Compressing Rank-Structured Matrices via Randomized Sampling

Abstract

Randomized sampling has recently been proven a highly efficient technique for computing approximate factorizations of matrices that have low numerical rank. This paper describes an extension of such techniques to a wider class of matrices that are not themselves rank-deficient but have off-diagonal blocks that are---specifically, the classes of so-called hierarchically off-diagonal low rank (HODLR) matrices and hierarchically block separable (HBS) matrices (a.k.a. hierarchically semiseparable (HSS) matrices). Such matrices arise frequently in numerical analysis and signal processing, in particular in the construction of fast methods for solving differential and integral equations numerically. These structures admit algebraic operations (matrix-vector multiplications, matrix factorizations, matrix inversion, etc.) to be performed very rapidly, but only once a data-sparse representation of the matrix has been constructed. This paper demonstrates that if an $N\times N$ matrix, and its transpose, can be appli...