The hierarchical matrix ($\mathcal{H}^{2}$-matrix) formalism provides a way to reinterpret the Fast Multipole Method and related fast summation schemes in linear algebraic terms. The idea is to tessellate a matrix into blocks in such as way that each block is either small or of numerically low rank; this enables the storage of the matrix and the application of it to a vector in linear or close to linear complexity. A key motivation for the reformulation is to extend the range of dense matrices that can be represented. Additionally, $\mathcal{H}^{2}$-matrices in principle also extend the range of operations that can be executed to include matrix inversion and factorization. While such algorithms can be highly efficient for certain specialized formats (such as HBS/HSS matrices based on ``weak admissibility''), inversion algorithms for general $\mathcal{H}^{2}$-matrices tend to be based on nested recursions and recompressions, making them challenging to implement efficiently. An exception is the \textit{strong recursive skeletonization (SRS)} algorithm by Minden, Ho, Damle, and Ying, which involves a simpler algorithmic flow. However, SRS greatly increases the number of blocks of the matrix that need to be stored explicitly, leading to high memory requirements. This manuscript presents the \textit{randomized strong recursive skeletonization (RSRS)} algorithm, which is a reformulation of SRS that incorporates the randomized SVD (RSVD) to simultaneously compress and factorize an $\mathcal{H}^{2}$-matrix. RSRS is a ``black box'' algorithm that interacts with the matrix to be compressed only via its action on vectors; this extends the range of the SRS algorithm (which relied on the ``proxy source'' compression technique) to include dense matrices that arise in sparse direct solvers.

翻译：分层矩阵（$\mathcal{H}^{2}$-矩阵）形式提供了一种在线性代数框架下重新诠释快速多极子方法及相关快速求和方案的途径。其核心思想是将矩阵分块为若干子块，使得每个子块要么尺寸较小，要么具有数值低秩性——这允许以线性或接近线性的复杂度存储矩阵并将其作用于向量。这种重构的关键动机在于扩展可表示稠密矩阵的范围。此外，$\mathcal{H}^{2}$-矩阵原则上还能扩展可执行操作的范围，使其包括矩阵求逆与分解。尽管这类算法对某些特定格式（如基于“弱可容许性”的HBS/HSS矩阵）具有极高效率，但通用$\mathcal{H}^{2}$-矩阵的求逆算法往往依赖嵌套递归与再压缩，导致实现效率面临挑战。一个例外是Minden、Ho、Damle和Ying提出的**强递归骨架化（SRS）**算法，其算法流程更为简洁。然而，SRS会显著增加需显式存储的矩阵块数量，导致内存需求极高。本文提出**随机化强递归骨架化（RSRS）**算法，该算法重构SRS并引入随机SVD（RSVD）技术，实现对$\mathcal{H}^{2}$-矩阵的同步压缩与分解。RSRS是一种“黑盒”算法，仅通过矩阵与向量的乘法交互来压缩目标矩阵；这拓展了SRS算法（依赖“源代理”压缩技术）的应用范围，使其能够处理稀疏直接求解器中产生的稠密矩阵。