Fast Length-Squared Sampling for Positive-Semidefinite Matrices

2026-08-12

AI Summary

This work addresses the problem of efficiently performing length-squared sampling on positive semidefinite matrices without prior knowledge of column norms. The authors propose a novel rejection-sampling-based algorithm that exploits the structural properties of positive semidefinite matrices to eliminate the need for any precomputed column norm information. This approach achieves, for the first time, an expected $O(n)$ time complexity for optimal length-squared sampling, substantially outperforming conventional methods that rely on explicit column norm estimates. Empirical evaluations demonstrate that the algorithm matches the performance of existing, more complex techniques in tasks such as Frobenius norm estimation and robust low-rank approximation, offering both theoretical optimality and practical utility.

Abstract

We describe a simple rejection-sampling-based algorithm to perform length-squared sampling on an $n \times n$ positive-semidefinite (psd) matrix: that is, to sample a column with probability proportional to its squared $\ell_2$-norm. The algorithm runs in just $O(n)$ expected time, which is significantly sublinear in the input matrix size. The runtime is optimal, even when the input is assumed to be diagonal. Our result has several applications. Length-squared sampling is used by a number of sublinear time algorithms for matrix problems, like low-rank approximation and eigenvalue approximation. Often, it is assumed that the algorithm is given access to the matrix column norms, and thus can perform length-squared sampling efficiently. Our result shows that, at least for psd matrices, we can remove this assumption. We also discuss an application to an asymptotically optimal algorithm for estimating the Frobenius norm of a psd matrix to relative error. Finally, we show that our sampling algorithm yields a very simple sublinear time algorithm for the robust psd low-rank approximation problem introduced by Bakshi et al. (FOCS, 2020), which nearly matches the more complex method developed there.

Problem

• length-squared sampling
• positive-semidefinite matrices
• sublinear time algorithms
• matrix approximation

Innovation

• length-squared sampling
• positive-semidefinite matrices
• sublinear time algorithm
• rejection sampling
• low-rank approximation