Geometry-Oblivious FMM for Compressing Dense SPD Matrices

We present GOFMM (geometry-oblivious FMM), a novel method that creates a hierarchical low-rank approximation, or “compression,” of an arbitrary dense symmetric positive definite (SPD) matrix. For many applications, GOFMM enables an approximate matrix-vector multiplication in N logN or even N time, where N is the matrix size. Compression requires N logN storage and work. In general, our scheme belongs to the family of hierarchical matrix approximation methods. In particular, it generalizes the fast multipole method (FMM) to a purely algebraic setting by only requiring the ability to sample matrix entries. Neither geometric information (i.e., point coordinates) nor knowledge of how the matrix entries have been generated is required, thus the term “geometry-obliclious.” Also, we introduce a shared-memory parallel scheme for hierarchical matrix computations that reduces synchronization barriers. We present results on the Intel Knights Landing and Haswell architectures, and on the NVIDIA Pascal architecture for a variety of matrices.CCS CONCEPTS• Theory of computation $\rightarrow$ Numeric approximation algorithms; Sketching and sampling; • Mathematics of computing $\rightarrow$ Mathematical software performance; Kernel density estimators; • Computing methodologies $\rightarrow$ Linear algebra algorithms; Parallel algorithms; Kernel methods; • Computer systems organization $\rightarrow$Multicore architectures; Heterogeneous (hybrid) systems;