Linear-Complexity Black-Box Randomized Compression of Hierarchically Block Separable Matrices

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 $A$ by analyzing the matrices $A\Omega$ and $A^{*}\Psi$. 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 a typical scaling. 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) and fully black-box in nature (in the sense that no matrix entry evaluation is required).