[Paper Review] A Higher-Order Cheeger's Inequality

Shayan Oveis Gharan, Luca Trevisan|arXiv (Cornell University)|Jul 13, 2011

Graph theory and applications15 references3 citations

TL;DR

This paper resolves a long-standing conjecture by proving a higher-order Cheeger inequality that characterizes when a graph can be partitioned into $k$ subsets, each with a sparse internal cut, based on the presence of $k$ eigenvalues close to 1 in the normalized adjacency matrix. The result provides a theoretical foundation for spectral clustering methods using top $k$ eigenvectors to embed and partition graphs.

ABSTRACT

A basic fact in algebraic graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue 1 in the normalized adjacency matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to 1. Cheeger's inequality provides an "approximate" version of the latter fact, and it states that a graph has a sparse cut (it is "almost disconnected") if and only if there are at least two eigenvalues that are close to one. It has been conjectured that an analogous characterization holds for higher multiplicities, that is there are $k$ eigenvalues close to 1 if and only if the vertex set can be partitioned into $k$ subsets, each defining a sparse cut. In this paper we resolve this conjecture. Our result provides a theoretical justification for clustering algorithms that use the top $k$ eigenvector to embed the vertices into $\R^k$, and then apply geometric considerations to the embedding.

Motivation & Objective

• To resolve the conjecture that $k$ eigenvalues close to 1 in the normalized adjacency matrix correspond to a $k$-way partitioning of the graph into sparse cuts.
• To establish a theoretical justification for clustering algorithms that use the top $k$ eigenvectors to embed vertices into $\mathbb{R}^k$ and apply geometric clustering.
• To extend Cheeger's inequality—originally for $k=2$—to higher multiplicities, providing a quantitative connection between spectral properties and graph structure.
• To formalize the relationship between the multiplicity of eigenvalues near 1 and the existence of multiple sparse cuts in a graph.

Proposed method

• Formalize the higher-order Cheeger inequality by analyzing the spectral properties of the normalized adjacency matrix.
• Use variational principles and spectral gap arguments to relate eigenvalue multiplicities near 1 to the existence of $k$ disjoint subsets with small internal expansion.
• Define a notion of $k$-way expansion that generalizes the standard Cheeger constant to multiple components.
• Apply techniques from spectral graph theory and isoperimetric inequalities to bound the expansion of each component in a $k$-partition.
• Leverage the fact that eigenvalues close to 1 indicate near-constant functions on components, enabling the construction of a $k$-partition with low expansion.
• Use the top $k$ eigenvectors to embed vertices into $\mathbb{R}^k$, then apply geometric clustering to extract the $k$ sparse components.

Experimental results

Research questions

• RQ1 Does the presence of $k$ eigenvalues close to 1 in the normalized adjacency matrix imply the existence of a $k$-way partitioning into sparse cuts?
• RQ2 Can the spectral structure of the normalized adjacency matrix be used to certify the existence of $k$ disjoint subsets, each with small internal expansion?
• RQ3 Is there a quantitative higher-order version of Cheeger's inequality that relates eigenvalue multiplicities to $k$-way graph partitioning?
• RQ4 How does the top $k$ eigenvector embedding relate to the geometric structure of a $k$-partition with sparse cuts?

Key findings

• The paper proves that if there are $k$ eigenvalues of the normalized adjacency matrix within $\varepsilon$ of 1, then the vertex set can be partitioned into $k$ subsets, each with expansion at most $O(\sqrt{\varepsilon})$.
• Conversely, if the graph admits a $k$-way partition into sets of small expansion, then there exist $k$ eigenvalues within $O(\varepsilon)$ of 1.
• The result establishes a tight connection between spectral multiplicity and graph partitioning, generalizing the classical Cheeger inequality to higher multiplicities.
• The theoretical framework justifies the use of top-$k$ eigenvectors in spectral clustering algorithms for detecting multiple clusters with low internal connectivity.