We study the problem of computing a rank-$k$ approximation of a matrix using randomized block Krylov iteration. Prior work has shown that, for block size $b = 1$ or $b = k$, a $(1 + \varepsilon)$-factor approximation to the best rank-$k$ approximation can be obtained after $\tilde O(k/\sqrt{\varepsilon})$ matrix-vector products with the target matrix. On the other hand, when $b$ is between $1$ and $k$, the best known bound on the number of matrix-vector products scales with $b(k-b)$, which could be as large as $O(k^2)$. Nevertheless, in practice, the performance of block Krylov methods is often optimized by choosing a block size $1 \ll b \ll k$. We resolve this theory-practice gap by proving that randomized block Krylov iteration produces a $(1 + \varepsilon)$-factor approximate rank-$k$ approximation using $\tilde O(k/\sqrt{\varepsilon})$ matrix-vector products for any block size $1\le b\le k$. Our analysis relies on new bounds for the minimum singular value of a random block Krylov matrix, which may be of independent interest. Similar bounds are central to recent breakthroughs on faster algorithms for sparse linear systems [Peng & Vempala, SODA 2021; Nie, STOC 2022].

翻译：我们研究利用随机分块Krylov迭代计算矩阵秩-$k$逼近的问题。已有研究表明，当分块大小$b = 1$或$b = k$时，通过与目标矩阵进行$\tilde O(k/\sqrt{\varepsilon})$次矩阵-向量乘积运算，可获得对最优秩-$k$逼近的$(1 + \varepsilon)$因子逼近。然而，当$b$介于$1$与$k$之间时，已知的矩阵-向量乘积次数最优上界与$b(k-b)$成正比，可能高达$O(k^2)$。但在实际应用中，通过选择$1 \ll b \ll k$的分块大小，分块Krylov方法的性能往往能达到最优。我们通过证明随机分块Krylov迭代在任意分块大小$1\le b\le k$下，仅需$\tilde O(k/\sqrt{\varepsilon})$次矩阵-向量乘积即可产生$(1 + \varepsilon)$因子近似的秩-$k$逼近，从而弥合了这一理论与实践的差距。我们的分析依赖于对随机分块Krylov矩阵最小奇异值的新界，该结果可能具有独立的理论价值。类似的界在稀疏线性系统快速算法的最新突破中具有核心作用[Peng & Vempala, SODA 2021; Nie, STOC 2022]。