Krylov subspace methods are a ubiquitous tool for computing near-optimal rank $k$ approximations of large matrices. While "large block" Krylov methods with block size at least $k$ give the best known theoretical guarantees, block size one (a single vector) or a small constant is often preferred in practice. Despite their popularity, we lack theoretical bounds on the performance of such "small block" Krylov methods for low-rank approximation. We address this gap between theory and practice by proving that small block Krylov methods essentially match all known low-rank approximation guarantees for large block methods. Via a black-box reduction we show, for example, that the standard single vector Krylov method run for $t$ iterations obtains the same spectral norm and Frobenius norm error bounds as a Krylov method with block size $\ell \geq k$ run for $O(t/\ell)$ iterations, up to a logarithmic dependence on the smallest gap between sequential singular values. That is, for a given number of matrix-vector products, single vector methods are essentially as effective as any choice of large block size. By combining our result with tail-bounds on eigenvalue gaps in random matrices, we prove that the dependence on the smallest singular value gap can be eliminated if the input matrix is perturbed by a small random matrix. Further, we show that single vector methods match the more complex algorithm of [Bakshi et al. `22], which combines the results of multiple block sizes to achieve an improved algorithm for Schatten $p$-norm low-rank approximation.

翻译：Krylov子空间方法是计算大型矩阵近最优秩$k$逼近的通用工具。尽管块大小至少为$k$的"大分块"Krylov方法具有已知最优理论保证，但在实践中通常首选块大小为1（单向量）或小常数的Krylov方法。尽管此类"小分块"Krylov方法广受欢迎，但目前缺乏其在低秩逼近中性能的理论界。我们通过证明小分块Krylov方法在低秩逼近中本质上能够匹配大分块方法的所有已知保证，弥合了理论与实践的差距。通过黑箱归约，我们表明：例如，运行$t$次迭代的标准单向量Krylov方法，其谱范数和Frobenius范数误差界与大分块大小$\ell \geq k$、运行$O(t/\ell)$次迭代的Krylov方法相当（仅在对序贯奇异值最小间隙的对数依赖程度上存在差异）。即对于给定数量的矩阵-向量乘积，单向量方法与任意大分块方法本质上同样有效。将我们的结果与随机矩阵特征值间隙的尾界相结合，我们证明：若输入矩阵被小随机矩阵扰动，则可消除对最小奇异值间隙的依赖。进一步，我们表明单向量方法能够匹配[Bakshi等 '22]中更复杂的算法——该算法通过融合多个分块大小的结果，实现了改进的Schatten $p$-范数低秩逼近方法。