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Inhomogeneous Hypergraph Clustering with Applications

Pan Li, Olgica Milenkovic

arXiv:1709.01249v4cs.LGstat.ML

TL;DR

Existing hypergraph partitioning treats all cuts of a hyperedge as equally costly, limiting its ability to represent structurally different subsets. The paper introduces inhomogeneous hypergraph partitioning, which uses cut-specific costs and graph-based spectral algorithms, and reports theoretical guarantees and improvements across several applications.

  • Problem

    Existing hypergraph partitioning assumes that different cuts of a hyperedge incur the same cost, despite subsets of vertices having different structural importance.

  • Method

    The paper assigns vector weights to hyperedge cuts, projects each inhomogeneous hyperedge onto a graph, merges the projections, and applies normalized-Laplacian spectral clustering.

  • Results

    The method provides theoretical performance guarantees and is evaluated in biological network studies, ranking structure learning, subspace clustering, and motif clustering.

  • Takeaways & Limitations

    Inhomogeneous partitioning represents higher-order relations with non-uniform cut costs and supports applications including structure learning, subspace segmentation, and motif clustering.

  • Takeaways & Limitations

    The projection optimization may produce negative edge weights, while imposing nonnegativity constraints can make the optimization problem infeasible.

Abstract

from arXiv · show

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.

1 Introduction

The paper motivates hypergraph clustering as a way to represent higher-order relations that pairwise graphs cannot capture, then proposes inhomogeneous partitioning to assign different costs to different hyperedge cuts. It applies this approach to several network-learning and clustering problems while providing theoretical guarantees.

  • Higher-order relations among subsets of objects require hypergraphs because pairwise similarities may not capture their structure.
  • Homogeneous hypergraph partitioning assigns one proportional cost to a cut hyperedge, preventing different vertices or subsets from having distinct structural contributions.
  • In metabolic reactions, reactants and products play different roles, so cuts that isolate different metabolites can preserve different pairings.
  • Inhomogeneous partitioning assigns vector weights to hyperedge cuts and allows non-uniform expansion of a hyperedge, unlike Clique Expansion.
  • The proposed methods combine graph transformations with normalized-Laplacian spectral clustering and are evaluated in biological networks, ranking structure learning, and subspace clustering.

2 Preliminaries and Problem Formulation

This section defines inhomogeneous hyperedges through cut-dependent weight functions and formulates normalized-cut objectives for two-way and k-way partitions. The definitions extend graph partitioning when every hyperedge has size two.

  • A hypergraph consists of vertices and hyperedges, with each hyperedge being a subset of the vertex set.
  • An inhomogeneous hyperedge assigns a nonnegative function to subsets, where the function gives the cost of cutting the hyperedge into a subset and its complement.
  • Consistent cut weights satisfy w_e(∅) = 0 and w_e(S) = w_e(e/S), while singleton cut weights must be specified.
  • The framework defines vertex degrees, subset volumes, hyperedge boundaries, and normalized-cut objectives for two-way partitioning.
  • The same construction extends to k-way inhomogeneous partitions and agrees with graph definitions when every hyperedge has size two.

3 Inhomogeneous Hypergraph Clustering Algorithms

The method projects each inhomogeneous hyperedge onto a weighted graph, merges these projections, and applies normalized spectral clustering. Submodularity supplies feasible nonnegative projections and approximation guarantees, while arbitrary weights can make the optimization infeasible or produce negative graph weights.

  • Algorithm: The algorithm projects each inhomogeneous hyperedge onto a subgraph, merges the subgraphs, and applies spectral clustering using the normalized Laplacian.The projection step introduces the inhomogeneous constraints; merging and spectral clustering follow standard graph-based procedures.
  • Algorithm: The projection optimization finds graph-edge weights that best approximate the hyperedge’s split costs through a complete graph on its vertices.Each projected edge weight is a linear function of the hyperedge’s inhomogeneous weight function.
  • Guarantees: If the optimization is feasible and projected weights are nonnegative, spectral clustering achieves a normalized-cut guarantee controlled by β*=max_e β(e).The guarantee compares the resulting normalized cut α* with the optimal hypergraph value α_H.
  • Limitations: Arbitrary inhomogeneous weights may make the optimization infeasible, while unconstrained solutions may contain negative projected weights that invalidate the spectral guarantee when truncated.Setting negative weights to zero can perform well empirically, but the modification invalidates Theorem 3.1.
  • Guarantees: Submodular hyperedge weights guarantee a nonnegative feasible solution; for 2≤δ(e)≤7, the displayed construction is feasible with β(e) values listed in Table 1.Submodularity is defined by we(S1)+we(S2)≥we(S1∩S2)+we(S1∪S2).
  • Guarantees: For triangle hyperedges, β=1, so submodular inhomogeneous triangle clustering incurs no projection distortion.The result extends the corresponding homogeneous triangle guarantee to the inhomogeneous case.
  • Limitations: The linear-map results are proven only for small hyperedge sizes, although the authors conjecture they extend to all δ(e).Theorem 3.5 gives a min-max optimal linear approximation ratio of Ω(2δ(e)/δ(e)^2).

