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We propose a theoretical framework of multi-way similarity to modelreal-valued data into hypergraphs for clustering via spectral embedding. Forgraph cut based spectral clustering, it is common to model real-valued datainto graph by modeling pairwise similarities using kernel function. This isbecause the kernel function has a theoretical connection to the graph cut. Forproblems where using multi-way similarities are more suitable than pairwiseones, it is natural to model as a hypergraph, which is generalization of agraph. However, although the hypergraph cut is well-studied, there is not yetestablished a hypergraph cut based framework to model multi-way similarity. Inthis paper, we formulate multi-way similarities by exploiting the theoreticalfoundation of kernel function. We show a theoretical connection between ourformulation and hypergraph cut in two ways, generalizing both weighted kernel$k$-means and the heat kernel, by which we justify our formulation. We alsoprovide a fast algorithm for spectral clustering. Our algorithm empiricallyshows better performance than existing graph and other heuristic modelingmethods.

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