  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.
