Hypergraph Modeling via Spectral Embedding Connection: Hypergraph Cut, Weighted Kernel k-Means, and Heat Kernel

Abstract

We propose a theoretical framework of multi-way similarity to model real-valued data into hypergraphs for clustering via spectral embedding. For graph cut based spectral clustering, it is common to model real-valued data into graph by modeling pairwise similarities using kernel function. This is because the kernel function has a theoretical connection to the graph cut. For problems where using multi-way similarities are more suitable than pairwise ones, it is natural to model as a hypergraph, which is generalization of a graph. However, although the hypergraph cut is well-studied, there is not yet established a hypergraph cut based framework to model multi-way similarity. In this paper, we formulate multi-way similarities by exploiting the theoretical foundation of kernel function. We show a theoretical connection between our formulation and hypergraph cut in two ways, generalizing both weighted kernel k-means and the heat kernel, by which we justify our formulation. We also provide a fast algorithm for spectral clustering. Our algorithm empirically shows better performance than existing graph and other heuristic modeling methods.

Generalizing p-Laplacian: spectral hypergraph theory and a partitioning algorithm

“…There are several future directions. A fruitful direction would be to explore if our p-Laplacian can converge to the continuous p-Laplace operator in the limit of infinite data, similarly to the graph case (Belkin and Niyogi, 2003) and the hypergraph case (Saito, 2022). Moreover, similarly to the previous studies (Hein et al, 2013;Saito et al, 2018), semi-supervised learning using S p as a regularizer would be valuable for a future study.…”