A randomized algorithm for computing a data sparse representation of a given rank structured matrix $A$ (a.k.a. an $H$-matrix) is presented. The algorithm draws on the randomized singular value decomposition (RSVD), and operates under the assumption that algorithms for rapidly applying $A$ and $A^{*}$ to vectors are available. The algorithm analyzes the hierarchical tree that defines the rank structure using graph coloring algorithms to generate a set of random test vectors. The matrix is then applied to the test vectors, and in a final step the matrix itself is reconstructed by the observed input-output pairs. The method presented is an evolution of the "peeling algorithm" of L. Lin, J. Lu, and L. Ying, "Fast construction of hierarchical matrix representation from matrix-vector multiplication," JCP, 230(10), 2011. For the case of uniform trees, the new method substantially reduces the pre-factor of the original peeling algorithm. More significantly, the new technique leads to dramatic acceleration for many non-uniform trees since it constructs sample vectors that are optimized for a given tree. The algorithm is particularly effective for kernel matrices involving a set of points restricted to a lower dimensional object than the ambient space, such as a boundary integral equation defined on a surface in three dimensions.

翻译：本文提出一种用于计算给定秩结构矩阵（亦称$H$-矩阵）数据稀疏表示的随机算法。该算法基于随机奇异值分解（RSVD）技术，其运行前提是存在能够快速计算矩阵$A$与$A^{*}$对向量作用的算法。本算法通过图着色算法分析定义秩结构的层次树以生成随机测试向量集，随后将矩阵作用于测试向量，最终根据观测到的输入-输出对重构矩阵本身。该方法是对L. Lin、J. Lu和L. Ying在《Journal of Computational Physics》230(10)卷（2011年）所著《基于矩阵-向量乘法的层次矩阵表示快速构造》中"剥离算法"的演进发展。对于均匀树情形，新方法显著降低了原始剥离算法的前置系数；更重要的是，由于能针对给定树结构构建优化采样向量，该技术对众多非均匀树实现了大幅加速。本算法特别适用于涉及低维流形点集的核矩阵，例如定义在三维空间曲面上的边界积分方程。