We propose a flexible framework for defining the 1-Laplacian of a hypergraph
that incorporates edge-dependent vertex weights. These weights are able to
reflect varying importance of vertices within a hyperedge, thus conferring the
hypergraph model higher expressivity than homogeneous hypergraphs. We then
utilize the eigenvector associated with the second smallest eigenvalue of the
hypergraph 1-Laplacian to cluster the vertices. From a theoretical standpoint
based on an adequately defined normalized Cheeger cut, this procedure is
expected to achieve higher clustering accuracy than that based on the
traditional Laplacian. Indeed, we confirm that this is the case using
real-world datasets to demonstrate the effectiveness of the proposed spectral
clustering approach. Moreover, we show that for a special case within our
framework, the corresponding hypergraph 1-Laplacian is equivalent to the
1-Laplacian of a related graph, whose eigenvectors can be computed more
efficiently, facilitating the adoption on larger datasets.

提出了一种灵活的超图 1-Laplacian 定义框架，包括依赖于边的顶点权重，以反映超边内不同顶点的重要性，增强了超图模型的表现能力。利用超图 1-Laplacian 的第二个最小特征向量进行聚类，可以实现比传统 Laplacian 更高的聚类精度，而且该方法可以在现实数据集上得到验证。在特定情况下，超图 1-Laplacian 等效于相关图的 1-Laplacian，可以更有效地计算特征向量，方便应用于更大的数据集。

基于 1-Laplacian 的边相关节点权重的超图谱聚类

Hypergraphs with Edge-Dependent Vertex Weights: Spectral Clustering  based on the 1-Laplacian

As a powerful tool for modeling complex relationships, hypergraphs are
gaining popularity from the graph learning community. However, commonly used
frameworks in deep hypergraph learning focus on hypergraphs with
edge-independent vertex weights (EIVWs), without considering hypergraphs with
edge-dependent vertex weights (EDVWs) that have more modeling power. To
compensate for this, we present General Hypergraph Spectral Convolution (GHSC),
a general learning framework that not only handles EDVW and EIVW hypergraphs,
but more importantly, enables theoretically explicitly utilizing the existing
powerful Graph Convolutional Neural Networks (GCNNs) such that largely ease the
design of Hypergraph Neural Networks. In this framework, the graph Laplacian of
the given undirected GCNNs is replaced with a unified hypergraph Laplacian that
incorporates vertex weight information from a random walk perspective by
equating our defined generalized hypergraphs with simple undirected graphs.
Extensive experiments from various domains including social network analysis,
visual objective classification, and protein learning demonstrate the
state-of-the-art performance of the proposed framework.

提出了 GHSC 框架，它不仅处理具有边独立顶点权重（EIVWs）的超图，还处理具有边相关顶点权重（EDVWs）的超图，并能够利用现有的 GCNN，从而大大简化了超图神经网络的设计。