Hypergraph $p$-Laplacian: A Differential Geometry View
This work addresses a problem for researchers in machine learning and data analysis by providing a more effective tool for hypergraph-based tasks, though it appears incremental as it builds on existing graph Laplacian concepts.
The paper tackles the problem of analyzing hypergraphs, where edges can connect any number of nodes, by proposing a novel hypergraph $p$-Laplacian that generalizes the analogy between graph Laplacian and differential geometry. The result shows that the proposed $p$-Laplacian outperforms standard hypergraph Laplacians in experiments on semi-supervised learning and normalized cut settings.
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.