Compression Properties for large Toeplitz-like matrices

TL;DR

This paper explains the compression properties of Toeplitz-like matrices using displacement structure, providing bounds on numerical ranks and enabling efficient hierarchical low-rank approximations for superfast solvers.

Contribution

It offers explicit bounds on numerical ranks of submatrices in Toeplitz-like matrices and develops efficient compression strategies based on displacement structure.

Findings

Explicit bounds on numerical ranks of submatrices

Efficient displacement-based compression strategies

Enhanced superfast rank-structured solvers

Abstract

Toeplitz matrices are abundant in computational mathematics, and there is a rich literature on the development of fast and superfast algorithms for solving linear systems involving such matrices. Any Toeplitz matrix can be transformed into a matrix with off-diagonal blocks that are of low numerical rank.This compressibility is relied upon in practice in a number of superfast Toeplitz solvers. In this paper, we show that the compression properties of these matrices can be thoroughly explained using their displacement structure. We provide explicit bounds on the numerical ranks of important submatrices that arise when applying HSS, HODLR and other approximations with hierarchical low-rank structure to transformed Toeplitz and Toeplitz-like matrices. Our results lead to very efficient displacement-based compression strategies that can be used to formulate adaptive superfast rank-structured solvers.