A Cheeger Inequality for Hypergraphs and Its Applications

1 INTRODUCTION

• The work addresses limited provable guarantees for clustering and partitioning datasets whose multiway relationships cannot be represented pairwise.The authors identify the underdevelopment of partitioning-oriented spectral theory for hypergraphs as a reason for the gap.Source: §1 · p. 1
• The framework establishes hypergraph Cheeger inequalities using Banerjee’s adjacency matrix and diffusion conductance, including a sharper conductance upper bound for non-covering hypergraphs.It also proves that the second-smallest normalized-Laplacian eigenvalue of every non-covering hypergraph is at most one.Source: §1 · pp. 2–3
• The paper constructs a new family of hypergraph expanders that attains the Alon–Boppana bound for hypergraphs.Source: §1 · p. 3
• The resulting Cheeger bounds are tight on both sides for cycle and cube hypergraphs.Source: §1 · p. 3
• The framework provides conductance and running-time guarantees for two-way spectral partitioning and higher-order multiway partitioning on hypergraphs.Source: §1 · pp. 2–3

2 RELATED WORK

• Hypergraph adjacency and Laplacian research spans matrix, tensor, random-walk, and nonlinear-operator approaches, with tensor algebra restricted to uniform hypergraphs [BCM17].Banerjee’s edge-size-normalized adjacency matrix is described as suitable for non-uniform hypergraphs.Source: §2 · pp. 3–4
• Prior hypergraph Cheeger inequalities include results restricted to uniform hypergraphs [Mul21] [XZ24] and a nonlinear-diffusion result without the usual higher-order eigenvalues.Banerjee’s result covers non-uniform hypergraphs, but the authors describe its bounds as weaker; their framework develops inequalities through the normalized-Laplacian approach.Source: §2 · pp. 4–5
• Existing local-clustering work includes personalized PageRank and nonlinear diffusion approaches, the latter requiring a differential-equation solution described as computationally demanding.Source: §2 · p. 5
• Earlier hypergraph partitioning methods were generally non-spectral and used clique or star expansions, while this paper adapts spectral partitioning methods to hypergraphs.Source: §2 · p. 5
• Only a few hypergraph-expander constructions were known, while high-dimensional expanders concern simplicial complexes, which are closed under inclusion [SFM23] [ZLB23] [Lub18].The paper adds a hypergraph-expander class tight for the Alon–Boppana bound.Source: §2 · p. 5

3.1 Definitions

• A hypergraph is covering when every pair of distinct vertices shares a hyperedge; its rank and co-rank are the largest and smallest hyperedge sizes.The degree of a vertex is the sum of weights of its incident hyperedges.Source: §3.1 · p. 6
• Hypergraph conductance and diffusion conductance are distinct generalizations of 2-graph conductance, but coincide when every edge has size two.Source: §3.1 · p. 6

3.2 Cheeger Inequalities for 2-graphs

• The preliminaries recall the ordinary and higher-order Cheeger inequalities for 2-graphs as foundations for the hypergraph results.Source: §3.2 · pp. 7, 9
• Higher-order expansion is defined over disjoint vertex subsets, with a corresponding diffusion expansion obtained by replacing conductance with diffusion conductance.Source: §3.2 · p. 8

3.3 An Algebraically Equivalent 2-Graph for Hypergraphs

• Under the stated independence condition on vertex weights, the framework formulates the averaging-based diffusion equivalence and equates graph expansion with hypergraph diffusion expansion.For a vertex set, graph conductance equals diffusion conductance; the corresponding higher-order expansions also match.Source: §3.3 · pp. 9–10
• The AE 2-graph has the same adjacency, Laplacian, and normalized Laplacian matrices as its hypergraph, and its graph conductance equals the hypergraph’s diffusion conductance.The AE graph is a weighted clique expansion.Source: §3.3 · p. 9

Cheeger’s inequality

• For connected hypergraphs, the section improves Banerjee’s Cheeger bound to φh(H) ≤ 2ν2(H), then derives a further-improved inequality for non-covering hypergraphs.The non-covering result is stated to be tight in Section 6.Source: §4 · pp. 10–11

Related spectral bounds

• The section compares its fixed normalized-Laplacian bounds with a nonlinear diffusion-operator bound whose eigenvalue is not directly comparable.That operator changes with the evolving distribution of mass across vertices.Source: §4 · p. 10

Improved inequality for non-covering hypergraphs

• The non-covering result provides a sharper relation between conductance and the second-smallest normalized-Laplacian eigenvalue.It extends the result of Xu and Zhou from κ-uniform hypergraphs to non-covering hypergraphs.Source: §4 · p. 11

Tight examples

• The cycle 2-graph is tight for the upper bounds in (7) and (13), and (13) gives a strictly better conductance upper bound for non-complete 2-graphs.The cited comparison concerns 2-graphs.Source: §4 · p. 11

Algorithmic implications

• The authors analyze Fiedler’s two-way global partitioning algorithm using the eigenvector for the second-smallest normalized-Laplacian eigenvalue.The analysis uses the section’s Cheeger inequality.Source: §4 · p. 11

