We consider the problem of estimating the spectrum of a symmetric bounded entry (not necessarily PSD) matrix via entrywise sampling. This problem was introduced by [Bhattacharjee, Dexter, Drineas, Musco, Ray '22], where it was shown that one can obtain an $\epsilon n$ additive approximation to all eigenvalues of $A$ by sampling a principal submatrix of dimension $\frac{\text{poly}(\log n)}{\epsilon^3}$. We improve their analysis by showing that it suffices to sample a principal submatrix of dimension $\tilde{O}(\frac{1}{\epsilon^2})$ (with no dependence on $n$). This matches known lower bounds and therefore resolves the sample complexity of this problem up to $\log\frac{1}{\epsilon}$ factors. Using similar techniques, we give a tight $\tilde{O}(\frac{1}{\epsilon^2})$ bound for obtaining an additive $\epsilon\|A\|_F$ approximation to the spectrum of $A$ via squared row-norm sampling, improving on the previous best $\tilde{O}(\frac{1}{\epsilon^{8}})$ bound. We also address the problem of approximating the top eigenvector for a bounded entry, PSD matrix $A.$ In particular, we show that sampling $O(\frac{1}{\epsilon})$ columns of $A$ suffices to produce a unit vector $u$ with $u^T A u \geq \lambda_1(A) - \epsilon n$. This matches what one could achieve via the sampling bound of [Musco, Musco'17] for the special case of approximating the top eigenvector, but does not require adaptivity. As additional applications, we observe that our sampling results can be used to design a faster eigenvalue estimation sketch for dense matrices resolving a question of [Swartworth, Woodruff'23], and can also be combined with [Musco, Musco'17] to achieve $O(1/\epsilon^3)$ (adaptive) sample complexity for approximating the spectrum of a bounded entry PSD matrix to $\epsilon n$ additive error.

翻译：我们研究通过逐项采样估计对称有界元素（未必半正定）矩阵谱的问题。该问题由[Bhattacharjee, Dexter, Drineas, Musco, Ray '22]首次提出，他们证明了通过采样维度为$\frac{\text{poly}(\log n)}{\epsilon^3}$的主子矩阵可获得$A$所有特征值的$\epsilon n$加性逼近。我们改进了其分析，证明仅需采样维度为$\tilde{O}(\frac{1}{\epsilon^2})$的主子矩阵（与$n$无关）。这与已知下界匹配，从而在$\log\frac{1}{\epsilon}$因子范围内解决了该问题的采样复杂度。利用类似技术，我们通过行范数平方采样获得了谱$\epsilon\|A\|_F$加性逼近的紧致界$\tilde{O}(\frac{1}{\epsilon^2})$，改进了先前最佳的$\tilde{O}(\frac{1}{\epsilon^{8}})$界。我们还处理了有界元素半正定矩阵$A$的顶部特征向量逼近问题。特别地，我们证明采样$O(\frac{1}{\epsilon})$列$A$足以产生满足$u^T A u \geq \lambda_1(A) - \epsilon n$的单位向量$u$。这与[Musco, Musco'17]在特殊情况下逼近顶部特征向量的采样界结果一致，但无需自适应采样。作为额外应用，我们指出采样结果可用于设计稠密矩阵的快速特征值估计草图以解决[Swartworth, Woodruff'23]的问题，也可与[Musco, Musco'17]结合实现有界元素半正定矩阵谱$\epsilon n$加性误差逼近的$O(1/\epsilon^3)$（自适应）采样复杂度。