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“大块”Krylov方法具有最广为人知的理论保证，但实践中常优先采用块大小为1（单向量）或小常数的“小块”方法。然而，尽管此类方法应用广泛，我们对其在低秩逼近中的性能仍缺乏理论界。本文通过证明小块Krylov方法本质上能达到与大块方法相同的已知低秩逼近保证，弥补了这一理论与实践的差距。通过黑箱归约，我们表明：例如，运行$t$次的标准单向量Krylov方法所获得的谱范数和Frobenius范数误差界，与块大小为$\ell \geq k$的Krylov方法运行$O(t/\ell)$次所得到的界一致（仅差关于最小连续奇异值间隔的对数依赖）。换言之，对于给定矩阵-向量乘积次数，单向量方法在效果上等价于任何大块大小的选择。进一步，将我们的结果与随机矩阵特征值间隙的尾界相结合，可证明若输入矩阵被小随机矩阵扰动，则对最小奇异值间隙的依赖可完全消除。此外，我们证明单向量方法可匹配[Bakshi等人 '22]提出的更复杂算法——该算法通过组合多种块大小的结果实现了Schatten $p$-范数低秩逼近的改进算法。