Construction

• Any family of 2-graph expanders can be transformed into a corresponding family of hypergraph expanders.Applying the construction to Ramanujan graphs yields a family tight for the Alon–Boppana bound.Source: §5 · p. 11

K(r) construction

• For each 0 ≤ r ≤ k−2, a k-regular 2-graph expander family also yields a K(r) family of regular (k−r)-uniform hypergraphs.This is a separate expander-family construction stated alongside the H(r) construction.Source: §5 · p. 12

H(r) construction

• For each 0 ≤ r ≤ k−2, a k-regular 2-graph expander family yields an H(r) family of regular (k−r)-uniform hypergraphs.The construction’s theorem states the existence of these families.Source: §5 · pp. 12–13

Optimal hypergraph expanders

• The constructed H(0) family from Ramanujan graphs is an optimal hypergraph expander family, asymptotically tight for the Alon–Boppana bound.The section states that Corollary 5.5 gives hypergraph expander constructions achieving that bound.Source: §5 · pp. 12–13

6 TIGHTNESS RESULTS

• There are examples where diffusion conductance equals conductance and others where it is Θ(Υ_H) times conductance.Together, these examples establish tightness on both sides of Eq. (2).Source: §6 · p. 14

6.1 Tightness example for the lower bounds of Theorem 4.1 and Theorem 4.3: The cube hypergraph

• The cube hypergraph is tight for the lower bounds of both Cheeger inequalities and the upper bound of Eq. (2).Its construction also yields the stated tightness for the conductance and diffusion-conductance bounds.Source: §6.1 · pp. 14–15

6.2 Tightness example for the upper bound of the diffusion conductance Cheeger’s inequality Theorem 4.3: The cycle hypergraph

• The cycle hypergraph is tight for the upper bound of the diffusion-conductance Cheeger inequality and for the upper bounds of Corollary 4.4 up to a factor of k.The example generalizes the role of the cycle in 2-graphs.Source: §6.2 · pp. 15–16

6.3 Tightness example for the lower bound of Eq. (2): The hyperbicycle

• The hyperbicycle is tight for the lower bound of Eq. (2), with conductance equal to diffusion conductance on its minimizing partition.The two conductances on U are both Θ(1).Source: §6.3 · pp. 16–17

• Theorem 7.1 gives a higher-order Cheeger inequality for hypergraphs in terms of normalized-Laplacian eigenvalues and conductance.It applies for integer ℓ with 1 < ℓ < n.Source: §7 · p. 18
• The paper also states a higher-order Cheeger inequality in terms of diffusion conductance.This version appears as Lemma 7.2.Source: §7 · p. 18
• The diffusion-conductance result follows by applying the 2-graph higher-order inequality to the hypergraph’s AE graph and using equality of the corresponding multiway quantities.The proof uses ρ_g,ℓ(G_H) = ρ̂_h,ℓ(H).Source: §7 · p. 18

8.1 A global partitioning algorithm for hypergraphs

• The global algorithm sorts the second normalized-Laplacian eigenvector and returns the minimum-conductance prefix sweep cut.Its conductance is bounded in terms of diffusion conductance, and its running time matches the 2-graph case.Source: §8.1 · p. 18

8.2 A multi-way partitioning algorithm for hypergraphs

• The multi-way algorithm maps vertices using the ℓ smallest-eigenvalue eigenvectors, clusters by radial projection distance, then takes minimum-conductance sweep cuts.It uses ℓ-means to form ℓ sets before sorting each set by the mapped-vector norm.Source: §8.2 · pp. 18–19
• Each returned multi-way set has conductance at most O(ℓ^2) times the ℓ-th normalized-Laplacian eigenvalue.The resulting multiway conductance is at most O(ℓ^2) times the diffusion multiway conductance.Source: §8.2 · p. 19

• The proof starts from the second normalized-Laplacian eigenvector and its positive support, setting the vector to zero outside that support.It then derives an inequality using the eigenvector relation and the positive-part vector.Source: §9 · pp. 19–20
• Ordering the positive-part vector’s values creates nested threshold sets ending at the positive support, with that support the smaller-volume side in the considered case.The proof treats the opposite smaller-volume case similarly.Source: §9 · p. 20
• The proof uses that Q lies in [0, 1], where Q(2 − Q) is increasing, to proceed with the eigenvalue bound.Source: §9 · pp. 20–21

• The framework establishes Cheeger-type inequalities for general non-uniform hypergraphs, including an improved inequality for non-covering hypergraphs.Source: §10 · p. 21
• It also yields higher-order Cheeger inequalities and theoretical guarantees for Fiedler’s spectral partitioning algorithm in hypergraphs.Source: §10 · p. 21
• A constructed family of optimal hypergraph expanders is tight for the Alon–Boppana bound.Source: §10 · p. 21
• Whether Corollary 4.4’s upper bound is tight when hypergraph rank is a parameter remains open.Source: §10 · p. 22
• The current expander method guarantees rank at most degree; constructing expanders with Υ_H = Θ(d_H) remains an open question.Source: §10 · p. 22