4 Related Work and Discussion

The framework generalizes several existing hypergraph projection methods while retaining a distinct inhomogeneous projection formulation. Its discussion contrasts these connections with methods whose approximation or computational properties remain unresolved.

  • Connections: The method’s projection, edge-weight merging, and spectral clustering components connect it to standard hypergraph clustering pipelines, while its inhomogeneous partition and projection are novel contributions.The framework also places earlier homogeneous methods within the inhomogeneous formulation.
  • Connections: Zhou’s normalized hypergraph cut is recovered by choosing we(S)=wH_e|S|(δ(e)−|S|)/δ(e), yielding the same boundary cost and projection weights.Under this choice, β(e)=1 and the projected edge weights match Zhou’s projection.
  • Connections: For uniform hypergraphs, the framework includes Clique expansion and Star expansion as special cases.The related degree definitions also coincide for uniform hypergraphs.
  • Comparison: Clique Averaging uses a one-shot ℓ2-minimization projection with n^2 variables and n^δ(e) observations, and its constant-approximation and quadratic-approximation properties are unknown.The method also has practical implementation and computational-complexity concerns.
  • Motif clustering: For motif clustering, the prior algorithm’s solution is identical in the corresponding setting, and Theorem 3.1 establishes a quadratic-factor approximation for motifs with more than three vertices.The earlier work had established the Cheeger inequality only for three-vertex motifs.
  • Discussion: Projection-based methods avoid the NP-hard task of learning spectra of certain hypergraph Laplacians, whereas alternative approaches use total variance, tensor spectra, or nonlinear Laplacians.These alternatives do not rely on hypergraph projection.

5 Applications

The applications use inhomogeneous hypergraph partitioning to encode higher-order structure in motifs and ranking comparisons, with spectral graph partitioning providing the computational step. Across examples, the method recovers meaningful hierarchical or category-aligned clusters and compares favorably with homogeneous or alternative approaches.

  • Motif clustering: Inhomogeneous motif clustering assigns different costs within network motifs, addressing the limitation that homogeneous clustering treats motif vertices and edges uniformly.Motifs are treated as hyperedges, allowing structural roles within a motif to influence partitioning.
  • Motif clustering: Five hierarchical clusters in the Florida Bay food web align with layers from producers to top-level consumers, with several clusters exceeding 80% functional similarity.Iterative InH-partitioning separates producers, primary consumers, secondary consumers, invertebrates, forage fishes, predators, and birds into biologically coherent groups.
  • Motif clustering: 127 of 128 Florida Bay vertices were covered by the chosen motif, whereas homogeneous triangular-motif clustering covered only 62 of 128 vertices.The authors attribute the difference to motif choice and to inhomogeneous weights revealing hierarchical structure that homogeneous fan weights obscure.
  • Ranking structure learning: The ranking application represents each candidate as a vertex and uses mutual information as an inhomogeneous singleton cost on three-candidate hyperedges.This formulation accommodates asymmetric mutual information, which prevents using homogeneous partitions for the same objective.
  • Ranking structure learning: In ranking structure learning, InH-partition outperforms Apar for the independent cluster and overall performance, while remaining robust to missing triples.It performs comparably for F.F., slightly worse for F.G., and remains robust when triples are independently subsampled.
  • Algorithmic procedure: The computational pipeline projects each inhomogeneous hyperedge into a graph, merges the projections, and applies normalized-Laplacian spectral clustering.The spectral routine constructs a degree matrix and normalized Laplacian, orders vertices using the second eigenvector, and searches sweep cuts.
  • Algorithmic procedure: The spectral graph procedure has a quadratic-factor guarantee relative to the optimal graph normalized cut.The guarantee is stated in terms of the algorithmic cut value, optimal graph cut value, and the second-smallest normalized-Laplacian eigenvalue.

C Proof of Theorem 3.1

The proof establishes the approximation relationship between the projected graph and the original inhomogeneous hypergraph, then combines it with the spectral graph-partitioning guarantee.

  • Approximation transfer: The projected graph G=(V,Eo,w) constantly approximates the original hypergraph when the optimization problem is feasible for every InH-hyperedge.The approximation is parameterized by β*=max_e β(e).
  • Approximation transfer: Nonnegative projected edge weights allow Theorem B.1 to be applied to Algorithm 1 on the graph G.Combining the spectral guarantee with the projection relation completes the proof.

D Proof of Theorem 3.3

The proof of Theorem 3.3 verifies feasibility and nonnegative projected weights using symmetry and submodularity, but the verification is established only for hyperedges of size at most seven.

  • Nonnegative projection weights: Submodularity and symmetry are used to cancel positive and negative coefficient terms and establish nonnegative projected edge weights.The proof partitions subsets into bijectively related groups and applies submodular inequalities to the corresponding weights.
  • Feasibility: Feasibility is checked by bounding the boundary volume for every nontrivial subset S of a hyperedge.Symmetry reduces the verification to subsets with |S|≤δ(e)/2.
  • Limitation: The argument is verified case by case for all hyperedge sizes δ(e)≤7 rather than by a general proof for arbitrary degree.The authors conjecture that the result also holds for degrees greater than seven.
  • Case analysis: The proof repeatedly applies symmetry and submodularity inequalities to convert weight inequalities into inequalities on projected boundary volumes.These inequalities are then used in the case analyses for larger hyperedges.
  • Base case: For δ(e)=2, the projection is a self-projection and singleton boundary volume equals the corresponding inhomogeneous cut cost.This gives β(e)=1 in the two-vertex case.

