Year: 2024
Type: article
Abstract: . A randomized algorithm for computing a compressed representation of a given rank-structured matrix [Formula: see text] is presented. The algorithm interacts with [Formula: see text] only through its action on vectors. Specifically, it draws two tall thin matrices [Formula: see text] from a suitable distribution, and then reconstructs [Formula: see text] from the information contained in the set [Formula: see text]. For the specific case of a “Hierarchically Block Separable (HBS)” matrix (a.k.a. Hierarchically Semi-Separable matrix) of block rank [Formula: see text], the number of samples [Formula: see text] required satisfies [Formula: see text], with [Formula: see text] 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 [Formula: see text] 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.
DOI: 10.1137/22m1528574
Authors: James Levitt, Per-Gunnar Martinsson
Institution: The University of Texas at Austin
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4397012344