Source-linked AI summary
Multi-way spectral partitioning and higher-order Cheeger inequalities
James R. Lee, Shayan Oveis Gharan, Luca Trevisan
TL;DR
The paper asks whether k small Laplacian eigenvalues characterize partitions into k sparse pieces, extending the two-way Cheeger picture. It resolves this conjecture through algorithmic spectral partitioning and rounding methods, and proves nearly optimal expansion–eigenvalue tradeoffs, while leaving an open question for k much larger than log n.
Problem
The paper addresses the conjectured higher-order analogue of Cheeger’s inequality: whether k eigenvalues near zero correspond to a partition into k subsets defining sparse cuts.
Method
The paper develops algorithmic spectral partitioning using bottom-k eigenvector embeddings, radial projection distances, random partitions, dimension reduction, and combined thresholding.
Results
The paper resolves the conjecture, proves small-set expansion at most O(√(λk log k)) for sets of size about n/k, and obtains a quantitatively optimal noisy-hypercube tradeoff.
Takeaways & Limitations
The results provide theoretical justification for clustering with the bottom k Laplacian eigenvectors and establish a nearly optimal connection between λk and expansion of small sets.
Takeaways & Limitations
The state of affairs for k ≫ log n in the noisy-hypercube setting remains an open question.
Abstract
from arXiv · showhide
A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero. It has been conjectured that an analogous characterization holds for higher multiplicities, i.e., there are $k$ eigenvalues close to zero if and only if the vertex set can be partitioned into $k$ subsets, each defining a sparse cut. We resolve this conjecture. Our result provides a theoretical justification for clustering algorithms that use the bottom $k$ eigenvectors to embed the vertices into $\mathbb R^k$, and then apply geometric considerations to the embedding. We also show that these techniques yield a nearly optimal tradeoff between the expansion of sets of size $\approx n/k$, and the $k$th smallest eigenvalue of the normalized Laplacian matrix, denoted $λ_k$. In particular, we show that in every graph there is a set of size at most $2n/k$ which has expansion at most $O(\sqrt{λ_k \log k})$. This bound is tight, up to constant factors, for the "noisy hypercube" graphs.
1 Introduction
The paper resolves the higher-order analogue of Cheeger’s inequality, linking the kth-smallest Laplacian eigenvalue to k-way sparse partitioning. It develops algorithmic spectral embeddings and rounding methods, with stronger bounds for structured graph families and nearly optimal small-set expansion guarantees.
- Higher-order Cheeger inequalities: Theorem 1.1 gives a strong quantitative version of ρG(k) = 0 if and only if λk = 0.This extends the zero-eigenvalue characterization from connected components to higher-order partitions.
- Algorithmic implications: The proof is algorithmic and justifies clustering with the bottom k Laplacian eigenvectors followed by geometric partitioning.The paper replaces a k-means step with random geometric partitioning and leaves direct analysis of k-means open.
- Small-set expansion: Theorem 1.2 bounds expansion for small sets using λk, with improved bounds for planar graphs and graphs excluding a fixed minor.The general bound has necessary k-dependence, while structured graph families can achieve stronger guarantees.
- Small-set expansion: The bound is quantitatively optimal for noisy hypercube graphs, connecting λk with expansion of sets of size approximately n/k.The paper resolves the related conjecture up to a factor of 2, and up to 1 + δ with a changed leading constant.
- Large gaps in the spectrum: If λ4k ≥ C(log k)^2λ2k, the paper proves a spectral-gap-based result supporting easier partitioning into k pieces than k + 1.The implicit constant in the associated upper bound is independent of k.
- Spectral partitioning methods: The method uses radial projection distance, random partitions, dimension reduction, and combined partitioning and thresholding to construct sparse pieces.For planar graphs, an induced shortest-path pseudometric combines spectral and intrinsic geometry to obtain dimension-independent bounds.
2 Preliminaries
This section establishes the weighted-graph and normalized-Laplacian framework, then recalls variational links between eigenvalues, Rayleigh quotients, and disjointly supported functions. It also introduces metric partitions used later in spectral arguments.
- Weighted degrees and edge weights define the graph’s weighted measure and the associated Hilbert space ℓ2(V,w).
- The normalized Laplacian is constructed from the degree and adjacency operators, with eigenvalues ordered from the smallest upward.
- The kth eigenvalue is bounded by twice the largest Rayleigh quotient among k disjointly supported functions.
- Applying this principle to indicator functions connects disjoint vertex sets’ conductance to the kth Laplacian eigenvalue.
