Eigenvector computation, e.g., k-SVD for finding top-k singular 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 of k-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 of k-SVD to one of the ground-truth solutions, regardless of what target matrix and how large target rank k are given. Theoretical results are evaluated and verified by experiments on synthetic or real data.

特征向量计算（例如，用于找到顶部k个奇异子空间的k -SVD ）通常对许多科学和工程任务至关重要。最近，人们开始对以奇异值差距依赖性分析相关方法产生兴趣。特别地，当间隙消失时，认为k -SVD的收敛被无间隙的亚线性速率所限制。在理论和经验上，我们在这项工作中都认为情况不一定如此，这使我们对这个重大问题有了新的认识。具体来说，我们在仔细分析中利用了最近提出的结构缺口，以建立k的统一线性收敛-SVD为真值解之一，无论给出什么目标矩阵和大目标等级k。理论结果是通过对合成或真实数据进行的实验来评估和验证的。