局部逼近有界项矩阵的最大特征向量

中文摘要 AI 辅助

我们基于Swartworth和Woodruff [SODA 25]的工作，提供了一种局部计算算法，用于逼近对称矩阵\(A \in \mathbb{R}^{n \times n}\)（其元素在\(-1\)到\(1\)之间）的最大特征向量\(x \in \mathbb{R}^n\)。他们展示了如何使用\(\tilde{O}(1/\varepsilon^4)\)次查询将特征值逼近到加法\(\varepsilon n\)误差。我们的局部计算算法在\(|\lambda_{\min}(A)| = O(\lambda_{\max}(A))\)时，预处理复杂度为\(\tilde{O}(1/\varepsilon^4)\)，每个坐标的查询复杂度为\(\tilde{O}(1/\varepsilon^2)\)以实现加法\(\varepsilon n\)逼近。当\(\lambda_{\min}(A)\)远超过\(\lambda_{\max}(A)\)时，预处理复杂度至多为\(\tilde{O}(1/\varepsilon^{6.\overline{6}})\)，每次查询为\(\tilde{O}(1/\varepsilon^{3.\overline{3}})\)。此外，我们给出了输出近似最大特征向量所需查询总数的下界\(\Omega(n/\varepsilon^2)\)。作为应用，我们用该算法为Goldreich、Goldwasser、Ron [JACM 98]的稠密图模型中的最稀疏割和最大割问题提供局部计算算法。通过访问（近似归一化邻接矩阵的）最大特征向量，我们实现了Cheeger不等式和Trevisan算法[SICOMP 12]的局部版本，以在多项式时间内获得“平方根最优”逼近。

英文摘要

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.

发表机构

• University of Pennsylvania（宾夕法尼亚大学）

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