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

TL;DR

This paper introduces a theoretical framework for hypergraph modeling using spectral embedding, connecting multi-way similarities with hypergraph cut, and demonstrates improved clustering performance over existing methods.

Contribution

It establishes a novel hypergraph cut framework based on multi-way similarities derived from kernel functions, generalizing weighted kernel k-means and heat kernel methods.

Findings

The proposed method outperforms existing graph-based clustering approaches.

A fast spectral clustering algorithm is developed and validated.

Theoretical connections justify the hypergraph modeling approach.

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.