Entrywise convergence of iterative methods for eigenproblems
This work addresses the need for efficient eigenvector computation in machine learning and statistics, offering incremental improvements in iterative methods for specific applications.
The paper tackles the problem of computing eigenvectors for large-scale problems by analyzing subspace iteration convergence under the ℓ₂→∞ norm, providing deterministic bounds and a practical stopping criterion. The results demonstrate comparable performance on downstream tasks with fewer iterations, saving substantial computational time.
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.