We study the problem of approximating the eigenspectrum of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with bounded entries (i.e., $\|A\|_{\infty} \leq 1$). We present a simple sublinear time algorithm that approximates all eigenvalues of $A$ up to additive error $\pm \epsilon n$ using those of a randomly sampled $\tilde{O}(\frac{1}{\epsilon^4}) \times \tilde O(\frac{1}{\epsilon^4})$ principal submatrix. Our result can be viewed as a concentration bound on the full eigenspectrum of a random principal submatrix. It significantly extends existing work which shows concentration of just the spectral norm [Tro08]. It also extends work on sublinear time algorithms for testing the presence of large negative eigenvalues in the spectrum [BCJ20]. To complement our theoretical results, we provide numerical simulations, which demonstrate the effectiveness of our algorithm in approximating the eigenvalues of a wide range of matrices.

中文翻译：

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通过随机采样的亚线性时间特征值逼近

我们研究了近似对称矩阵 $A \in \mathbb{R}^{n \times n}$ 的特征谱的问题，其中包含有界条目（即 $\|A\|_{\infty} \leq 1$ ）。我们提出了一个简单的次线性时间算法，它使用随机采样的 $\tilde{O}(\frac{1}{\epsilon^4} ) \times \tilde O(\frac{1}{\epsilon^4})$ 主子矩阵。我们的结果可以看作是随机主子矩阵的完整特征谱上的集中边界。它显着扩展了现有工作，仅显示频谱范数 [Tro08] 的集中。它还扩展了亚线性时间算法的工作，用于测试频谱中是否存在大的负特征值 [BCJ20]。为了补充我们的理论结果，我们提供了数值模拟，