A randomized algorithm for computing a compressed representation of a given rank-structured matrix $A \in \mathbb{R}^{N\times N}$ is presented. The algorithm interacts with $A$ only through its action on vectors. Specifically, it draws two tall thin matrices $\Omega,\,\Psi \in \mathbb{R}^{N\times s}$ from a suitable distribution, and then reconstructs $A$ from the information contained in the set $\{A\Omega,\,\Omega,\,A^{*}\Psi,\,\Psi\}$. For the specific case of a "Hierarchically Block Separable (HBS)" matrix (a.k.a. Hierarchically Semi-Separable matrix) of block rank $k$, the number of samples $s$ required satisfies $s = O(k)$, with $s \approx 3k$ being representative. While a number of randomized algorithms for compressing rank-structured matrices have previously been published, the current algorithm appears to be the first that is both of truly linear complexity (no $N\log(N)$ factors in the complexity bound) and fully "black box" in the sense that no matrix entry evaluation is required. Further, all samples can be extracted in parallel, enabling the algorithm to work in a "streaming" or "single view" mode.

翻译：本文提出一种用于计算给定秩结构矩阵 $A \in \mathbb{R}^{N\times N}$ 压缩表示的随机化算法。该算法仅通过矩阵对向量的作用与 $A$ 交互。具体而言，算法从适当分布中抽取两个高瘦矩阵 $\Omega,\,\Psi \in \mathbb{R}^{N\times s}$，然后从集合 $\{A\Omega,\,\Omega,\,A^{*}\Psi,\,\Psi\}$ 包含的信息重构 $A$。对于块秩为 $k$ 的"层次块可分离（HBS）"矩阵（亦称层次半可分离矩阵），所需样本量 $s$ 满足 $s = O(k)$，其中 $s \approx 3k$ 具有代表性。尽管此前已发表若干秩结构矩阵压缩的随机化算法，但当前算法似乎是首个同时具备真正线性复杂度（复杂度边界中不含 $N\log(N)$ 因子）且完全"黑盒化"的算法——即无需任何矩阵元求值。此外，所有样本可并行抽取，使算法能够在"流式"或"单次扫描"模式下运行。