Sublinear Time Eigenvalue Approximation via Random Sampling
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.07647
 Bibcode:
 2021arXiv210907647B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Mathematics  Numerical Analysis
 EPrint:
 22 pages, 4 figures, typos corrected