A polylogarithmic higher-order Cheeger inequality

2026-09-28

AI Summary

This study investigates the quantitative relationship between higher-order graph eigenvalues and the maximum conductance of k-way cuts. Methodologically, it combines regularized spectral embedding with independent local truncation techniques to construct precisely k disjoint vertex sets and non-negative functions. Furthermore, a principal direction condition is innovatively introduced to control covariance loss, significantly optimizing the measurement of boundary and volume. Ultimately, this work establishes logarithmic-polynomial tight bounds: an upper bound of O([1+log(k+1)]^5√λ_k) for conductance and O([1+log(k+1)]^{10}λ_k) for the Rayleigh quotient, thereby proving the theoretically optimal limits for higher-order Cheeger inequalities.

Abstract

Let $\lambda_k(G)$ be the $k$th eigenvalue of the normalized Laplacian of a finite undirected weighted graph, and let $\rho_G(k)$ be the minimum possible maximum conductance of $k$ disjoint nonempty vertex sets. We prove \[ \rho_G(k)\le C[1+\log(k+1)]^5\sqrt{\lambda_k(G)} \] for an absolute constant $C$. The construction gives exactly $k$ sets and a bound in terms of $\lambda_k(G)$, with all boundaries and volumes measured in the original graph. More strongly, it yields $k$ nonnegative functions with pairwise disjoint supports and Rayleigh quotients $O([1+\log(k+1)]^{10}\lambda_k(G))$. The proof uses independent local cutoffs whose lost covariance is controlled by conditioning on the loss along a principal direction in each cell. A regularized spectral embedding bounds cutoff energy on the entire original low eigenspace, while an adaptive construction reduces the remaining coefficient dimension by at least half at each stage. A dimension argument then converts almost rank-one local covariances into exactly $k$ scalar witnesses.

Problem

• Higher-order Cheeger inequality
• Graph conductance
• Normalized Laplacian
• Spectral graph theory
• Eigenvalues

Innovation

• higher-order Cheeger inequality
• normalized Laplacian
• spectral embedding
• Rayleigh quotient
• local cutoffs