: A Differential Geometry View

TL;DR

This paper introduces a hypergraph $p$-Laplacian inspired by differential geometry, enabling advanced analysis of hypergraphs and improving semi-supervised learning and hypergraph cut methods.

Contribution

It generalizes the graph Laplacian to hypergraphs with a novel $p$-Laplacian and develops related semi-supervised learning and normalized cut techniques.

Findings

The hypergraph $p$-Laplacian outperforms standard hypergraph Laplacians in experiments.

Proposed methods effectively analyze hypergraphs with complex connections.

Theoretical connections to hypergraph cut problems are established.

Abstract

The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph $p$-Laplacian. Unlike the existing two-node graph Laplacians, this generalization makes it possible to analyze hypergraphs, where the edges are allowed to connect any number of nodes. Moreover, we propose a semi-supervised learning method based on the proposed hypergraph $p$-Laplacian, and formalize them as the analogue to the Dirichlet problem, which often appears in physics. We further explore theoretical connections to normalized hypergraph cut on a hypergraph, and propose normalized cut corresponding to hypergraph $p$-Laplacian. The proposed $p$-Laplacian is shown to outperform standard hypergraph Laplacians in the experiment on a hypergraph semi-supervised learning and normalized cut setting.