Sublinear Time Eigenvalue Approximation via Random Sampling

TL;DR

This paper introduces a sublinear time algorithm for approximating the eigenvalues of symmetric matrices using random sampling of submatrices, with improved error bounds and practical effectiveness demonstrated through simulations.

Contribution

It presents the first non-uniform sampling method for eigenvalue approximation with enhanced error bounds and new concentration bounds for matrices with bounded entries.

Findings

Achieves eigenvalue approximation with additive error b1 a9 n in sublinear time.

Provides improved error bounds using non-uniform sampling based on sparsity and Frobenius norm.

Demonstrates effectiveness of the algorithms through numerical simulations.

Abstract

We study the problem of approximating the eigenspectrum of a symmetric matrix $\mathbf A \in \mathbb{R}^{n \times n}$ with bounded entries (i.e., $\|\mathbf A\|_{\infty} \leq 1$). We present a simple sublinear time algorithm that approximates all eigenvalues of $\mathbf{A}$ up to additive error $\pm \epsilon n$ using those of a randomly sampled $\tilde {O}\left (\frac{\log^3 n}{\epsilon^3}\right ) \times \tilde O\left (\frac{\log^3 n}{\epsilon^3}\right )$ principal submatrix. Our result can be viewed as a concentration bound on the complete eigenspectrum of a random submatrix, significantly extending known bounds on just the singular values (the magnitudes of the eigenvalues). We give improved error bounds of $\pm \epsilon \sqrt{\text{nnz}(\mathbf{A})}$ and $\pm \epsilon \|\mathbf A\|_F$ when the rows of $\mathbf A$ can be sampled with probabilities proportional to their sparsities or their squared $\ell_2$ norms respectively. Here $\text{nnz}(\mathbf{A})$ is the number of non-zero entries in $\mathbf{A}$ and $\|\mathbf A\|_F$ is its Frobenius norm. Even for the strictly easier problems of approximating the singular values or testing the existence of large negative eigenvalues (Bakshi, Chepurko, and Jayaram, FOCS '20), our results are the first that take advantage of non-uniform sampling to give improved error bounds. From a technical perspective, our results require several new eigenvalue concentration and perturbation bounds for matrices with bounded entries. Our non-uniform sampling bounds require a new algorithmic approach, which judiciously zeroes out entries of a randomly sampled submatrix to reduce variance, before computing the eigenvalues of that submatrix as estimates for those of $\mathbf A$. We complement our theoretical results with numerical simulations, which demonstrate the effectiveness of our algorithms in practice.