- Random partitions are characterized by bounded cluster diameters, padding probabilities, and Lipschitz separation guarantees.
- Euclidean, excluded-minor, and bounded-genus metrics admit random-partition bounds with dimension- or geometry-dependent parameters.
3 Localizing eigenfunctions
The section develops a localization framework that turns the bottom k eigenfunctions into separated, mass-preserving functions with controlled Rayleigh quotients. Applying metric partitioning yields multi-way spectral partitioning and higher-order Cheeger bounds.
- 3 Localizing eigenfunctions: Disjointly supported functions with Rayleigh quotient at most k^O(1)λ_k can be constructed in any weighted graph.
- 3 Localizing eigenfunctions: The radial projection distance treats vertices as close when their Euclidean embedding distance is small relative to their norms.
- 3 Localizing eigenfunctions: An eigenfunction embedding from an orthonormal system is spreading, preventing its ℓ2 mass from concentrating excessively on a small region.
- 3 Localizing eigenfunctions: Localization restricts an embedding to a neighborhood while retaining its mass and controlling edge stretching.
- 3 Localizing eigenfunctions: Separated regions containing substantial embedding mass yield disjointly supported functions, with scalar coordinates preserving a no-larger Rayleigh quotient.
- 3 Localizing eigenfunctions: The resulting non-expanding k-partition applies to weighted graphs and improves for excluded-minor and bounded-genus graphs.
- 3 Localizing eigenfunctions: For every δ, at least ⌈(1−δ)k⌉ disjoint sets can be obtained from the eigenfunction embedding, with expansion controlled by λ_k.
4 Improved quantitative bounds
This section develops dimension-reduction and partitioning tools that produce many disjoint low-expansion sets or functions from spectral embeddings. The resulting theorems give improved quantitative bounds and connect spectral gaps to multi-way partitioning.
- Multi-way partitioning: Theorem 4.1 produces at least ⌈(1 −δ)k⌉ disjoint sets from any ℓ2(V, w)-orthonormal system.The sets arise under the theorem’s weighted-graph and δ assumptions.
- Dimension reduction: Random Gaussian projection reduces the embedding dimension while preserving the relevant Rayleigh-quotient and spreading properties with constant probability.The map Γk,h uses i.i.d. k-dimensional Gaussians and projects into Rh; the target dimension is O(δ^-2 log k).
- Dimension reduction: With probability at least 25/32, the projected embedding remains (∆/4, (1+7δ)η)-spreading.This combines the local spreading claim with the dimension-reduction estimates.
- Multi-way partitioning: Lemma 4.8 converts a (∆, 1/4k)-spreading embedding into at least ⌈(1−δ)k⌉ disjoint sets.These sets are then used in the proof of the multi-way Cheeger result.
- Spectral gaps: Theorem 4.10 and Corollary 4.11 use spectral gaps to obtain at least (1−3δ)k disjoint nonempty sets or disjointly supported functions with controlled expansion.The relevant spectral-gap condition compares λ(1+δ)k with c(log k)^2λk.
- Quantitative bounds: For sets of size at most Cn/k, the resulting expansion bound has logarithmic dependence on k, while the correct dependence for finding only k/2 sets is Θ(√log k).The asymptotic dependence for a full k-partition remains unresolved, especially when k ≫ log n.
5 Conclusion
The conclusion presents a randomized spectral-clustering algorithm based on eigenvector embeddings, Gaussian projection, random partitioning, and Cheeger sweeps. It also identifies open questions about practical dimension reduction, k-means analysis, and the optimal dependence on k.
- Algorithm: A Gaussian projection maps the spectral embedding into Rh with h = O(log k) before geometric partitioning.The projected map is F* = Γ2k,h ◦ F.
- Complexity: The algorithm runs in O(n·poly(k)) time, with every step except random partitioning nearly linear and the partitioning step taking O(n · 2h).The method can use any orthonormal vectors with small Rayleigh quotient and can exploit fast Laplacian solvers.
- Open questions: The conclusion leaves open whether dimension reduction improves practical clusterings and whether k-means can be rigorously analyzed in place of random geometric partitioning.These questions concern projected points and the k-means heuristic used in related spectral-clustering approaches.
- Open questions: The optimal asymptotic dependence on k for Theorem 1.1 remains open, and full k-partitioning can have polynomial dependence on k in some graph families.For finding k/2 disjoint non-expanding sets, the stated dependence is Θ(√log k).