Far-field compression for fast kernel summation methods in high dimensions

Abstract

We consider fast kernel summations in high dimensions: given a large set of points in d dimensions (with d 3) and a pair-potential function (the kernel function), we compute a weighted sum of all pairwise kernel interactions for each point in the set. Direct summation is equivalent to a (dense) matrix-vector multiplication and scales quadratically with the number of points. Fast kernel summation algorithms reduce this cost to log-linear or linear complexity.Treecodes and Fast Multipole Methods (FMMs) deliver tremendous speedups by constructing approximate representations of interactions of points that are far from each other. In algebraic terms, these representations correspond to low-rank approximations of blocks of the overall interaction matrix. Existing approaches require an excessive number of kernel evaluations with increasing d and number of points in the dataset.To address this issue, we use a randomized algebraic approach in which we first sample the rows of a block and then construct its approximate, low-rank interpolative decomposition. We examine the feasibility of this approach theoretically and experimentally. We provide a new theoretical result showing a tighter bound on the reconstruction error from uniformly sampling rows than the existing state-of-the-art. We demonstrate that our sampling approach is competitive with existing (but prohibitively expensive) methods from the literature. We also construct kernel matrices for the Laplacian, Gaussian, and polynomial kernels -all commonly used in physics and data analysis. We explore the numerical properties of blocks of these matrices, and show that they are amenable to our approach. Depending on the data set, our randomized algorithm can successfully compute low rank approximations in high dimensions. We report results for data sets with ambient dimensions from four to 1,000.

Hierarchical Particle Mesh: An FFT-accelerated Fast Multipole Method

“…For example, a set of potential values at discrete points on a sphere or cube surrounding the target and source cell to parameterize the internal (for a target cell) or external (for a source cell) gravitational fields have been used extensively in the past (c.f. Makino 1999;Kawai & Makino 2001;Ying et al 2004;Ying 2006;Rogers 2015;March & Biros 2017). Other parameterizations such as Chebyshev polynomials (Fong & Darve Particles in the FMM cell Effective masses on the gridlet associated with the FMM cell Anatomy of a gridlet (shown for a 2D 2 × 2 case).…”

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“…where two groups of points are separated). Recently a randomized algebraic approach in a similar setup was suggested in [35]. Instead, we use Algorithm 1 as a tool to rapidly evaluate…”

Accelerating isogeometric boundary element analysis for 3‐dimensional elastostatics problems through black‐box fast multipole method with proper generalized decomposition

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“…Some methods have been proposed to ameliorate the cost of this stage and have typically been constructed through a low-rank operation calculated by wavelet decomposition, 44,45 singular value decomposition (SVD), 41,46 QR decomposition, 47 "skeletonization" technique, 48,49 QR factorization with ACA, 50,51 and subsampling strategy. 52 As a canonical method to extract the optimal approximation basis, the computational cost of SVD is O(mn 2 ) for an m × n matrix and is a significant part of the whole bbFMM calculation. The proper generalized decomposition (PGD) method 53,54 provides a quasioptimal basis based on an iterative procedure with a low computational effort (O(2mn)), rather than directly solving eigenfunctions as in SVD.…”