Does block size matter in randomized block Krylov low-rank approximation?

Abstract: 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].