  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.
