Entrywise convergence of iterative methods for eigenproblems

TL;DR

This paper analyzes the convergence of iterative eigenvector algorithms using the $\,\ell_{2 \to \infty}$ norm, providing deterministic bounds, practical stopping criteria, and demonstrating computational efficiency improvements.

Contribution

It introduces a novel convergence analysis for subspace iteration in the $\,\ell_{2 \to \infty}$ norm, including practical stopping rules and empirical validation.

Findings

Comparable downstream task performance with fewer iterations

Deterministic bounds for convergence in the $\,\ell_{2 \to \infty}$ norm

Effective stopping criteria for iterative eigenvector algorithms

Abstract

Several problems in machine learning, statistics, and other fields rely on computing eigenvectors. For large scale problems, the computation of these eigenvectors is typically performed via iterative schemes such as subspace iteration or Krylov methods. While there is classical and comprehensive analysis for subspace convergence guarantees with respect to the spectral norm, in many modern applications other notions of subspace distance are more appropriate. Recent theoretical work has focused on perturbations of subspaces measured in the $\ell_{2 \to \infty}$ norm, but does not consider the actual computation of eigenvectors. Here we address the convergence of subspace iteration when distances are measured in the $\ell_{2 \to \infty}$ norm and provide deterministic bounds. We complement our analysis with a practical stopping criterion and demonstrate its applicability via numerical experiments. Our results show that one can get comparable performance on downstream tasks while requiring fewer iterations, thereby saving substantial computational time.