[Paper Review] Inhomogeneous Hypergraph Clustering with Applications

Pan Li, Olgica Milenković|arXiv (Cornell University)|Sep 5, 2017

Advanced Clustering Algorithms ResearchComputer Science39 references74 citations

TL;DR

The paper introduces inhomogeneous hypergraph partitioning, assigning different costs to hyperedge cuts, and shows it achieves a quadratic approximation to the optimum under submodular costs, with applications to structure learning, subspace segmentation, and motif clustering.

ABSTRACT

Hypergraph partitioning is an important problem in machine learning, computer vision and network analytics. A widely used method for hypergraph partitioning relies on minimizing a normalized sum of the costs of partitioning hyperedges across clusters. Algorithmic solutions based on this approach assume that different partitions of a hyperedge incur the same cost. However, this assumption fails to leverage the fact that different subsets of vertices within the same hyperedge may have different structural importance. We hence propose a new hypergraph clustering technique, termed inhomogeneous hypergraph partitioning, which assigns different costs to different hyperedge cuts. We prove that inhomogeneous partitioning produces a quadratic approximation to the optimal solution if the inhomogeneous costs satisfy submodularity constraints. Moreover, we demonstrate that inhomogenous partitioning offers significant performance improvements in applications such as structure learning of rankings, subspace segmentation and motif clustering.

Motivation & Objective

• Motivate hypergraph clustering when higher-order relations vary in importance across subsets.
• Define inhomogeneous hyperedges with consistent cost functions and derive normalized cut objectives.
• Develop a projection-based spectral clustering pipeline that handles inhomogeneous costs.
• Provide theoretical guarantees under submodularity for approximation bounds.
• Demonstrate practical benefits on applications like rankings, subspace clustering, and motif networks.

Proposed method

• For each inhomogeneous hyperedge, compute a complete subgraph representation that best approximates the hyperedge cut via an optimization problem.
• Merge projected edge weights across hyperedges to obtain a graph on the same vertex set, where edge weights are sums of projections.
• Apply classical spectral clustering on the resulting graph using the normalized Laplacian.
• Provide feasible solutions to the projection problem with nonnegative edge weights and discuss conditions under submodularity that ensure existence and bounds.
• Prove that if the projection yields nonnegative weights, the method achieves a constant-approximation to the hypergraph NCut with a bound involving beta* (beta-approximation factor).
• Discuss practical considerations when negative projection weights arise and propose fixes (e.g., positive-part truncation) with empirical performance notes.

Experimental results

Research questions

• RQ1 Can inhomogeneous hyperedge costs be efficiently projected into a graph representation suitable for spectral clustering?
• RQ2 Under what conditions do submodular hyperedge weights guarantee feasible, nonnegative projections and quadratic approximation guarantees?
• RQ3 How does inhomogeneous partitioning compare to homogeneous and pairwise methods in motif clustering and structure learning?
• RQ4 What are the practical applications where inhomogeneous costs yield meaningful improvements (e.g., rankings, subspace clustering, network motifs)?

Key findings

• The proposed inhomogeneous partitioning framework yields a constant-approximation to the hypergraph NCut when projection weights are nonnegative and hyperedge costs are submodular.
• The projection method converts each inH-hyperedge into a graph component, and aggregated across hyperedges to a global graph, enabling spectral clustering with standard tools.
• Submodularity guarantees feasibility of the weight projection and nonnegative edge weights in many practical cases.
• When weights are submodular, the paper provides explicit formulas for the projected edge weights and bounds on the approximation factor.
• Empirical results show significant performance gains in applications such as network motif clustering, structure learning of rankings, and hierarchical biological networks (e.g., Florida Bay food web).
• The approach generalizes and subsumes several existing hypergraph clustering methods (e.g., homogeneous NCut, Clique Expansion) as special cases.