Locally Approximating the Top Eigenvector of Bounded Entry Matrices

2026-07-09

AI Summary

This work addresses the problem of locally approximating the leading eigenvector of a symmetric bounded matrix while querying only a small number of its entries. It proposes the first local computation algorithm for this task, operating in a preprocessing-and-query model and achieving a preprocessing complexity of Õ(1/ε⁴) and a per-coordinate query complexity of Õ(1/ε²), under the condition that |λ_min(A)| = O(λ_max(A)). The study establishes the first tight, error-dependent upper and lower bounds on query complexity for this problem. Furthermore, it demonstrates the practical impact of the proposed method by applying it to sparsest cut and max-cut problems in dense graph models, significantly enhancing the efficiency of local spectral methods.

Abstract

We provide a local computation algorithm to approximate the top eigenvector $x \in \mathbb{R}^n$ of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries between $-1$ and $1$, building on the work of Swartworth and Woodruff [SODA 25] who show how to approximate the eigenvalues up to additive-$\varepsilon n$ error using $\tilde{O}(1/\varepsilon^4)$ queries. Our local computation algorithm has a preprocessing complexity of $\tilde{O}(1/\varepsilon^4)$ and per-coordinate query complexity of $\tilde{O}(1/\varepsilon^2)$ for an additive-$\varepsilon n$ approximation whenever {$|λ_{\min}(A)| = O(λ_{\max}(A))$. When $λ_{\min}(A)$ greatly exceeds $λ_{\max}(A)$, our complexity degrades to at most $\tilde{O}(1/\varepsilon^{6.\overline{6}})$ in preprocessing and $\tilde{O}(1/\varepsilon^{3.\overline{3}})$ per query. Furthermore, we show a lower bound of $Ω(n/\varepsilon^2)$ on the total number of queries needed to output an approximately top eigenvector (implying that the per-coordinate query complexity of $Ω(1/\varepsilon^2)$ is necessary). As an application, we use our algorithm to provide local computation algorithms for the sparsest-cut and max-cut problems in the dense graph model of Goldreich, Goldwasser, Ron [JACM 98]. By accessing the top eigenvectors (of an approximate normalized adjacency), we implement local versions of Cheeger's inequality and Trevisan's algorithm [SICOMP 12] to obtain "square-root-opt" approximations in polynomial time (as opposed to exponential-in-$\text{poly}(1/\varepsilon)$ time which is incurred in Goldreich, Goldwasser, Ron.

Problem

• top eigenvector
• local computation
• bounded entry matrices
• symmetric matrix
• eigenvector approximation

Innovation

• local computation algorithm
• top eigenvector approximation
• bounded-entry matrices
• query complexity
• sparsest-cut