Multi-way spectral partitioning and higher-order Cheeger inequalities (1111.1055v6)

Abstract: 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 $\lambda_k$. In particular, we show that in every graph there is a set of size at most $2n/k$ which has expansion at most $O(\sqrt{\lambda_k \log k})$. This bound is tight, up to constant factors, for the "noisy hypercube" graphs.

• The paper resolves a central conjecture linking low Laplacian eigenvalues to the feasibility of multi-way graph partitioning.
• It rigorously employs spectral embedding using the Laplacian's bottom eigenvectors to extend Cheeger-type inequalities with quantitative expansion bounds.
• The results inspire efficient clustering algorithms with theoretical guarantees, advancing network analysis and applications in various domains.

Multi-way Spectral Partitioning and Higher-order Cheeger Inequalities

The paper "Multi-way spectral partitioning and higher-order Cheeger inequalities" by Lee, Oveis Gharan, and Trevisan addresses a central conjecture in spectral graph theory regarding the multi-way spectral partitioning of graphs. The authors provide provable guarantees for clustering algorithms using low eigenvectors of the Laplacian, extending Cheeger-type inequalities to multi-way expansions, which has implications for algorithm performance and understanding graph structures.

Key Contributions

1. Resolution of a Conjecture: The paper resolves a conjecture regarding the characterization of graph partitions by higher-order eigenvalues. Specifically, the presence of $k$ eigenvalues close to zero correlates with the possibility of partitioning the graph into $k$ subsets, each representing a sparse cut.
2. Spectral Embedding and Clustering: The paper provides a theoretical basis for clustering algorithms that embed graph vertices into $\mathbb{R}^k$ using the Laplacian's bottom $k$ eigenvectors. The authors present rigorous methods ensuring this embedding reflects the graph's multi-way expansion properties.
3. Quantitative Expansion Bounds: The authors advance a significant theoretical connection between the expansion of sets sized approximately $n/k$ and the $k$th smallest Laplacian eigenvalue $\lambda_k$. They demonstrate that in any graph, there are at least $k/2$ disjoint sets with an expansion upper bound of $O(\sqrt{\lambda_k \log k})$, a result paralleling efforts by other researchers but with improved performance metrics.
4. Algorithmic Contributions: The results are not only of theoretical interest but also yield new algorithms for spectral partitioning with close-to-optimal bounds. The key techniques extend Cheeger's inequality to multi-way settings, adapting partition algorithms based on the spectral properties of the Laplacian.
5. Handling Special Graph Classes: The exploration of planar and minor-excluded graphs shows more efficient bounds relative to general graphs. For instance, planar graphs exhibit a bound improvement due to the inherent structural features that these classes naturally afford.
6. Implications of Spectrum Gaps: The paper also investigates the conditions under which large gaps in the spectrum inform cluster separability, a critical insight for spectral clustering methodologies.

Implications and Future Directions

The theoretical advancements provide a stronger foundation for algorithms in graph partitioning, clustering, and other applications reliant on spectral methods. The presented results are crucial in enhancing the capabilities of spectral clustering techniques — particularly in domains such as social network analysis, bioinformatics, and computer vision, where identifying latent community structures or cluster formations within data is essential.

Moving forward, the research invites exploration into efficient computational techniques, especially in high-dimensional scenarios and on larger graphs. The potential refinements in dimension reduction methods and the amalgamation of these with intrinsic graph features (like those leveraged for planar graphs) pose intriguing directions for further research.

The noted improvements in partitioning algorithms can influence the design and analysis of large-scale systems in distributed computing contexts or optimized designs for complex networks—strengthening both theoretical insights and practical implementations.

In conclusion, this paper advances our understanding of spectral graph theory and multi-way partitioning, creating avenues for innovation in both theoretical frameworks and real-world applications across diverse scientific and engineering disciplines.