We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an $n\times n$ matrix $\mathbf{A}$, accessible only though matrix-vector products with $\mathbf{A}$ and $\mathbf{A}^{\mathsf{T}}$. We prove that, for the rank-$k$ HODLR approximation problem, our method achieves a $(1+\beta)^{\log(n)}$-optimal approximation in expected Frobenius norm using $O(k\log(n)/\beta^3)$ matrix-vector products. In particular, the algorithm obtains a $(1+\varepsilon)$-optimal approximation with $O(k\log^4(n)/\varepsilon^3)$ matrix-vector products, and for any constant $c$, an $n^c$-optimal approximation with $O(k \log(n))$ matrix-vector products. Apart from matrix-vector products, the additional computational cost of our method is just $O(n \operatorname{poly}(\log(n), k, \beta))$. We complement the upper bound with a lower bound, which shows that any matrix-vector query algorithm requires at least $\Omega(k\log(n) + k/\varepsilon)$ queries to obtain a $(1+\varepsilon)$-optimal approximation. Our algorithm can be viewed as a robust version of widely used "peeling" methods for recovering HODLR matrices and is, to the best of our knowledge, the first matrix-vector query algorithm to enjoy theoretical worst-case guarantees for approximation by any hierarchical matrix class. To control the propagation of error between levels of hierarchical approximation, we introduce a new perturbation bound for low-rank approximation, which shows that the widely used Generalized Nystr\"om method enjoys inherent stability when implemented with noisy matrix-vector products. We also introduce a novel randomly perforated matrix sketching method to further control the error in the peeling algorithm.

翻译：我们提出了一种随机算法，用于生成一个接近最优的层次非对角低秩（HODLR）逼近，以逼近一个仅能通过其与向量及转置的矩阵-向量积来访问的 $n\times n$ 矩阵 $\mathbf{A}$。我们证明，对于秩-$k$ 的 HODLR 逼近问题，我们的方法在期望 Frobenius 范数下实现了 $(1+\beta)^{\log(n)}$-最优逼近，仅需使用 $O(k\log(n)/\beta^3)$ 次矩阵-向量积。特别地，该算法能以 $O(k\log^4(n)/\varepsilon^3)$ 次矩阵-向量积获得 $(1+\varepsilon)$-最优逼近，并且对于任意常数 $c$，能以 $O(k \log(n))$ 次矩阵-向量积获得 $n^c$-最优逼近。除了矩阵-向量积之外，我们方法的额外计算成本仅为 $O(n \operatorname{poly}(\log(n), k, \beta))$。我们用一个下界补充了上界，该下界表明任何矩阵-向量查询算法至少需要 $\Omega(k\log(n) + k/\varepsilon)$ 次查询才能获得 $(1+\varepsilon)$-最优逼近。我们的算法可被视为广泛使用的用于恢复 HODLR 矩阵的"剥离"方法的鲁棒版本，并且据我们所知，是第一个在理论上具有最坏情况保证、适用于任何层次矩阵类逼近的矩阵-向量查询算法。为了控制误差在层次逼近的各层级间传播，我们引入了一个新的低秩逼近扰动界，它表明广泛使用的广义 Nystr\"om 方法在使用带噪声的矩阵-向量积实现时具有固有的稳定性。我们还引入了一种新颖的随机穿孔矩阵素描方法，以进一步控制剥离算法中的误差。