Optimal Eigenvalue Approximation via Sketching

Given a symmetric matrix $A$, we show from the simple sketch $GAG^T$, where $G$ is a Gaussian matrix with $k = O(1/\epsilon^2)$ rows, that there is a procedure for approximating all eigenvalues of $A$ simultaneously to within $\epsilon \|A\|_F$ additive error with large probability. Unlike the work of (Andoni, Nguyen, SODA, 2013), we do not require that $A$ is positive semidefinite and therefore we can recover sign information about the spectrum as well. Our result also significantly improves upon the sketching dimension of recent work for this problem (Needell, Swartworth, Woodruff FOCS 2022), and in fact gives optimal sketching dimension. Our proof develops new properties of singular values of $GA$ for a $k \times n$ Gaussian matrix $G$ and an $n \times n$ matrix $A$ which may be of independent interest. Additionally we achieve tight bounds in terms of matrix-vector queries. Our sketch can be computed using $O(1/\epsilon^2)$ matrix-vector multiplies, and by improving on lower bounds for the so-called rank estimation problem, we show that this number is optimal even for adaptive matrix-vector queries.