Compressing Rank-Structured Matrices via Randomized Sampling

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 datasparse representation of the matrix has been constructed. This paper demonstrates that if an N ×N matrix, and its transpose, can be applied to a vector in O(N) time, and if the ranks of the off-diagonal blocks are bounded by an integer k, then the cost for constructing an HODLR representation is O(k2 N (logN)2), and the cost for constructing an HBS representation is O(k2 N logN) (assuming that the matrix is compressible in the respective format). The point is that when legacy codes (based on, e.g., the fast multipole method) can be used for the fast matrix-vector multiply, the proposed algorithm can be used to obtain the data-sparse representation of the matrix, and then well-established techniques for HODLR/HBS matrices can be used to invert or factor the matrix. The proposed scheme is also useful in simplifying the implementation of certain operations on rankstructured matrices such as matrix-matrix multiplication, low rank update, and addition.