Hypergraph Partitioning using Tensor Eigenvalue Decomposition

TL;DR

This paper introduces a tensor eigenvalue-based method for hypergraph partitioning that captures super-dyadic interactions, extending spectral techniques to hypergraphs and improving partitioning accuracy over existing reduction-based methods.

Contribution

It proposes a novel tensor eigenvalue decomposition approach for hypergraph partitioning, overcoming limitations of graph reduction methods and enabling more accurate hyperedge cuts.

Findings

The method captures super-dyadic interactions more effectively.

It provides a tighter bound on hypergraph Laplacian eigenvalues.

Improves min-cut solutions on hypergraphs and standard graphs.

Abstract

Hypergraphs have gained increasing attention in the machine learning community lately due to their superiority over graphs in capturing super-dyadic interactions among entities. In this work, we propose a novel approach for the partitioning of k-uniform hypergraphs. Most of the existing methods work by reducing the hypergraph to a graph followed by applying standard graph partitioning algorithms. The reduction step restricts the algorithms to capturing only some weighted pairwise interactions and hence loses essential information about the original hypergraph. We overcome this issue by utilizing the tensor-based representation of hypergraphs, which enables us to capture actual super-dyadic interactions. We prove that the hypergraph to graph reduction is a special case of tensor contraction. We extend the notion of minimum ratio-cut and normalized-cut from graphs to hypergraphs and show the relaxed optimization problem is equivalent to tensor eigenvalue decomposition. This novel formulation also enables us to capture different ways of cutting a hyperedge, unlike the existing reduction approaches. We propose a hypergraph partitioning algorithm inspired from spectral graph theory that can accommodate this notion of hyperedge cuts. We also derive a tighter upper bound on the minimum positive eigenvalue of even-order hypergraph Laplacian tensor in terms of its conductance, which is utilized in the partitioning algorithm to approximate the normalized cut. The efficacy of the proposed method is demonstrated numerically on simple hypergraphs. We also show improvement for the min-cut solution on 2-uniform hypergraphs (graphs) over the standard spectral partitioning algorithm.