We study p-Laplacians and spectral clustering for a recently proposed hypergraph model that incorporates edge-dependent vertex weights (EDVW). These weights can reflect different importance of vertices within a hyperedge, thus conferring the hypergraph model higher expressivity and flexibility. By constructing submodular EDVW-based splitting functions, we convert hypergraphs with EDVW into submodular hypergraphs for which the spectral theory is better developed. In this way, existing concepts and theorems such as p-Laplacians and Cheeger inequalities proposed under the submodular hypergraph setting can be directly extended to hypergraphs with EDVW. For submodular hypergraphs with EDVW-based splitting functions, we propose an efficient algorithm to compute the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the vertices, achieving higher clustering accuracy than traditional spectral clustering based on the 2-Laplacian. More broadly, the proposed algorithm works for all submodular hypergraphs that are graph reducible. Numerical experiments using real-world data demonstrate the effectiveness of combining spectral clustering based on the 1-Laplacian and EDVW.

翻译：我们研究了一种最近提出的超图模型中的p-拉普拉斯算子与谱聚类问题，该模型引入了边依赖顶点权重（EDVW）。这些权重能够反映超边内不同顶点的重要性差异，从而赋予超图模型更高的表达力与灵活性。通过构造基于EDVW的子模分裂函数，我们将含EDVW的超图转化为子模超图——后者在谱理论方面已有更完善的发展。由此，现有在子模超图框架下提出的概念与定理（例如p-拉普拉斯算子与Cheeger不等式）可直接推广至含EDVW的超图。针对具有基于EDVW分裂函数的子模超图，我们提出了一种高效算法，用于计算超图1-拉普拉斯算子第二小特征值对应的特征向量。进而利用该特征向量对顶点进行聚类，其聚类准确率显著高于基于2-拉普拉斯算子的传统谱聚类方法。更广泛地说，所提出的算法适用于所有可归约为图的子模超图。基于真实数据集的数值实验表明，将基于1-拉普拉斯算子的谱聚类与EDVW相结合具有显著有效性。