A unified linear convergence analysis ofk-SVD

Eigenvector computation, e.g.,k-SVD for finding top-ksingular subspaces, is often of central importance to many scientific and engineering tasks. There has been resurgent interest recently in analyzing relevant methods in terms of singular value gap dependence. Particularly, when the gap vanishes, the convergence ofk-SVD is considered to be capped by a gap-free sub-linear rate. We argue in this work both theoretically and empirically that this is not necessarily the case, refreshing our understanding on this significant problem. Specifically, we leverage the recently proposed structured gap in a careful analysis to establish a unified linear convergence ofk-SVD to one of the ground-truth solutions, regardless of what target matrix and how large target rankkare given. Theoretical results are evaluated and verified by experiments on synthetic or real data.