E Proof of Theorem 3.4

The proof of Theorem 3.4 reduces the optimal projection-weight problem using permutation and complement symmetries, then solves a linear program to obtain optimal coefficients and lower bounds.

  • Variable reduction: The projection optimization problem has 3⌊δ(e)/2⌋−1 variables.The variable count follows from the symmetry structure of the linear mapping coefficients.
  • Variable reduction: Permutation invariance makes each coefficient depend only on |{v, ṽ}∩S| and |S|.This reduces the coefficient representation to a function of two parameters rather than individual vertices and subsets.
  • Symmetry constraints: Complement symmetry imposes paired coefficient constraints such as φ̃(0,s)=φ̃(2,δ(e)−s) and φ̃(1,s)=φ̃(1,δ(e)−s).These identities further reduce the feasible coefficient family.
  • Linear-program formulation: The resulting optimization is a linear program with uncountably many constraints because the admissible submodular functions are uncountable.A finite collection of submodular functions is used to obtain efficiently computable lower bounds for βo.
  • Linear-program formulation: Solving the linear program yields optimal βΩ values matching those listed in Table 1 for the analyzed hyperedge sizes.The cases δ(e)=2 and 3 are simple, while δ(e)≥4 requires identifying suitable sets Ω and, for δ(e)=7, a larger verification table.

F Proof of Theorem 3.5

The proof counts admissible subsets under several vertex-placement cases, then uses symmetry and substitution identities to establish the required decomposition.

  • Cases where both selected vertices lie in the complement contribute zero when both are among {u, ũ}.
  • Symmetry and substitutions k → δ(e)−k, k → k+1, and k → δ(e)−(k+1) yield the identities used in the proof.
  • The resulting expression decomposes into weighted sums involving we(S), |S|(δ(e)−|S|), and corresponding indicator terms.
  • These identities complete the proof.

G Complexity analysis

The algorithm projects each inhomogeneous hyperedge, merges the projections into a graph, and applies spectral clustering; complexity depends on these three steps and on the largest hyperedge size.

  • The algorithm has three computational steps: hyperedge projection, subgraph merging, and spectral clustering on the derived graph.
  • Let δ* = max_e∈E δ(e) denote the largest hyperedge size used in the complexity analysis.
  • When only singleton weights are known, perfect projections can be obtained with β(e) = 1.
  • For δ(e) > 3, a perfect projection is not unique because the projection problem is underdetermined.
  • The projected weights must satisfy a special balancing condition across the weights associated with different vertices.

I.1 Structure Learning of Ranking Data

The ranking experiments compare InH-partition with APar on synthetic data and use real election data to study candidate clustering; InH-partition generally performs strongly, especially under sparse triple sampling and harder score settings.

  • Synthetic data: Synthetic rankings are generated from riffled-independent sets with n = 16 and varying target size q, then compared using InH-partition and APar.
  • Synthetic data: InH-partition outperforms APar for almost all sample sizes, although APar can be better when q = 4 and the sample size is large.
  • Synthetic data: InH-partition reaches an 80% success rate with triple-sampling probability r < 0.2, whereas APar requires r > 0.7.
  • Synthetic data: With s_i^3 parameters, APar performs poorly while InH-partition changes little despite the harder mutual-information estimation setting.
  • Real data: The Irish election dataset contains rankings of 14 candidates and is used for the paper’s ranking-learning tasks.

I.2 Subspace segmentation

Subspace segmentation uses hypergraph clustering to group data lying in intrinsically embedded subspaces. The paper replaces homogeneous hyperedge costs with inhomogeneous deviations that distinguish how individual points fit their companions, improving segmentation in supported settings.

  • Subspace segmentation partitions data according to their intrinsically embedded subspaces, and hypergraph-clustering methods have shown strong performance for this task.
  • Homogeneous methods assign one deviation-based weight to each hyperedge, whereas the proposed inhomogeneous deviation measures each vertex's distance to the affine subspace generated by the remaining vertices.The vertex-specific deviations construct inhomogeneous costs through w_e({v}) = exp[−d_InH_e({v})^2/θ^2].
  • The choice of a good deviation remains an open problem that may depend on the dataset, motivating comparisons across multiple homogeneous and inhomogeneous deviations.
  • In k-line experiments, InH-partition achieved 50% of H-partition's misclassification errors when noise was small (θ_n < 0.01).At higher noise, it remained better for k = 2 but deteriorated as the number of classes increased because informative hyperedges became less likely.
  • On the Hopkins155 motion-segmentation dataset, InH-partition outperformed hypergraph-based benchmarks and all compared methods except SSC.The method had significantly lower complexity and was easier to use and implement than SSC, while the authors identify hyperedge sampling as a possible improvement area.
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