Tight Sampling Bounds for Eigenvalue Approximation

TL;DR

This paper improves sampling bounds for approximating eigenvalues and eigenvectors of symmetric matrices, achieving near-optimal sample complexity and simplifying previous methods for spectral estimation.

Contribution

It provides tight, dimension-independent sampling bounds for eigenvalue and eigenvector approximation, resolving previous gaps and matching lower bounds.

Findings

Sample complexity for eigenvalue approximation is reduced to rac{1}{\u03b5^2} with no dependence on n.

Achieves a rac{1}{b5^2} bound for spectral approximation via squared row-norm sampling.

Sampling O(b5^{-1}) columns suffices for near-optimal top eigenvector approximation.

Abstract

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.